Systematic Search for Solvable Potentials from Biconfluent, Doubly Confluent, and Triconfluent Heun Equations

Abstract

A transformation method was applied to the biconfluent (BHE), doubly confluent (DHE), and triconfluent (THE) Heun equations to generate and classify exactly solvable quantum mechanical potentials derived from them. With this, the range of potentials solvable in terms of the confluent hypergeometric function can be extended. The resulting potentials contained five independently tunable terms and two terms originating from the Schwartzian derivative that depended only on the parameters of the $z\left(x\right)$ transformation function. The polynomial solutions of these potentials contain expansion coefficients obtained from three-term (BHE and DHE) and four-term (THE) recurrence relations. For the simplest $z\left(x\right)$ transformation functions, the Lemieux–Bose potentials have been recovered for the BHE and DHE. The coupling parameters of these potentials and also of five potentials derived from the THE have been expressed in terms of the parameters of the respective differential equations. The present scheme offers a general framework into which a number of earlier results can be integrated in a systematic way. These include special cases of potentials obtained from less general versions of the Heun-type equations and individual solvable potentials obtained from various methods that do not necessarily refer to the Heun-type equations considered here. Several potentials derived here were found to coincide with or reduce to potentials found earlier from the quasi-exactly solvable (QES) formalism. Based on their mathematical form, their physically relevant features (domain of definition, asymptotic behaviour, single- or multi-well structure) were discussed, and possible fields of applications were pointed out.

1. Introduction

Searching for quantum mechanical potentials for which the Schrödinger equation can be solved exactly has been an active topic in the past century. Exactly solvable potentials are important tools that describe the phenomena of the quantum world either as toy models or bases on which more advanced perturbative or numerical models can be developed and tested. In addition to this, the internal coherence and consistency of the closed mathematical formulation of these models is found interesting by many researchers, especially because exactly solvable problems are often related to various symmetry concepts.

In the simplest case, quantum mechanical potentials are considered in one dimension, which means that either the physical problem is defined in this limited spatial domain or it can be reduced to this form after the separation of the coordinate variables. In both cases, the task is to solve an eigenvalue problem represented by the Schrödinger equation and the boundary conditions prescribed by the physical system. Furthermore, in most cases, one faces the time-independent Schrödinger equation, when one has to find the energy eigenfunctions and the corresponding energy eigenvalues. From the mathematical point of view, finding the solutions of a second-order differential equation is not an especially challenging problem; nevertheless, a variety of methods has been introduced to tackle it.

In the early years of quantum mechanics, the Schrödinger equation was solved for a number of simple potentials (e.g., the harmonic oscillator, Morse, Pöschl–Teller potentials) by employing a well-chosen variable transformation that took the Schrödinger equation into the differential equation of some special function of mathematical physics, the solutions of which were known from the literature. Another approach was factorizing the second-order differential operator into the product of two first-order ones [1]. Later, both approaches were developed into a systematic procedure: the factorization method was generalized [2] and was eventually developed into supersymmetric quantum mechanics (SUSYQM) [3,4], while the variable transformation method (sometimes called point canonical transformation) was presented in a general form applicable to any special function satisfying a second-order differential equation [5] and was specified for the hypergeometric and confluent hypergeometric functions, introducing the Natanzon class potentials [6]. These potentials have been analyzed in great detail, and the transformation method also helped to introduce a classification scheme of the relatively simple shape-invarant potentials [7] and more complex ones [8]. The bound-state solutions are typically expressed in terms of classical orthogonal (i.e., Jacobi and generalized Laguerre) polynomials [9]. These efforts took advantage of the fact that the theory of both the (confluent) hypergeometric functions and the classical orthogonal polynomials has reached rather high levels in mathematics well before the introduction of quantum mechanics a century ago. Natanzon (confluent) potentials are, therefore, well understood (see, e.g., Ref. [10] and Chap. 7 of Ref. [11]).

A number of potentials that clearly do not belong to the Natanzon (confluent) class have also been found using various methods. These often contained Natanzon (confluent) potentials as special cases, so efforts to construct a mathematical framework that generalizes the (confluent) hypergeometric differential equation and its solutions emerged in a natural way. One of the approaches focused on the application of the Heun-type equations, i.e., the Heun, confluent, biconfluent, doubly confluent, and triconfluent Heun equations [12]. A systematic study on generating exactly solvable potentials by the transformation of these equations has been presented in Ref. [13], although it focused mainly on the general structure of the potentials and not on their solutions and their energy spectrum. Later, a renaissance of this field was initiated by Ishkhanyan, who discussed the Heun [14], the confluent Heun [15], and the biconfluent Heun [16] equations, also looking into the solutions of the related potential problems. The typical approach of these works was expanding the solutions in terms of known special functions, e.g., hypergeometric and Hermite functions. In a review [17], the general form of potentials derived from each Heun-type equation was presented, without discussing any particular potential or the bound-state solutions.

Although these Heun-type equations have been known since the 19th century [18] (similarly to the Darboux transformation [19], which can be considered the precursor of the factorization method and SUSYQM), their theory is much less elaborate than that of the (confluent) hypergeometric differential equation. For this reason, their application in the construction of exactly solvable quantum mechanical potentials is less widespread. A further important issue is that the form of the biconfluent, doubly confluent, and triconfluent eguations is not uniform in the literature. Most authors follow the notation of the monograph [12], where the canonical forms of these equations contain four parameters in the first two cases and three in the third one. These expressions are typically derived from the Heun equation by eliminating its finite singularities. Further forms of these equations are also in use: they are found to be more practical in certain situations. Here, we follow the notation of Ref. [20], where the formalism contains a consistent set of five parameters in each equation. An advantage of using this formalism is that it allows implementing certain limiting cases, for example.

These circumstances serve as the inspiration of our present work, in which we summarize the status of this field combining the available physical and mathematical results, supplement them with further explanation on some less well-explored details, and present a unified framework into which a number of previously not connected results are synthetized. Here, we focus on the biconfluent, doubly confluent, and triconfluent Heun equations and also present the confluent hypergeometric differential equation as a reference system. The common feature of these differential equations is that they have one regular singular point at $z=0$. The Heun and confluent Heun equations have more regular singular points, (similarly to the hypergeometric differential equation), which requires a different methodological approach, so we leave the discussion of these problems to a later study.

The arrangement of the paper is as follows. In Section 2, a general method of constructing solvable potentials is outlined. This method is then applied to the biconfluent, doubly confluent, and triconfluent Heun equations in Section 3, first in a general form recovering and generalizing earlier results [10,13,17], and then for concrete potentials and their polynomial solutions, also explaining their connection to potentials obtained from other approaches, occasionally together with known and possible fields of application. Finally, the results are summarized and some conclusions are drawn in Section 4.

2. Transformation to the One-Dimensional Schrödinger Equation

We consider a rather general form of the second-order differential equation of an $F\left(z\right)$ special function of mathematical physics:

$${\displaystyle \frac{{\mathrm{d}}^{2}F}{\mathrm{d}{z}^2}}+Q\left(z\right){\displaystyle \frac{\mathrm{d}F}{\mathrm{d}z}}+R\left(z\right)F\left(z\right)=0.$$

(1)

The functions $Q\left(z\right)$ and $R\left(z\right)$ are well-defined for each $F\left(z\right)$. We wish to transform [5] this equation into the one-dimensional Schrödinger equation

$${\displaystyle \frac{{\mathrm{d}}^2\psi }{\mathrm{d}{x}^2}}+(E-V\left(x\right))\psi \left(x\right)=0$$

(2)

by applying a general variable transformation $z\left(x\right)$. Here, E is the energy and $V\left(x\right)$ is the potential function, using the units of $2m=1$ and $\hslash =1$. We assume that the $z\left(x\right)$ function mapping the domain of definition of the Schrödinger equation to z is monotonous, so its inverse $x\left(z\right)$ is well-defined. The coordinate x can be defined on the full x axis: $x\in (-\infty ,\infty )$, on the positive semi-axis: $x\in [0,\infty )$, or on a finite domain of it: $x\in [a,b]$, depending on the nature of the actual physical problem.

The solution of the Schrödinger equation is assumed to take the form

$$\psi \left(x\right)=f\left(x\right)F\left(z\right(x\left)\right),$$

(3)

where $f\left(x\right)$ is a function to be determined later. Substituting Equation (3) into Equation (2) and comparing the appropriate terms, we find that

$$\begin{array}{cc}E-V\left(x\right)=& {\displaystyle \frac{z^{‴}\left(x\right)}{2z^{\prime }\left(x\right)}}-{\displaystyle \frac{3}{4}}{\left({\displaystyle \frac{z^{″}\left(x\right)}{z^{\prime }\left(x\right)}}\right)}^2\hfill \\ & +{\left(z^{\prime }\left(x\right)\right)}^2\left[R\left(z\left(x\right)\right)-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}Q}{\mathrm{d}z}}-{\displaystyle \frac{1}{4}}{Q}^2\left(z\left(x\right)\right)\right].\end{array}$$

(4)

The expression inside the square brackets is called the Bose invariant [21,22], and it is uniquely defined for any second-order differential equation of the type (1). The $f\left(x\right)$ function is also determined by the procedure, so the solutions of the Schrödinger equation are found to take the form

$$\psi \left(x\right)\sim {\left(z^{\prime }\left(x\right)\right)}^{-{\displaystyle \frac{1}{2}}}exp\left({\displaystyle \frac{1}{2}}{\int }^{z\left(x\right)}Q\left(z\right)\mathrm{d}z\right)F\left(z\left(x\right)\right).$$

(5)

Note that all the key quantities are determined solely by the functions $Q\left(z\right)$ and $R\left(z\right)$ that define the special function $F\left(z\right)$, and by the $z\left(x\right)$ transformation function. In principle, any reasonable $z\left(x\right)$ function leads to an energy eigenvalue E and a potential $V\left(x\right)$; however, such a general choice may not guarantee that different solutions of the same potential will be obtained. A possible solution to the problem of generating energy-independent solvable potentials was proposed in Ref. [5], where it was noted that E on the left-hand side of Equation (4) has to be balanced by a constant term on the right-hand side, and this prescription defines a differential equation for $z\left(x\right)$. Generally, this constant has to originate from the expression containing the terms from the $Q\left(z\right)$ and $R\left(z\right)$ functions, as these terms depend on the parameters of the special function $F\left(z\right)$. The first two terms of Equation (4) with higher derivatives of $z\left(x\right)$ originate from the Schwartzian derivative

$$\{z,x\}\equiv {\displaystyle \frac{z^{‴}\left(x\right)}{z^{\prime }\left(x\right)}}-{\displaystyle \frac{3}{2}}{\left({\displaystyle \frac{z^{″}\left(x\right)}{z^{\prime }\left(x\right)}}\right)}^2,$$

(6)

and contain only parameters of the $z\left(x\right)$ function.

The differential equation defining the $z\left(x\right)$ function can generally be written as

$${\left({\displaystyle \frac{\mathrm{d}z}{\mathrm{d}x}}\right)}^2\mathsf{\Phi }\left(z\right)=C,$$

(7)

where the $\mathsf{\Phi }\left(z\right)$ function contains one or more terms appearing in parenthesis in the last term of Equation (4). Remembering that $z\left(x\right)$ is expected to be monotonous inside its domain of definition in order to lead to unique mapping, $z^{\prime }$ is not allowed to change signs there. This implies that ${\left(z^{\prime }\left(x\right)\right)}^2$ cannot be zero, which indicates from Equation (7) that $C/\mathsf{\Phi }\left(z\right)$ cannot be non-zero either. The integration of Equation (7) leads to the expression

$$\int {\mathsf{\Phi }}^{1/2}\left(z\right)\mathrm{d}z=C^{1/2}(x+x_0),$$

(8)

which represents the inverse of the $z\left(x\right)$ function. When $x\left(z\right)$ is non-invertible, the potential is called implicit. Even in this case, all the calculations can be carried out in a parametric form, and exact formulae can be obtained for both the solutions (5), the potential $V\left(x\right)$, and the energy eigenvalues E. $x_0$ represents a constant of integration corresponding to a coordinate shift that has no effect on the energy spectrum. It is usually unimportant and can be chosen such that $z\left(0\right)=0$, for example. However, in certain situations (e.g., in $\mathcal{PT}$-symmetric quantum mechanics, see [11]), it can play an important role, e.g., by avoiding singularities.

Utilizing Equation (7), Equation (4) can be rewritten to an expression depending on $Q\left(z\right)$, $R\left(z\right)$, $\mathsf{\Phi }\left(z\right)$, C, and $z\left(x\right)$. The first two are defined by $F\left(z\right)$, while $\mathsf{\Phi }\left(z\right)$ and C (which determine $z\left(x\right)$) can be chosen with some liberty:

$$\begin{array}{cc}E-V\left(x\right)=& {\displaystyle \frac{z^{‴}\left(x\right)}{2z^{\prime }\left(x\right)}}-{\displaystyle \frac{3}{4}}{\left({\displaystyle \frac{z^{″}\left(x\right)}{z^{\prime }\left(x\right)}}\right)}^2\hfill \\ & +{\displaystyle \frac{C}{\mathsf{\Phi }\left(z\right(x\left)\right)}}\left[R\left(z\left(x\right)\right)-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}Q}{\mathrm{d}z}}-{\displaystyle \frac{1}{4}}{Q}^2\left(z\left(x\right)\right)\right]\end{array}$$

(9)

$$\begin{array}{cc}=& -{\displaystyle \frac{5C}{16{\mathsf{\Phi }}^3\left(z\right)}}{\left({\displaystyle \frac{\mathrm{d}\mathsf{\Phi }}{\mathrm{d}\mathrm{z}}}\right)}^2+{\displaystyle \frac{C}{4{\mathsf{\Phi }}^2\left(z\right)}}{\displaystyle \frac{{\mathrm{d}}^2\mathsf{\Phi }}{\mathrm{d}{\mathrm{z}}^2}}\hfill \\ & +{\displaystyle \frac{C}{\mathsf{\Phi }\left(z\right(x\left)\right)}}\left[R\left(z\left(x\right)\right)-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}Q}{\mathrm{d}z}}-{\displaystyle \frac{1}{4}}{Q}^2\left(z\left(x\right)\right)\right].\hfill \end{array}$$

(10)

In the last equation, the derivatives of $z\left(x\right)$ appearing in the Schwartzian derivative were substituted by expressions obtained from Equation (7) [22]. The two terms contain ${\mathsf{\Phi }}^j\left(z\right)$ with $j=-2$ and $-3$, while the main term contains ${\mathsf{\Phi }}^{-1}\left(z\right)$. In general, the terms originating from the Schwartzian derivative are independent of the main terms; however, for specific choices of $\mathsf{\Phi }\left(z\right)$, they can be absorbed into the latter. This important point will be discussed later on.

The formalism can be elaborated further if we assume that the general form of the expression containing $Q\left(z\right)$ and $R\left(z\right)$ in Equation (9) can be written as the linear combination of linearly independent z-dependent terms:

$$R\left(z\left(x\right)\right)-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}Q}{\mathrm{d}z}}-{\displaystyle \frac{1}{4}}{Q}^2\left(z\left(x\right)\right)=\sum _{i=1}^N{\alpha }_i{\phi }_i\left(z\left(x\right)\right).$$

(11)

One can now assume that the potential will contain the same terms, plus those originating from the Schwartzian derivative:

$$V\left(x\right)=-{\displaystyle \frac{1}{2}}\{z,x\}+{\displaystyle \frac{C}{\mathsf{\Phi }\left(z\right(x\left)\right)}}\sum _{i=1}^N{s}_i{\phi }_i\left(z\left(x\right)\right).$$

(12)

Furthermore, the general form of the $\mathsf{\Phi }\left(z\right)$ function must be composed of the same ${\phi }_i\left(z\right)$ terms in order to generate the constant term corresponding to E in Equation (9), i.e.,

$$\mathsf{\Phi }\left(z\left(x\right)\right)=\sum _{i=1}^N{p}_i{\phi }_i\left(z\left(x\right)\right).$$

(13)

Substituting Equations (11)–(13) into Equation (9), multiplying by $\mathsf{\Phi }\left(z\right(x\left)\right)$, and collecting the like terms, one obtains a set of N algebraic equations that connect the parameters appearing in $\mathsf{\Phi }\left(z\right)$, the potential $V\left(x\right)$, and in the expression obtained from $Q\left(z\right)$ and $R\left(z\right)$ in Equation (4):

$${\alpha }_i+s_i-{\displaystyle \frac{E}{C}}{p}_i=0,i=1\dots N.$$

(14)

These equations have to be satisfied simultaneously in order to obtain an exactly solvable potential. It is worthwhile to remember the origin of the parameters appearing in Equation (14). The $p_i$ parameters appear in $\mathsf{\Phi }\left(z\right)$, and are transferred into $z\left(x\right)$ from Equation (7) and thus from (8). The same applies to C and $x_0$, which are simply scaling factors. The ${\alpha }_i$ parameters originate from $Q\left(z\right)$ and $R\left(z\right)$, i.e., they depend on the parameters of the special function $F\left(z\right)$. All these parameters determine the $s_i$ coupling coefficients that appear in the potential $V\left(x\right)$. The coupling coefficients have to be chosen such that the resulting potential becomes energy-independent.

The complexitiy of this task depends on that of $\mathsf{\Phi }\left(z\right)$: the fewer non-zero $p_i$ parameters it contains, the easier it will be to generate a reasonable potential. When only one of the $p_i$ parameters, say $p_I$, is non-zero, then the expression for the energy, $E=C({\alpha }_I-s_I)/p_I$, follows immediately, while for all $i\ne I$, the coupling parameters will become $s_i=-{\alpha }_i$. It now remains to express ${\alpha }_i$ with the parameters of the special function $F\left(z\right)$ such that the energy dependence is not transferred into them from ${\alpha }_I$. This task becomes increasingly complicated if more than one of the $p_i$ parameters are chosen to be non-zero.

3. Application to the Biconfluent, Doubly Confluent, and Triconfluent Heun Equations

Here, we apply the transformation method to the biconfluent, doubly confluent, and triconfluent Heun equations [12,20], as well as to the confluent hypergeometric differential equation [9], as a reference. Similar analysis of the Heun and the confluent Heun equations is left for a further study in order to keep the discussion compact.

First, the general properties of the potential $V\left(x\right)$ appearing in the transformed Schrödinger equation will be studied, reviewing the results from earlier publications; then, the bound-state solutions will be discussed for each special function. In order to follow a systematic description, our notation will occasionally slightly differ from the one used in the literature.

3.1. The General Form of the Potentials

As a reference for the more complicated cases, first, the Natanzon confluent potentials are reviewed, and the main steps of the method are illustrated by the relatively simple example of the confluent hypergeometric differential equation. Then the three Heun-type equations are discussed, emphasizing the differences with respect to the reference problem. Here, the form of the differential equations is taken from Ref. [20], which is more general and more consistent than the forms displayed in the monograph [12], which are usually applied in the literature. The consequences of using different formalisms are discussed in each case.

3.1.1. $F\left(z\right)={}_{1}F_1(a;c;z)$: The Natanzon Confluent Potential Class

The transformation of the confluent hypergeometric differential equation

$${\displaystyle \frac{{\mathrm{d}}^{2}F}{\mathrm{d}{z}^2}}+\left({\displaystyle \frac{c}{z}}-1\right){\displaystyle \frac{\mathrm{d}F}{\mathrm{d}z}}-{\displaystyle \frac{a}{z}}F\left(z\right)=0$$

(15)

was considered in Natanzon’s original publication [6]. See also Ref. [23], where the expression “Natanzon confluent potentials” was introduced. Now Equation (4) reduces to

$$\begin{array}{cc}E-V\left(x\right)=& {\displaystyle \frac{z^{‴}\left(x\right)}{2z^{\prime }\left(x\right)}}-{\displaystyle \frac{3}{4}}{\left({\displaystyle \frac{z^{″}\left(x\right)}{z^{\prime }\left(x\right)}}\right)}^2\hfill \\ & +{\displaystyle \frac{{\left(z^{\prime }\left(x\right)\right)}^2}{z^2\left(x\right)}}\left[\left({\displaystyle \frac{c}{2}}-a\right)z\left(x\right)-{\displaystyle \frac{1}{4}}{z}^2\left(x\right)-{\displaystyle \frac{c}{2}}\left({\displaystyle \frac{c}{2}}-1\right)\right].\hfill \end{array}$$

(16)

There are three independent terms originating from $Q\left(z\right)$ and $R\left(z\right)$, and all of them contain integer powers of z. Factoring out the $z^{-2}\left(x\right)$ term, the $\mathsf{\Phi }\left(z\right)$ function in Equation (13) has the structure

$$\mathsf{\Phi }\left(z\left(x\right)\right)\equiv {\displaystyle \frac{\varphi \left(z\right(x\left)\right)}{z^2\left(x\right)}}\equiv {\displaystyle \frac{1}{z^2\left(x\right)}}(p_{1}z\left(x\right)+p_2{z}^2\left(x\right)+p_3),$$

(17)

while the potential is

$$V\left(x\right)=-{\displaystyle \frac{1}{2}}\{z,x\}+{\displaystyle \frac{C}{\varphi \left(z\right(x\left)\right)}}(s_{1}z\left(x\right)+s_2{z}^2\left(x\right)+s_3).$$

(18)

The actual form of Equation (14) contains three algebraic equations

$$\begin{array}{ccc}\hfill {\displaystyle \frac{c}{2}}-a+s_1-{\displaystyle \frac{E}{C}}{p}_1& =& 0,\hfill \\ \hfill -{\displaystyle \frac{1}{4}}+s_2-{\displaystyle \frac{E}{C}}{p}_2& =& 0,\hfill \\ \hfill -{\displaystyle \frac{c}{2}}\left({\displaystyle \frac{c}{2}}-1\right)+s_3-{\displaystyle \frac{E}{C}}{p}_3& =& 0,\hfill \end{array}$$

(19)

while, according to Equation (5), the solutions are

$$\psi \left(x\right)\sim z^{(c-1)/2}\left(x\right){\varphi }^{1/4}\left(z\left(x\right)\right)exp{(-z\left(x\right)/2)}_1{F}_1(a;c;z\left(x\right)).$$

(20)

For $a=-N$, where N is a non-negative integer, the confluent hypergeometric function reduces to the generalized Laguerre polynomial [9] $L_N^{(c-1)}\left(z\left(x\right)\right)$, which appears in the bound-state solution of the related potential problems. These will be discussed in the next subsection.

3.1.2. Potentials Solvable in Terms of the Biconfluent Heun Equation

The biconfluent Heun equation

$${\displaystyle \frac{{\mathrm{d}}^{2}F}{\mathrm{d}{z}^2}}+\left({\displaystyle \frac{\gamma }{z}}+\delta +ϵz\right){\displaystyle \frac{\mathrm{d}F}{\mathrm{d}z}}+\left(\alpha -{\displaystyle \frac{q}{z}}\right)F\left(z\right)=0$$

(21)

solved formally by the functions $Hb(\alpha ,\gamma ,\delta ,ϵ,q;z)$ has one regular singularity and an irregular singularity of rank 2 [12,20], with the former placed conventionally at $z=0$ and the latter at infinity. Equation (21) contains five parameters and has been taken from [20]. The four-parameter version, called the canonical equation of this class, appearing in Equation (1.2.5) on p. 194 of Ref. [12], can be obtained from the substitution $\{\gamma ,\delta ,ϵ,\alpha ,q\}\to \{\alpha +1,-\beta ,-2,\gamma -\alpha -2,(\delta +(\alpha +1)\beta )/2\}$. Reducing the number of parameters to four by setting $ϵ$ to a fixed value corresponds to a scale transformation of the variable z. However, this choice prevents the direct reduction of (21) to the confluent hypergeometric differential Equation (15), which also complicates the reduction of potentials obtained from Equation (21) to the Natanzon confluent potentials discussed in Section 3.1.1.

The actual form of Equation (4) in this case is

$$\begin{array}{cc}E-V\left(x\right)=& {\displaystyle \frac{z^{‴}\left(x\right)}{2z^{\prime }\left(x\right)}}-{\displaystyle \frac{3}{4}}{\left({\displaystyle \frac{z^{″}\left(x\right)}{z^{\prime }\left(x\right)}}\right)}^2\hfill \\ & +{\displaystyle \frac{{\left(z^{\prime }\left(x\right)\right)}^2}{z^2\left(x\right)}}\left[-{\displaystyle \frac{\gamma }{2}}\left({\displaystyle \frac{\gamma }{2}}-1\right)-\left(q+{\displaystyle \frac{\gamma \delta }{2}}\right)z\left(x\right)\right.\hfill \\ & \left.+\left(\alpha -{\displaystyle \frac{ϵ}{2}}-{\displaystyle \frac{{\delta }^2}{4}}-{\displaystyle \frac{\gamma ϵ}{2}}\right)z^2\left(x\right)-{\displaystyle \frac{\delta ϵ}{2}}{z}^3\left(z\right)-{\displaystyle \frac{{ϵ}^2}{4}}{z}^4\left(x\right)\right].\hfill \end{array}$$

(22)

There are five significant terms in Equation (22), as the structures of the $Q\left(z\right)$ and $R\left(z\right)$ functions also contain three and two significant terms, which contain integer powers of z.

It is reasonable to define the $\mathsf{\Phi }\left(z\right)$ function as

$$\mathsf{\Phi }\left(z\left(x\right)\right)\equiv {\displaystyle \frac{\varphi \left(z\right(x\left)\right)}{z^2\left(x\right)}}\equiv {\displaystyle \frac{1}{z^2\left(x\right)}}\left(p_1+p_{2}z\left(x\right)+p_3{z}^2\left(x\right)+p_4{z}^3\left(x\right)+p_5{z}^4\left(x\right)\right),$$

(23)

and a similar structure can be assigned to the significant terms of the potential too:

$$V\left(x\right)=-{\displaystyle \frac{1}{2}}\{z,x\}+{\displaystyle \frac{C}{\varphi \left(z\right(x\left)\right)}}\left(s_1+s_{2}z\left(x\right)+s_3{z}^2\left(x\right)+s_4{z}^3\left(x\right)+s_5{z}^4\left(x\right)\right).$$

(24)

With these choices, Equation (14) turns into

$$\begin{array}{ccc}\hfill -{\displaystyle \frac{\gamma }{2}}\left({\displaystyle \frac{\gamma }{2}}-1\right)+s_1-p_1{\displaystyle \frac{E}{C}}& =& 0,\hfill \\ \hfill -\left(q+{\displaystyle \frac{\gamma \delta }{2}}\right)+s_2-p_2{\displaystyle \frac{E}{C}}& =& 0,\hfill \\ \hfill +\left(\alpha -{\displaystyle \frac{ϵ}{2}}-{\displaystyle \frac{{\delta }^2}{4}}-{\displaystyle \frac{\gamma ϵ}{2}}\right)+s_3-p_3{\displaystyle \frac{E}{C}}& =& 0,\hfill \\ \hfill -{\displaystyle \frac{\delta ϵ}{2}}+s_4-p_4{\displaystyle \frac{E}{C}}& =& 0,\hfill \\ \hfill -{\displaystyle \frac{{ϵ}^2}{4}}+s_5-p_5{\displaystyle \frac{E}{C}}& =& 0.\hfill \end{array}$$

(25)

The general solutions become

$$\psi \left(x\right)\sim {\varphi }^{1/4}\left(z\left(x\right)\right){\left(z\left(x\right)\right)}^{(\gamma -1)/2}exp\left({\displaystyle \frac{\delta }{2}}z\left(x\right)+{\displaystyle \frac{ϵ}{4}}{z}^2\right)Hb(\alpha ,\gamma ,\delta ,ϵ,q;z\left(x\right)).$$

(26)

It can be seen that the formulae reduce to those obtained for the confluent hypergeometric function ${}_{1}F_1(a;c;z)$ for $\alpha =0$, $\gamma =c$, $\delta =-1$, $ϵ=0$, and $q=a$, with $s_4=s_5=p_4=p_5=0$. In particular, the last two equations of Equation (25) vanish, and the remaining three recover Equation (19). Potentials generated from the four-parameter version of the biconfluent Heun equation [12] have been discussed in Ref. [17]. The potentials displayed in Equation (68) of that work correspond to taking $p_5=0$ and $s_5=-4$ in Equations (23) and (24).

3.1.3. Potentials Solvable in Terms of the Doubly Confluent Heun Equation

The doubly confluent Heun equation [12,20]

$${\displaystyle \frac{{\mathrm{d}}^{2}F}{\mathrm{d}{z}^2}}+\left({\displaystyle \frac{\gamma }{z^2}}+{\displaystyle \frac{\delta }{z}}+ϵ\right){\displaystyle \frac{\mathrm{d}F}{\mathrm{d}z}}+\left({\displaystyle \frac{\alpha }{z}}-{\displaystyle \frac{q}{z^2}}\right)F\left(z\right)=0$$

(27)

has two rank 1 irregular singularities at $z=0$ and at infinity. Its solutions are formally written as $Hd(\alpha ,\gamma ,\delta ,ϵ,q;z)$. Here again, we considered the five-parameter version taken from [20]. The four-parameter version appearing in Equation (1.4.40) on page 142 of Ref. [12] generated from the symmetruc canonical form of this equation (Equation (0.0.1) on page 131) can be obtained from the substitution $\{\gamma ,\delta ,ϵ,\alpha ,q\}\to \{\alpha ,2(1-{\beta }_{-1}),\alpha ,\alpha ({\beta }_1-{\beta }_{-1}+1),-{\alpha }^2/2+\gamma -{({\beta }_{-1}-1/2)}^2\}$. Note that the first and third parameters take on the same value in the substitution. Equation (4) takes the form

$$\begin{array}{cc}E-V\left(x\right)=& {\displaystyle \frac{z^{‴}\left(x\right)}{2z^{\prime }\left(x\right)}}-{\displaystyle \frac{3}{4}}{\left({\displaystyle \frac{z^{″}\left(x\right)}{z^{\prime }\left(x\right)}}\right)}^2\hfill \\ & +{\left(z^{\prime }\left(x\right)\right)}^2\left[-{\displaystyle \frac{{\gamma }^2}{4}}{\displaystyle \frac{1}{z^4}}+\left(\gamma -{\displaystyle \frac{\gamma \delta }{2}}\right){\displaystyle \frac{1}{z^3}}\right.\hfill \\ & \left.+\left(-q+{\displaystyle \frac{\delta }{2}}-{\displaystyle \frac{{\delta }^2}{4}}-{\displaystyle \frac{\gamma ϵ}{2}}\right){\displaystyle \frac{1}{z^2}}+\left(\alpha -{\displaystyle \frac{\delta ϵ}{2}}\right){\displaystyle \frac{1}{z}}-{\displaystyle \frac{{ϵ}^2}{4}}\right].\hfill \end{array}$$

(28)

Here again, there are five significant terms, which inspire the following form of the $\mathsf{\Phi }\left(z\right)$ function

$$\mathsf{\Phi }\left(z\left(x\right)\right)\equiv {\displaystyle \frac{\varphi \left(z\right(x\left)\right)}{z^4\left(x\right)}}\equiv {\displaystyle \frac{1}{z^4\left(x\right)}}\left(p_1+p_{2}z\left(x\right)+p_3{z}^2\left(x\right)+p_4{z}^3\left(x\right)+p_5{z}^4\left(x\right)\right)$$

(29)

and of the potential

$$V\left(x\right)=-{\displaystyle \frac{1}{2}}\{z,x\}+{\displaystyle \frac{C}{\varphi \left(z\right(x\left)\right)}}\left(s_1+s_{2}z\left(x\right)+s_3{z}^2\left(x\right)+s_4{z}^3\left(x\right)+s_5{z}^4\left(x\right)\right).$$

(30)

The set of equations connecting the $p_i$, $s_i$, and C parameters, as well as the energy E with the parameters of the doubly confluent Heun equation are

$$\begin{array}{ccc}\hfill -{\displaystyle \frac{{\gamma }^2}{4}}+s_1-p_1{\displaystyle \frac{E}{C}}& =& 0,\hfill \\ \hfill \gamma -{\displaystyle \frac{\gamma \delta }{2}}+s_2-p_2{\displaystyle \frac{E}{C}}& =& 0,\hfill \\ \hfill -q+{\displaystyle \frac{\delta }{2}}-{\displaystyle \frac{{\delta }^2}{4}}-{\displaystyle \frac{\gamma ϵ}{2}}+s_3-p_3{\displaystyle \frac{E}{C}}& =& 0,\hfill \\ \hfill \alpha -{\displaystyle \frac{\delta ϵ}{2}}+s_4-p_4{\displaystyle \frac{E}{C}}& =& 0,\hfill \\ \hfill -{\displaystyle \frac{{ϵ}^2}{4}}+s_5-p_5{\displaystyle \frac{E}{C}}& =& 0.\hfill \end{array}$$

(31)

The general form of the solutions is then

$$\psi \left(x\right)\sim {\varphi }^{1/4}\left(z\left(x\right)\right){\left(z\left(x\right)\right)}^{(\delta -1)/2}exp\left(-\gamma /\left(2z\right(x\left)\right)+z\left(x\right)ϵ/2\right)Hd(\alpha ,\gamma ,\delta ,ϵ,q;z\left(x\right)).$$

(32)

The formulae reduce to those relevant to the confluent hypergeometric function ${}_{1}F_1(a;c;z)$ for $\alpha =-a$, $\gamma =0$, $\delta =c$, $ϵ=-1$, and $q=0$, with $p_1=p_2=s_1=s_2=0$.

A study in Ref. [17] generated the same potential from the normal form of the generalized double confluent Heun equation, Equation (1.1.10) on p. 134 of Ref. [12]. This equation has five parameters and its Bose invariant contains the same five independent terms as the Bose invariant of Equation (27). The general potential appearing in Equation (73) of Ref. [17] coincides with that in Equation (30), here for the parameter choices $B_2=-{\gamma }^2/4$, $B_{-1}=\gamma (1-\delta /2)$, $B_0=-q+\delta /2-{\delta }^2/4-\gamma ϵ/2$, $B_1+1/4=\alpha -\delta ϵ/2$, and $B_2=-{ϵ}^2/4$. No further discussion concerning the allowed parameter ranges is presented in Ref. [17], but the correspondence with our results indicates that $B_{-2}\le 0$ and $B_2\le 0$ have to hold if $\gamma$ and $ϵ$ are real, as expected from the structure of the wave functions (32).

3.1.4. Potentials Solvable in Terms of the Triconfluent Heun Equation

The five-parameter tricofluent Heun equation [20]

$${\displaystyle \frac{{\mathrm{d}}^{2}F}{\mathrm{d}{z}^2}}+\left(\gamma +\delta z+ϵz^2\right){\displaystyle \frac{\mathrm{d}F}{\mathrm{d}z}}+\left(\alpha z-q\right)F\left(z\right)=0$$

(33)

has one irregular singularity of rank 3 at infinity. The solutions of Equation (33) are written as $Ht(\alpha ,\gamma ,\delta ,ϵ,q;z)$. The monograph [12] discusses fewer-parameter versions of the same equation. The canonical form of the triconfluent Heun equation displayed in Equation (1.2.1) on p. 254 is obtained from Equation (33) with the substitution

$$\{\gamma ,\delta ,ϵ,\alpha ,q\}\to \{-\gamma ,0,-3,\beta -3,-\alpha \}.$$

(34)

The reparametrization of this equation without a first-order derivative term in Equation (1.2.3) is called the representative form of the triconfluent Heun equation. The actual form of Equation (4)

$$\begin{array}{cc}E-V\left(x\right)=& {\displaystyle \frac{z^{″}\left(x\right)}{2z^{\prime }\left(x\right)}}-{\displaystyle \frac{3}{4}}{\left({\displaystyle \frac{z^{″}\left(x\right)}{z^{\prime }\left(x\right)}}\right)}^2\hfill \\ & +{\left(z^{\prime }\left(x\right)\right)}^2\left[-{\displaystyle \frac{{ϵ}^2}{4}}{z}^4-{\displaystyle \frac{\delta ϵ}{2}}{z}^3-\left({\displaystyle \frac{{\delta }^2}{4}}+{\displaystyle \frac{\gamma ϵ}{2}}\right)z^2-\left(ϵ+{\displaystyle \frac{\gamma \delta }{2}}-\alpha \right)z-\left(q+{\displaystyle \frac{\delta }{2}}\right)\right]\hfill \end{array}$$

(35)

has, again, five significant terms, and the coefficients of these terms depend on five parameters.

Now, the five-term expression for $\mathsf{\Phi }\left(z\right)$ can be chosen as

$$\mathsf{\Phi }\left(z\left(x\right)\right)=p_1{z}^4\left(x\right)+p_2{z}^3\left(x\right)+p_3{z}^2\left(x\right)+p_{4}z\left(x\right)+p_5,$$

(36)

while the potential can be written as

$$V\left(x\right)=-{\displaystyle \frac{1}{2}}\{z,x\}+{\displaystyle \frac{C}{\mathsf{\Phi }\left(z\right(x\left)\right)}}\left(s_1{z}^4\left(x\right)+s_2{z}^3\left(x\right)+s_3{z}^2\left(x\right)+s_{4}z\left(x\right)+s_5\right).$$

(37)

The usual procedure then leads to the following set of equations:

$$\begin{array}{ccc}\hfill -{\displaystyle \frac{{ϵ}^2}{4}}+s_1-p_1{\displaystyle \frac{E}{C}}& =& 0,\hfill \\ \hfill -{\displaystyle \frac{\delta ϵ}{2}}+s_2-p_2{\displaystyle \frac{E}{C}}& =& 0,\hfill \\ \hfill -\left({\displaystyle \frac{{\delta }^2}{4}}+{\displaystyle \frac{\gamma ϵ}{2}}\right)+s_3-p_3{\displaystyle \frac{E}{C}}& =& 0,\hfill \\ \hfill -\left(ϵ+{\displaystyle \frac{\gamma \delta }{2}}-\alpha \right)+s_4-p_4{\displaystyle \frac{E}{C}}& =& 0,\hfill \\ \hfill -\left(q+{\displaystyle \frac{\delta }{2}}\right)+s_5-p_5{\displaystyle \frac{E}{C}}& =& 0.\hfill \end{array}$$

(38)

The general solutions are written as

$$\psi \left(x\right)\sim {\mathsf{\Phi }}^{1/4}\left(z\left(x\right)\right)exp\left(\gamma z\left(x\right)/2+\delta z^2\left(x\right)/4+ϵz^3\left(x\right)/6\right)Ht(\alpha ,\gamma ,\delta ,ϵ,q;z\left(x\right)).$$

(39)

Note that the term with $z^{-1}$ is missing from $Q\left(z\right)$ in Equation (33). One consequence of this fact is that the triconfluent Heun equation cannot be reduced to the confluent hypergeometric differential Equation (15), and another one is that there is no power-like term in the pre-factor solution of the solution (39) (see Equation (5)).

Reference [17] presents a general formula for potentials derived from the three-parameter representative form of the triconfluent Heun equation mentioned above. The potentials displayed in Equation (78) of that work correspond to taking $p_1=0$, $p_2=0$, $s_1=-9$, and $s_2=0$ in Equations (36) and (37).

3.2. Concrete Potentials and Their Solutions

In order to obtain concrete quantum mechanical potentials from the general results, the $z\left(x\right)$ function has to be determined from the actual form of Equation (7). For this, the $\mathsf{\Phi }\left(z\right)$ Function (13) has to be specified. In addition to determining $z\left(x\right)$, this function also appears directly in the expression of the potential (10). In particular, ${\mathsf{\Phi }}^{-1}\left(x\right)$ forms part of the main terms of the potential (i.e., those originating from $Q\left(z\right)$ and $R\left(z\right)$), while its derivatives contribute to the terms originating from the Schwartzian derivative. It is not trivial whether these latter terms coincide with those appearing in the main potential term. If this is not the case, then the potential will contain terms that do not depend on the parameters of the special function, only on the $p_i$ parameters from the variable transformation, and thus of $z\left(x\right)$. This implies that there will be terms with fixed coupling coefficients. Occasionally, potentials of this type are called conditionally solvable on grounds that they are solvable only under the condition that some of the potential terms have fixed coupling coefficients.

In order to clarify this situation, it is worthwhile to inspect the structure of the $\mathsf{\Phi }\left(z\right)$ functions obtained in Section 3 for each special function $F\left(z\right)$. The general expression was found to be

$$\mathsf{\Phi }\left(z\left(x\right)\right)={\displaystyle \frac{{\varphi }^{\left(\nu \right)}\left(z\right)}{z^t}},$$

(40)

where ${\varphi }^{\left(\nu \right)}\left(z\right)$ is a $\nu$’th order polynomial in z with $\nu +1$ terms, while t takes on the possible values 0, 2, or 4. These values are specified for each special function discussed here in

Table 1. The parameters of the specific $\mathsf{\Phi }\left(z\right)$ functions relevant to the special functions discussed in Section 3. Here, $\nu$ is the order of the polynomial appearing in the numerator of the expressions (17), (23), (29), and (36).

F	${}_{\mathbf{1}}\mathit{F}_{\mathbf{1}}$	$\mathbf{Hb}$	$\mathbf{Hd}$	$\mathbf{Ht}$
$\nu$	2	4	4	4
t	2	2	4	0

.

The actual structure of the ${\varphi }^{\left(\nu \right)}\left(z\right)$ polynomial depends on the $p_i$ parameters ($i=1,2,\dots \nu +1$), which can be chosen arbitrarily. For arbitrary $p_i$, the situation outlined in the previous paragraph can occur, i.e., potential terms with fixed coupling coefficients can appear from the Schwartzian derivative (see Equation (10)). Alternatively, one may chose the $p_i$ parameters in such a way that ${\varphi }^{\left(\nu \right)}\left(z\right)$ itself is written in the form $z^k$. In this case, eventually, the $\mathsf{\Phi }\left(z\right)$ function will have the structure

$$\mathsf{\Phi }\left(z\left(x\right)\right)=p{z}^{\mu }\equiv p{z}^{k-t}.$$

(41)

This $\mathsf{\Phi }\left(z\right)$ has to be substituted into the terms originating from the Schwartzian derivative (see Equations (9) and (10)).

The allowed values of k are $k=0,1,\dots \nu$, so those of $\mu$ are $\mu =-t,-t+1,\dots \nu -t$, depending on which term we pick from the polynomial ${\varphi }^{\left(\nu \right)}\left(z\right)$. The same terms will appear as the main terms of the potential (12). Substituting Equation (41) into Equation (10), the expression

$$-{\displaystyle \frac{1}{2}}\{z,x\}\equiv -{\displaystyle \frac{z^{‴}\left(x\right)}{2z^{\prime }\left(x\right)}}+{\displaystyle \frac{3}{4}}{\left({\displaystyle \frac{z^{″}\left(x\right)}{z^{\prime }\left(x\right)}}\right)}^2=-{\displaystyle \frac{C\mu (\mu +4)}{16p}}{z}^{-\mu -2}.$$

(42)

follows for the Schwartzian derivative term. The power of z in this expression is $-\mu -2=t-2-k$, while the powers of z in the main potential terms in Equation (12) range from $-k$ to $\nu -k$. This means that the potential term originating from the Schwartzian derivative will appear among the terms of the main potential terms as long as $0\le t-2\le \nu$ holds. It is seen from Table 1 that this condition holds for all the special functions, except for the triconfluent Heun equation, where we have $t-2=-2<0$. This means that the potentials derived for this special function will always contain a term, the coupling coefficient, of which will not depend on the parameters appearing in the triconfluent Heun equation, and which will be different from the main potential terms in Equation (37).

The specific choice (41) also guarantees that the integration of Equation (8) leads to an invertible $x\left(z\right)$ function, i.e., to an explicit form of $x\left(z\right)$:

$$C^{1/2}(x+x_0)=\left\{\begin{array}{cc}{\displaystyle \frac{2p^{1/2}}{\mu +2}}{z}^{\mu /2+1},\hfill & if\mu =k-t\ne -2\hfill \\ p^{1/2}ln\left(z\right),\hfill & \mathrm{if}\mu =\mathrm{k}-\mathrm{t}=-2\hfill \end{array}\right.$$

(43)

implying

$$z\left(x\right)=\left\{\begin{array}{cc}{\left[{\displaystyle \frac{(\mu +2)C^{1/2}(x+x_0)}{2p^{1/2}}}\right]}^{2/(\mu +2)}\hfill & if\mu =k-t\ne -2\hfill \\ C_{0}exp\left({\displaystyle \frac{C^{1/2}}{p^{1/2}}}x\right)\hfill & \mathrm{if}\mu =\mathrm{k}-\mathrm{t}=-2.\hfill \end{array}\right.$$

(44)

In the latter $\mu =-2$ case, the Schwartzian derivative (42) turns into the constant $C/\left(4p\right)$. In summary, all the special choices with a single-term polynomial ${\varphi }^{\left(\nu \right)}\left(z\right)=p{z}^k$ lead to a single potential term (or a constant) from the Schwartzian derivative, and this term appears among the main potential terms, with the exception of potentials derived from the triconfluent Heun equation. Combining Equations (42) and (44), one can even determine the explicit coordinate dependence of the Schwartzian derivative term if $\mu =k-t\ne -2$:

$$-{\displaystyle \frac{1}{2}}\{z,x\}=-{\displaystyle \frac{\mu (\mu +4)}{4{(\mu +2)}^2}}{(x+x_0)}^{-2}$$

(45)

So unless $k-t=-2$, when the Schwartzian derivative is constant, it contributes to the potential by an inverse quadratic function of the coordinate $x+x_0$. Furthermore, the coupling coefficient of this term always exceeds $-1/4$, which means that this singularity is either repulsive (if $|k-t+2|<2$), or attractive, but falls within the domain of weak attraction (if $|k-t+2|>2$) [24]. For the confluent hypergeometric, biconfluent Heun and doubly confluent Heun cases, this term can be combined with another potential term exhibiting inverse square coordinate dependence, while for the triconfluent Heun equation, it stands alone.

Table 2. The specific $\varphi \left(z\right)$ function defined in Equation (17) for the confluent hypergeometric equation, the $z\left(x\right)$ functions calculated from them using Equation (7), the resulting potentials (18), and energy eigenvalues E. For the sake of simplicity, the integration constant in Equation (8) was set to $x_0=0$.

$\mathit{\varphi }\left(\mathit{z}\right)$	$\mathit{z}\left(\mathit{x}\right)$	$\left[-\frac{1}{2}\{\mathit{z},\mathit{x}\}\right]$	$+\frac{\mathit{C}{\mathit{s}}_1}{\mathit{\varphi }\left(\mathit{z}\right)}\mathit{z}$	$+\frac{\mathit{C}{\mathit{s}}_2}{\mathit{\varphi }\left(\mathit{z}\right)}{\mathit{z}}^2$	$+\frac{\mathit{C}{\mathit{s}}_3}{\mathit{\varphi }\left(\mathit{z}\right)}$	E
$p_3$	$exp\left({\displaystyle \frac{C^{1/2}}{p_3^{1/2}}}x\right)$	$\left[{\displaystyle \frac{C}{4p_3}}\right]$	$+{\displaystyle \frac{C}{p_3}}\left(a-{\displaystyle \frac{c}{2}}\right)z$	+${\displaystyle \frac{C}{4p_3}}{z}^2$	$+{\displaystyle \frac{C{s}_3}{p_3}}$	${\displaystyle \frac{C}{p_3}}\left(s_3-{\displaystyle \frac{c}{2}}\left({\displaystyle \frac{c}{2}}-1\right)\right)$
$p_{1}z$	${\displaystyle \frac{C}{4p_1}}{x}^2$	$\left[{\displaystyle \frac{3C}{16p_1}}{\displaystyle \frac{1}{z}}\right]$	$+{\displaystyle \frac{C{s}_1}{p_1}}$	$+{\displaystyle \frac{C}{4p_1}}z$	$+{\displaystyle \frac{C}{p_1}}{\displaystyle \frac{c}{2}}\left({\displaystyle \frac{c}{2}}-1\right)z^{-1}$	${\displaystyle \frac{C}{p_1}}\left(s_1+{\displaystyle \frac{c}{2}}-a\right)$
$p_2{z}^2$	${\displaystyle \frac{C^{1/2}}{p_2^{1/2}}}x$	$\left[0\right]$	$+{\displaystyle \frac{C}{p_2}}\left(a-{\displaystyle \frac{c}{2}}\right){\displaystyle \frac{1}{z}}$	$+{\displaystyle \frac{C{s}_2}{p_2}}$	$+{\displaystyle \frac{C}{p_2}}{\displaystyle \frac{c}{2}}\left({\displaystyle \frac{c}{2}}-1\right){\displaystyle \frac{1}{z^2}}$	${\displaystyle \frac{C}{p_2}}\left(s_2-{\displaystyle \frac{1}{4}}\right)$

,

Table 3. The specific $\varphi \left(z\right)$ function defined in Equation (23) for the biconfluent Heun equation, the $z\left(x\right)$ functions calculated from them using Equation (7), the resulting potentials (24), and energy eigenvalues E. The coupling coefficients $s_i$ and E can be extracted from Equation (25) for each choice of $\varphi \left(z\right)$: $s_1={\displaystyle \frac{\gamma }{2}}\left({\displaystyle \frac{\gamma }{2}}-1\right)$, $s_2=\left(q+{\displaystyle \frac{\gamma \delta }{2}}\right)$, $s_3=\left(-\alpha +{\displaystyle \frac{ϵ}{2}}+{\displaystyle \frac{{\delta }^2}{4}}+{\displaystyle \frac{\gamma ϵ}{2}}\right)$, $s_4={\displaystyle \frac{\delta ϵ}{2}}$, $s_5={\displaystyle \frac{{ϵ}^2}{4}}$. For the sake of simplicity, the integration constant in Equation (8) was set to $x_0=0$.

$\mathit{\varphi }\left(\mathit{z}\right)$	$\mathit{z}\left(\mathit{x}\right)$	$\left[-\frac{1}{2}\{\mathit{z},\mathit{x}\}\right]$	$+\frac{\mathit{C}{\mathit{s}}_1}{\mathit{\varphi }\left(\mathit{z}\right)}$	$+\frac{\mathit{C}{\mathit{s}}_2\mathit{z}}{\mathit{\varphi }\left(\mathit{z}\right)}$	$+\frac{\mathit{C}{\mathit{s}}_3{\mathit{z}}^2}{\mathit{\varphi }\left(\mathit{z}\right)}$	$+\frac{\mathit{C}{\mathit{s}}_4{\mathit{z}}^3}{\mathit{\varphi }\left(\mathit{z}\right)}$	$+\frac{\mathit{C}{\mathit{s}}_5{\mathit{z}}^4}{\mathit{\varphi }\left(\mathit{z}\right)}$	E
$p_1$	$exp\left({\displaystyle \frac{C^{1/2}}{p_1^{1/2}}}x\right)$	$\left[{\displaystyle \frac{C}{4p_1}}\right]$	$+{\displaystyle \frac{C{s}_1}{p_1}}$	+${\displaystyle \frac{C{s}_2}{p_1}}z$	$+{\displaystyle \frac{C{s}_3}{p_1}}{z}^2$	$+{\displaystyle \frac{C{s}_4}{p_1}}{z}^3$	$+{\displaystyle \frac{C{s}_5}{p_1}}{z}^4$	${\displaystyle \frac{C}{p_1}}\left(s_1-{\displaystyle \frac{\gamma }{2}}\left({\displaystyle \frac{\gamma }{2}}-1\right)\right)$
$p_{2}z$	${\displaystyle \frac{C}{4p_2}}{x}^2$	$\left[{\displaystyle \frac{3C}{16p_2}}{\displaystyle \frac{1}{z}}\right]$	$+{\displaystyle \frac{C{s}_1}{p_2}}{\displaystyle \frac{1}{z}}$	$+{\displaystyle \frac{C{s}_2}{p_2}}$	$+{\displaystyle \frac{C{s}_3}{p_2}}z$	$+{\displaystyle \frac{C{s}_4}{p_2}}{z}^2$	$+{\displaystyle \frac{C{s}_5}{p_2}}{z}^3$	${\displaystyle \frac{C}{p_2}}\left[s_2-\left(q+{\displaystyle \frac{\gamma \delta }{2}}\right)\right]$
$p_3{z}^2$	${\displaystyle \frac{C^{1/2}}{p_3^{1/2}}}x$	$\left[0\right]$	$+{\displaystyle \frac{C{s}_1}{p_3}}{\displaystyle \frac{1}{z^2}}$	$+{\displaystyle \frac{C{s}_3}{p_3}}{\displaystyle \frac{1}{z}}$	$+{\displaystyle \frac{C{s}_3}{p_3}}$	$+{\displaystyle \frac{C{s}_4}{p_3}}z$	$+{\displaystyle \frac{C{s}_5}{p_3}}{z}^2$	${\displaystyle \frac{C}{p_3}}\left[s_3-\left(-\alpha +{\displaystyle \frac{ϵ}{2}}+{\displaystyle \frac{{\delta }^2}{4}}+{\displaystyle \frac{\gamma ϵ}{2}}\right)\right]$
$p_4{z}^3$	${\left({\displaystyle \frac{3C^{1/2}x}{2p_4^{1/2}}}\right)}^{2/3}$	$\left[-{\displaystyle \frac{5C}{16p_4}}{\displaystyle \frac{1}{z^3}}\right]$	$+{\displaystyle \frac{C{s}_1}{p_4}}{\displaystyle \frac{1}{z^3}}$	$+{\displaystyle \frac{C{s}_2}{p_4}}{\displaystyle \frac{1}{z^2}}$	$+{\displaystyle \frac{C{s}_3}{p_4}}{\displaystyle \frac{1}{z}}$	$+{\displaystyle \frac{C{s}_4}{p_4}}$	$+{\displaystyle \frac{C{s}_5}{p_4}}z$	${\displaystyle \frac{C}{p_4}}\left(s_4-{\displaystyle \frac{\delta ϵ}{2}}\right)$
$p_5{z}^4$	${\left({\displaystyle \frac{2C^{1/2}x}{p_5^{1/2}}}\right)}^{1/2}$	$\left[-{\displaystyle \frac{3C}{4p_5}}{\displaystyle \frac{1}{z^4}}\right]$	$+{\displaystyle \frac{C{s}_1}{p_5}}{\displaystyle \frac{1}{z^4}}$	$+{\displaystyle \frac{C{s}_2}{p_5}}{\displaystyle \frac{1}{z^3}}$	$+{\displaystyle \frac{C{s}_3}{p_5}}{\displaystyle \frac{1}{z^2}}$	$+{\displaystyle \frac{C{s}_4}{p_5}}{\displaystyle \frac{1}{z}}$	$+{\displaystyle \frac{C{s}_5}{p_5}}$	${\displaystyle \frac{C}{p_5}}\left(s_5-{\displaystyle \frac{{ϵ}^2}{4}}\right)$

,

Table 4. The specific $\varphi \left(z\right)$ function defined in Equation (29) for the doubly confluent Heun equation, the $z\left(x\right)$ functions calculated from them using Equation (7), the resulting potentials (30), and energy eigenvalues E. The coupling coefficients $s_i$ and E can be extracted from Equation (31) for each choice of $\varphi \left(z\right)$: $s_1={\displaystyle \frac{{\gamma }^2}{4}}$, $s_2=\gamma \left({\displaystyle \frac{\delta }{2}}-1\right)$, $s_3=\left(q-{\displaystyle \frac{\delta }{2}}+{\displaystyle \frac{{\delta }^2}{4}}+{\displaystyle \frac{\gamma ϵ}{2}}\right)$, $s_4=\left({\displaystyle \frac{ϵ\delta }{2}}-\alpha \right)$, $s_5={\displaystyle \frac{{ϵ}^2}{4}}$. For the sake of simplicity, the integration constant in Equation (8) was set to $x_0=0$.

$\mathit{\varphi }\left(\mathit{z}\right)$	$\mathit{z}\left(\mathit{x}\right)$	$\left[-\frac{1}{2}\{\mathit{z},\mathit{x}\}\right]$	$+\frac{\mathit{C}{\mathit{s}}_1}{\mathit{\varphi }\left(\mathit{z}\right)}$	$+\frac{\mathit{C}{\mathit{s}}_2\mathit{z}}{\mathit{\varphi }\left(\mathit{z}\right)}$	$+\frac{\mathit{C}{\mathit{s}}_3{\mathit{z}}^2}{\mathit{\varphi }\left(\mathit{z}\right)}$	$+\frac{\mathit{C}{\mathit{s}}_4{\mathit{z}}^3}{\mathit{\varphi }\left(\mathit{z}\right)}$	$+\frac{\mathit{C}{\mathit{s}}_5{\mathit{z}}^4}{\mathit{\varphi }\left(\mathit{z}\right)}$	E
$p_1$	$-{\left({\displaystyle \frac{C^{1/2}x}{p_1^{1/2}}}\right)}^{-1}$	$\left[0\right]$	$+{\displaystyle \frac{C{s}_1}{p_1}}$	+${\displaystyle \frac{C{s}_2}{p_1}}z$	$+{\displaystyle \frac{C{s}_3}{p_1}}{z}^2$	$+{\displaystyle \frac{C{s}_4}{2p_1}}{z}^3$	$+{\displaystyle \frac{C{s}_5}{p_1}}{z}^4$	${\displaystyle \frac{C}{p_1}}\left(s_1-{\displaystyle \frac{{\gamma }^2}{4}}\right)$
$p_{2}z$	${\left({\displaystyle \frac{C}{4p_2}}{x}^2\right)}^{-1}$	$\left[{\displaystyle \frac{3C}{16p_2}}z\right]$	$+{\displaystyle \frac{C{s}_1}{p_2}}{\displaystyle \frac{1}{z}}$	$+{\displaystyle \frac{C{s}_2}{p_2}}$	$+{\displaystyle \frac{C{s}_3}{p_2}}z$	$+{\displaystyle \frac{C{s}_4}{p_2}}{z}^2$	$+{\displaystyle \frac{C{s}_5}{p_2}}{z}^3$	${\displaystyle \frac{C}{p_2}}\left(s_2+\gamma -{\displaystyle \frac{\gamma \delta }{2}}\right)$
$p_3{z}^2$	$exp\left({\displaystyle \frac{C^{1/2}}{p_3^{1/2}}}x\right)$	$\left[-{\displaystyle \frac{C}{4p_3}}\right]$	$+{\displaystyle \frac{C{s}_1}{p_3}}{\displaystyle \frac{1}{z^2}}$	$+{\displaystyle \frac{C{s}_2}{p_3}}{\displaystyle \frac{1}{z}}$	$+{\displaystyle \frac{C{s}_3}{p_3}}$	$+{\displaystyle \frac{C{s}_4}{p_3}}z$	$+{\displaystyle \frac{C{s}_5}{p_3}}{z}^2$	${\displaystyle \frac{C}{p_3}}\left[s_3-\left(q-{\displaystyle \frac{\delta }{2}}+{\displaystyle \frac{{\delta }^2}{4}}+{\displaystyle \frac{\gamma ϵ}{2}}\right)\right]$
$p_4{z}^3$	${\displaystyle \frac{C}{4p_4}}{x}^2$	$\left[-{\displaystyle \frac{3C}{16p_4}}{z}^{-1}\right]$	$+{\displaystyle \frac{C{s}_1}{p_4}}{\displaystyle \frac{1}{z^3}}$	$+{\displaystyle \frac{C{s}_2}{p_4}}{\displaystyle \frac{1}{z^2}}$	$+{\displaystyle \frac{C{s}_3}{p_4}}{\displaystyle \frac{1}{z}}$	$+{\displaystyle \frac{C{s}_4}{p_4}}$	$+{\displaystyle \frac{C{s}_5}{p_4}}z$	${\displaystyle \frac{C}{p_4}}\left(s_4-{\displaystyle \frac{\delta ϵ}{2}}+\alpha \right)$
$p_5{z}^4$	${\displaystyle \frac{C^{1/2}}{p_5^{1/2}}}x$	$\left[0\right]$	$+{\displaystyle \frac{C{s}_1}{p_5}}{\displaystyle \frac{1}{z^4}}$	$+{\displaystyle \frac{C{s}_2}{p_5}}{\displaystyle \frac{1}{z^3}}$	$+{\displaystyle \frac{C{s}_3}{p_5}}{\displaystyle \frac{1}{z^2}}$	$+{\displaystyle \frac{C{s}_4}{p_5}}{\displaystyle \frac{1}{z}}$	$+{\displaystyle \frac{C{s}_5}{p_5}}$	${\displaystyle \frac{C}{p_5}}\left(s_5-{\displaystyle \frac{{ϵ}^2}{4}}\right)$

and

Table 5. The specific $\varphi \left(z\right)$ function defined in Equation (36) for the triconfluent Heun equation, the $z\left(x\right)$ functions calculated from them using Equation (7), the resulting potentials (37), and energy eigenvalues E. The coupling coefficients $s_i$ and E can be extracted from Equation (38) for each choice of $\varphi \left(z\right)$: $s_1={\displaystyle \frac{{ϵ}^2}{4}}$, $s_2={\displaystyle \frac{\delta ϵ}{2}}$, $s_3={\displaystyle \frac{{\delta }^2}{4}}+{\displaystyle \frac{\gamma ϵ}{2}}$, $s_4=ϵ+{\displaystyle \frac{\gamma \delta }{2}}-\alpha$, $s_5=q+{\displaystyle \frac{\delta }{2}}-{\displaystyle \frac{{\delta }^2}{4}}$. For the sake of simplicity, the integration constant in Equation (8) was set to $x_0=0$.

$\mathit{\varphi }\left(\mathit{z}\right)$	$\mathit{z}\left(\mathit{x}\right)$	$\left[-\frac{1}{2}\{\mathit{z},\mathit{x}\}\right]$	$+\frac{\mathit{C}{\mathit{s}}_1{\mathit{z}}^4}{\mathit{\varphi }\left(\mathit{z}\right)}$	$+\frac{\mathit{C}{\mathit{s}}_2{\mathit{z}}^3}{\mathit{\varphi }\left(\mathit{z}\right)}$	$+\frac{\mathit{C}{\mathit{s}}_3{\mathit{z}}^2}{\mathit{\varphi }\left(\mathit{z}\right)}$	$+\frac{\mathit{C}{\mathit{s}}_4\mathit{z}}{\mathit{\varphi }\left(\mathit{z}\right)}$	$+\frac{\mathit{C}{\mathit{s}}_5}{\mathit{\varphi }\left(\mathit{z}\right)}$	E
$p_5$	${\displaystyle \frac{C^{1/2}x}{p_5^{1/2}}}$	$\left[0\right]$	$+{\displaystyle \frac{C{s}_1}{p_5}}{z}^4$	+${\displaystyle \frac{C{s}_2}{p_5}}{z}^3$	$+{{\displaystyle \frac{C{s}_3{p}_5}{z}}}^2$	$+{\displaystyle \frac{C{s}_4}{p_5}}z$	$+{\displaystyle \frac{C{s}_5}{p_5}}$	${\displaystyle \frac{C}{p_5}}\left(s_5-q-{\displaystyle \frac{\delta }{2}}+{\displaystyle \frac{{\delta }^2}{4}}\right)$
$p_{4}z$	${\left({\displaystyle \frac{3C^{1/2}}{2p_4^{1/2}}}x\right)}^{2/3}$	$\left[{\displaystyle \frac{-5C}{16p_4}}{\displaystyle \frac{1}{z^3}}\right]$	$+{\displaystyle \frac{C{s}_1}{p_4}}{z}^3$	$+{\displaystyle \frac{C{s}_2}{p_4}}{z}^2$	$+{\displaystyle \frac{C{s}_3}{p_4}}z$	$+{\displaystyle \frac{C{s}_4}{p_4}}$	$+{\displaystyle \frac{C{s}_5}{p_4}}{\displaystyle \frac{1}{z}}$	${\displaystyle \frac{C}{p_4}}\left(s_4-ϵ-{\displaystyle \frac{\gamma \delta }{2}}+\alpha \right)$
$p_3{z}^2$	${\left({\displaystyle \frac{2C^{1/2}}{p_3^{1/2}}}x\right)}^{1/2}$	$\left[-{\displaystyle \frac{3C}{4p_3}}{\displaystyle \frac{1}{z^4}}\right]$	$+{\displaystyle \frac{C{s}_1}{p_3}}{z}^2$	$+{\displaystyle \frac{C{s}_2}{p_3}}z$	$+{\displaystyle \frac{C{s}_3}{p_3}}$	$+{\displaystyle \frac{C{s}_4}{p_3}}{\displaystyle \frac{1}{z}}$	$+{\displaystyle \frac{C{s}_5}{p_3}}{\displaystyle \frac{1}{z^2}}$	${\displaystyle \frac{C}{p_3}}\left(s_3-{\displaystyle \frac{{\delta }^2}{4}}-{\displaystyle \frac{\gamma ϵ}{2}}\right)$
$p_2{z}^3$	${\left({\displaystyle \frac{5C^{1/2}}{2p_2^{1/2}}}x\right)}^{2/5}$	$\left[-{\displaystyle \frac{21C}{16p_2}}{\displaystyle \frac{1}{z^5}}\right]$	$+{\displaystyle \frac{C{s}_1}{p_2}}z$	$+{\displaystyle \frac{C{s}_2}{p_2}}$	$+{\displaystyle \frac{C{s}_3}{p_2}}{\displaystyle \frac{1}{z}}$	$+{\displaystyle \frac{C{s}_4}{p_2}}{\displaystyle \frac{1}{z^2}}$	$+{\displaystyle \frac{C{s}_5}{p_2}}{\displaystyle \frac{1}{z^3}}$	${\displaystyle \frac{C}{p_2}}\left(s_2-{\displaystyle \frac{\delta ϵ}{2}}\right)$
$p_1{z}^4$	${\left({\displaystyle \frac{3C^{1/2}}{p_2^{1/2}}}x\right)}^{1/3}$	$\left[-{\displaystyle \frac{2C}{p_1}}{\displaystyle \frac{1}{z^6}}\right]$	$+{\displaystyle \frac{C{s}_1}{p_1}}$	$+{\displaystyle \frac{C{s}_2}{p_1}}{\displaystyle \frac{1}{z}}$	$+{\displaystyle \frac{C{s}_3}{p_1}}{\displaystyle \frac{1}{z^2}}$	$+{\displaystyle \frac{C{s}_4}{p_1}}{\displaystyle \frac{1}{z^3}}$	$+{\displaystyle \frac{C{s}_5}{p_1}}{\displaystyle \frac{1}{z^4}}$	${\displaystyle \frac{C}{p_1}}\left(s_1-{\displaystyle \frac{{ϵ}^2}{4}}\right)$

display the individual potentials obtained from the confluent hypergeometric, biconfluent Heun, doubly confluent Heun, and triconfluent Heun equations by chosing the specific single-term $\mathsf{\Phi }\left(z\right)$ function. Each of these potentials are characterized by a number of independent potential terms and a number of potential parameters that originate from the actual differential equation and the variable transformation $z\left(x\right)$. The latter parameters are C and $x_0$ from Equation (8), C being a uniform energy scale, while $x_0$ is a coordinate shift, which is usually an unimportant parameter and has no effect on the energy spectrum. The $\mathsf{\Phi }\left(z\right)$ function also depends on the $p_i$ parameters, as in Equation (13); however, now only a single term is considered in $\mathsf{\Phi }\left(z\right)$, so the related parameter can be absorbed into C as $C/p_i$, or can be chosen as $p_i=1$.

3.2.1. The Natanzon Confluent Potential Class

The potentials generated from the confluent hypergeometric equation are presented in Table 2. This equation depends on two parameters, a and c, so together with C and $x_0$, there are altogether four parameters. One of these, a, however, turns out to be reduced to negative integer number $a=-N$ in the practical applications, as will be discussed later. As it has been pointed out previously, the Schwartzian derivative produces a single potential term that coincides with one of the three independent potential terms. Furthermore, one of the three terms always reduces to a constant by construction, so these potentials are composed of two independently tunable potential terms. Note that the corresponding $s_i$ parameter appears in an additive energy constant both in the potential, and in the energy, so it is physically irrelevant.

The confluent hypergeometric function can be written as an infinite power expansion around $z=0$, the expansion coefficents of which are known in closed form [9]. The power series can be terminated by setting $a=-N$: in this case, the confluent hypergeometric function ${}_{1}F_1(a;c;z)$ is reduced to the generalized Laguerre polynomial $L_N^{(c-1)}\left(z\right)$ [9], apart from a constant factor. The three equations in Equation (19) lead to three well-known exactly solvable potentials when only one of the $p_i$ parameters is chosen to be non-zero.

From the substitution of $p_1=1$, $p_2=p_3=0$, $C=2\omega$, and $c=l+3/2$ into Equation (19), the actual values of $s_2$ and $s_3$ follow, together with the energy formula

$$E=\omega \left(N+l+{\displaystyle \frac{3}{2}}\right).$$

(46)

Then the second line of Table 2 recovers the radial harmonic oscillator in three dimensions [7]

$$V\left(x\right)={\displaystyle \frac{1}{4}}{\omega }^2{x}^2+{\displaystyle \frac{l(l+1)}{x^2}}.$$

(47)

Note that the $s_1$ parameter, which contributes to both the potential and the energy eigenvalues by the same constant, was chosen as $s_1=0$.

The $p_2=1$, $p_1=p_3=0$, $C={\mathrm{e}}^2/{(N+l+1)}^2$, $c=2l+2$, and $s_2=0$ choice for Equation (19), appearing in the third line of Table 2, recovers the radial Coulomb potential in three dimensions [7]

$$V\left(x\right)=-{\displaystyle \frac{{\mathrm{e}}^2}{x}}+{\displaystyle \frac{l(l+1)}{x^2}}$$

(48)

and the corresponding energy eigenvalues

$$E=-{\displaystyle \frac{{\mathrm{e}}^2}{4{(N+l+1)}^2}}.$$

(49)

In a similar fashion, taking $p_3=1$, $p_1=p_2=0$, $c=2s-2N$, and $s_3=-1/4$ recovers the Morse potential

$$V\left(x\right)=-Csexp(-\alpha x)+{\displaystyle \frac{C}{4}}$$

(50)

and its energy eigenvalues

$$E=-C{\left(s-{\displaystyle \frac{1}{2}}-N\right)}^2$$

(51)

provided that the $\alpha =-C^{1/2}$ choice is made.

Note that in the last two cases, some model parameters (c or C) depend on N, but the potential itself is independent of it. This reparametrization is avoided in the case of the harmonic oscillator, as N appears there only in the energy eigenvalues (46). The structure of the system of equations (19) offers a simple explanation for this difference. In particular, there is only one equation that contains N there, i.e., the first one when the $a=-N$ substitution is made. Keeping only the $p_1$ parameter appearing in this equation as non-zero implies that E is directly related to N in this case, while the $p_2=p_3=0$ choice eliminates E from the remaining two equations, which then set the $s_2$ and $s_3$ coupling coefficients in terms of the model parameters. This implies that one of the $p_i$ choices has special importance from the point of view of constructing exactly solvable potentials.

A further level of complexity is considering potentials generated by taking more than one $p_i$ parameter with a non-zero value. In this case, the energy eigenvalue E appears in several equations, so expressing it in terms of N and the parameters (c and C) can be more complicated, especially when the significant $s_i$, coupling coefficients are required to be independent of N. A further complication is that $\varphi \left(z\right)$, and thus, $\mathsf{\Phi }\left(z\right)$ in Equation (40) contains several terms, so the integral in Equation (8) may be more complicated to compute, and the $x\left(z\right)$ function may not be invertible to $z\left(x\right)$. Furthermore, the more complicated $z\left(x\right)$ function results in extra potential terms originating from the Schwartzian derivative, as can be seen from Equation (10).

Despite these complications, exactly solvable potentials of this kind exist. The one discussed in Ref. [25] is obtained from the choice $p_1\ne 0$, $p_2\ne 0$, and $p_3=0$, so it contains both the radial harmonic oscillator and the Coulomb potential as special cases. In particular, the actual form of Equation (7) is

$${\left({\displaystyle \frac{\mathrm{d}z}{\mathrm{d}x}}\right)}^2\sim C{\displaystyle \frac{z^2}{z^2+\theta z}}={\displaystyle \frac{C}{\theta }}{\displaystyle \frac{z^2}{z^2/\theta +z}}.$$

(52)

The limit $\theta \to 0$ corresponds to $\varphi \left(z\right)=z^2$, i.e., $p_2=1$ and the radial Coulomb potential, while $\theta \to \infty$ combined with the condition $C/\theta \equiv \tilde{C}=const.$ leads to $\varphi \left(z\right)=z$, i.e., $p_1=1$ and the radial harmonic oscillator. The potential is Coulomb-like asymptotically and approximates the harmonic oscillator near the origin. Its energy spectrum also shows Coulomb-like and harmonic oscillator-like features for large and small values of N, respectively. The $z\left(x\right)$ function is known only implicitly from $x\left(z\right)$, so this is an example for an implicit potential.

3.2.2. Potentials from the Biconfluent Heun Equation

Table 3 contains the potentials obtained from the biconfluent Heun equation. There are five parameters appearing in this equation, which determine the coupling coefficients of the five potential terms. Again, the Schwartzian derivative produces a single potential term that coincides with one of the main potential terms. Furthermore, here again, one of the five main potential terms becomes a constant, which manifests itself as an unimportant energy shift, leaving four significant potential terms. There are five different potentials in Table 3, each corresponding to a different choice of $\mathsf{\Phi }\left(z\right)$, and thus $\varphi \left(z\right)$. These potentials have been identified in Ref. [13] and have also been studied in Ref. [10]. These works, however, focused only on the general structure of the potentials and did not connect the coupling coefficients of the potential terms with the parameters of the biconfluent Heun equation. They did not discuss the solutions either. The latter issue has been considered in Ref. [26], where the polynomial solutions of the Heun-type equations have been analyzed, along with the constraints this implies on the parameters. Here, we combine these results with those concerning the potentials.

The general solutions $Hb(\alpha ,\gamma ,\delta ,ϵ,q;z)$ of (21) can be expanded into an infinite power series around the singular point $z=0$. The expansion coefficients can be computed from a three-term recursion relation [12]. Applying the notation used here, the recursion relation is [26]

$$(k+1)(\gamma +k)C_{k+1}=-(\delta k-q)C_k-[\alpha +ϵ(k-1)]C_{k-1}.$$

(53)

Assuming that $\gamma \ne -k$, the power series can be terminated, reducing it to a polynomial of order N. For this, two conditions have to be satisfied (after the substitution $k=N+1$):

$$\alpha =-Nϵ$$

(54)

and

$$C_{N+1}^{\left(N\right)}=0.$$

(55)

These conditions guarantee that the expansion coefficients $C_k^{\left(N\right)}$ vanish for $k>N$. Equation (55) is a closed expression containing the parameters appearing in Equation (21), which can be computed using the recursion relation. In particular, it is an $(N+1)$’th order algebraic equation of q, which has to be solved in order to generate the expansion coefficients. This can be done analytically up to $N=2$. In summary, the conditions (54) and (55) reduce the independent parameters from Equation (21) to three, plus a non-negative integer N. These developments imply that the equations containing $\alpha$ and q in Equation (25), i.e., the second and third ones, play a special role when exactly solvable potentials are constructed from the biconfluent Heun equation.

Before discussing the five potentials displayed in Table 3, it is worthwhile to identify their domain of definition. For this, their behavior for $z\to \pm \infty$ and for $z\to 0$ has to be analyzed, combining the potential terms with the given $z\left(x\right)$ function. In four cases (i.e., with the exception of $p_1\ne 0$), $z\left(x\right)$ is the power of x with a positive integer or a positive rational exponent, while the potential terms contain positive or negative integer powers of $z\left(x\right)$. Whenever there are terms with a negative power of z, a singularity occurs at $x=0$, so the potential is naturally defined on the positive x semi-axis, i.e., $x\in [0,\infty )$. This holds for the last four potentials in Table 3. However, for the second one with $p_2\ne 0$, the only singular potential term is the centrifugal one with $z^{-1}$. Canceling this with an appropriate choice of $s_1$, i.e., $\gamma$, this potential can be extended to $x\in (-\infty ,\infty )$, where it will be symmetric under reflection. The potential in the first line of Table 3 (with $p_1\ne 0$) is naturally defined on the whole x axis, because all the potential terms are exponentials.

It is also possible to estimate the general structure of the five potentials without referring to their concrete mathematical form. Each of them consists of four terms (powers of x or exponentials) that are monotonous functions of x. This means that the signs of their coupling coefficients determine the number and nature of their extremal points. If all four coupling coefficients have the same sign, then $V\left(x\right)$ itself will also be a monotonous function of x, so one expects only an absolute minimum. When the signs are mixed, one or two minima can be formed. The asymptotic behavior of the potential for $x\to \infty$ is determined by the highest positive power of z. For the first four potentials, this term is the one proportional to $s_5={ϵ}^2/4$, which is positive for real values of $ϵ$. This means that these potentials are confined asymptotically. (We assume that $C/p_i>0$ holds). The fifth potential contains terms with negative powers of x, so it vanishes asymptotically. Furthermore, as it can be seen from the exponential term of Equation (26), the solutions of all five potentials vanish asymptotically if $ϵ<0$ holds.

Taking $p_2=1$ and $C=1$ leads to $z\left(x\right)=x^2/4$, and combining this with the parameter set $\gamma =2s$, $\delta =-4b$, $ϵ=-16a$, and $s_2=0$ results in the sextic oscillator potential

$$V\left(x\right)={\displaystyle \frac{(2s-1/2)(2s-3/2)}{x^2}}+\left[b^2-4a(s+N+{\displaystyle \frac{1}{2}})\right]x^2+2ab{x}^4+a^2{x}^6.$$

(56)

The energy eigenvalues are

$$E=4bs-q,$$

(57)

while the corresponding eigenfunctions are

$$\psi \left(r\right)\sim x^{2s-1/2}exp\left(-{\displaystyle \frac{a{x}^4}{4}}-{\displaystyle \frac{b{x}^2}{2}}\right)P_N(x^2/4).$$

(58)

Here, $P_N\left(z\left(x\right)\right)$ is the polynomial solution of the differential equation (21) discussed before. Note that the coupling coefficient of the quadratic term in (56) depends on N, so the same potential corresponds to several combinations of $s+N$. The energy eigenvalues contain q, the allowed values of which are determined from an $(N+1)$’th order algebraic equation that follows from the requirement that an $(N+1)\times (N+1)$ determinant vanishes. The resulting $q_i$ values set the energy eigenvalues as in Equation (57). These have been calculated exactly up to $N=2$ in Ref. [27], where the sextic oscillator has been applied in a model describing collective nuclear excitations. For higher values of N, the roots of the algebraic equation have to be determined numerically. In the application in Ref. [27], the minima of the potential played a central role, and it was shown that the $(a,b)$ phase space contains three well-defined domains where the potential has one minimum at $x=0$, one minimum at $x>0$, or it has both.

As we have discussed previously, this potential is naturally defined on the positive semi-axis $x\in [0,\infty )$, but it can also be extended to the whole x axis too. For this, the centrifugal term in Equation (56) has to be canceled by the choices $s=1/4$ or $s=3/4$, i.e., $\gamma =1/2$ or $\gamma =3/2$. These two choices supply the even and odd solutions in the bound-state eigenfunctions (58) in one spatial dimension.

The sextic oscillator is known to be the archetype of quasi-exactly solvable (QES) potentials [28,29]. The equivalence of the QES treatment and that based on the transformation of the biconfluent Heun equation has been pointed out recently [30]. In that work, a more general treatment of the sextic oscillator was also presented, in which the solutions were expanded in terms of Hermite functions. The QES result is obtained when the Hermite functions reduce to Hermite polynomials. There have been further methods discussing the sextic oscillator using the Hill determinant method and continued fractions [31], as well as the Nikiforov–Uvarov method [32].

The $p_3=1$, $C=1$ choice gives the $z\left(x\right)=x$ function and the potential appearing in the third line of Table 3. This potential corresponds to the QES potential appearing in Equation (1.3.63) of Ref. [29]

$$\begin{array}{cc}V\left(x\right)=& a^2{x}^2+2abx-{\displaystyle \frac{A+b(d+2l-2c-1)}{x}}\hfill \\ & +\left[\left(l+{\displaystyle \frac{d-3}{2}}\right)\left(l+{\displaystyle \frac{d-1}{2}}\right)-c(d+2l-c-2)\right]{\displaystyle \frac{1}{x^2}}+b^2\hfill \end{array}$$

(59)

with the following choice of the parameters: $\gamma =d+2l-2c-1$, $\delta =2b$, $ϵ=2a$, $q=-A$, and $s_3=b^2$. This notation slightly differs from the original one in order to avoid confusion, and (59) also contains the centrifugal term (not included in the version in Ref. [29]), corresponding to radial potentials in d dimensions. The energy eigenvalues

$$E=a(2N+d+2l-2c)+b^2$$

(60)

presented in Ref. [29] are also recovered. Here again, the q values have to be determined as the roots of an $(N+1)$’th order algebraic equation. Since q appears in the coupling parameter of a potential term in this case, Equation (59) accounts for a series of potentials that differ in the coupling coefficient of the Coulombic ($x^{-1}$) term, while the energy eigenvalues (60) apply to each of them. Note that N only appears in the energy formula.

This potential has been discussed in terms of the Hill determinant method, for example [33]. It is also known as the Cornell potential [34], describing quark systems in a non-relativistic setting, although in that application, numerical and perturbative solutions were employed rather than exact or quasi-exact ones.

Setting $p_1=1$ and $d=-C^{1/2}$ gives the $z\left(x\right)=exp(-dx)$ function appearing in the first line of Table 3. A special case of this example is another QES potential presented in Equation (1.3.29) of Ref. [29]

$$V\left(x\right)=b^2-c(2b+d)exp(-dx)+[c^2-2a(b+dN+d)]exp(-2dx)+2acexp(-3dx)+a^{2}exp(-4dx)$$

(61)

provided that the parameters are chosen as $\gamma =2b/d+1$, $\delta =-2c/d$, $ϵ=-2a/d$, and $s_1=b^2/d^2-1/4$. This parameter set leads to $q=0$ and $E=0$.

Note that the three potentials discussed up to this point contain the three well-known potentials discussed in Section 3.1.1 in the limit that reduces the biconfluent Heun equation to the confluent hypergeometric differential equation.

The remaining two potentials in the last two lines of Table 3 contain terms with fractional powers of x. Taking $p_4=1$ and $C=1$, one obtains $z\left(x\right)={(3x/2)}^{2/3}$. The resulting potential contains, as a special case, the potential appearing in Equation (22) of Ref. [35]

$$V\left(x\right)={\displaystyle \frac{7}{36}}{\displaystyle \frac{1}{x^2}}-{\displaystyle \frac{2}{9}}{b}_1{x}^{-4/3}+{\displaystyle \frac{1}{9}}{b}_1^2{x}^{-2/3}+{\displaystyle \frac{1}{9}}{a}_1^2{x}^{2/3}$$

(62)

with the energy formula

$$E={\displaystyle \frac{4}{9}}\left(a_0-{\displaystyle \frac{1}{2}}{a}_1{b}_1\right).$$

(63)

This potential follows from the parameter set $\gamma =3$, $\delta ={(2/3)}^{2/3}(b_1-2a_0/a_1)$, $ϵ={(2/3)}^{4/3}{a}_1$, $q={(2/3)}^{2/3}(3a_0/a_1-2b_1)$, and $s_4=0$.

For $p_5=1$ and $C=1$, i.e., with $z\left(x\right)={\left(2x\right)}^{1/2}$, the potential in the last line of Table 3 contains, as a special case, the one in Equation (21) of Ref. [35]

$$V\left(x\right)=-{\displaystyle \frac{1}{4}}\left(a_0-{\displaystyle \frac{1}{2}}{a}_1{b}_1\right)x^{-1/2}+{\displaystyle \frac{3}{16}}{b}_1^2{x}^{-1}-{\displaystyle \frac{1}{8}}{b}_1{x}^{-3/2}$$

(64)

with the energy formula

$$E=-{\displaystyle \frac{1}{16}}{a}_1^2.$$

(65)

This potential is recovered using the parameter set $\gamma =3$, $\delta =2^{-1/2}(b_1-2a_0/a_1)$, $ϵ=a_1/2$, $q=2^{-1/2}(3a_0/a_1-2b_1)$, and $s_5=0$.

These two potentials have been derived from a differential equation closely related to the biconfluent Heun equation [35], although it contains an extra term proportional to z in addition to $\alpha -q/z$ in the non-derivative part of Equation (21).

We are unaware of any studies analyzing potentials and their solutions derived from the biconfluent Heun equation using multi-term $\mathsf{\Phi }\left(z\right)$ functions in Equation (23). These problems are certainly more involved technically, but their complex structure also allows various simplifications that may lead to exact solutions, even if for a limited number of states.

3.2.3. Potentials from the Doubly Confluent Heun Equation

The potentials obtained from the doubly confluent Heun equation are displayed in Table 4. Similarly to the biconfluent Heun equation, this equation also depends on five parameters, which define the coupling coefficients of the five potential terms. The term originating from the Schwartzian derivative matches again with one of the main potential terms, of which one is, again, a constant. There is, however, an important difference with respect to the biconfluent Heun equation: only three potentials are independent, because those originating from $\varphi \left(z\right)=p_1$ and $\varphi \left(z\right)=p_5{z}^4$ lead to the same (reparametrized) potential terms, and the same relation holds between the potentials derived from $\varphi \left(z\right)=p_{2}z$ and $\varphi \left(z\right)=p_4{z}^3$. This finding has been pointed out in Ref. [10] without any explanation. It was also not discussed in the earlier publication [13] identifying three general potentials derived from the doubly confluent Heun equation. This result can be interpreted in terms of the invariances of the doubly confluent Heun equation [12]. In particular, it turns out that if $Hd(\alpha ,\gamma ,\delta ,ϵ,q;z)$ is its solution in the present notation, then so is $z^{k}exp(\gamma /z-ϵz)Hd(t\gamma (2-k-\delta ),tϵ,2-2k-\delta ,t\gamma ,q-k(k-1-\delta );t/z)$, where $t=\pm 1$. This invariance property seems to be unique among the four equations discussed here.

The power series expansion of the solutions around $z=0$, again, leads to a three-term recursion relation:

$$(k+1)\gamma C_{k+1}=-[k(k-1)+\delta k-q]C_k-[\alpha +ϵ(k-1)]C_{k-1}.$$

(66)

Similarly to the case of the biconfluent Heun equation, the termination at $k=N$ leads to two conditions:

$$\begin{array}{c}\alpha =-Nϵ,\hfill \\ C_{N+1}^{\left(N\right)}=0.\hfill \end{array}$$

(67)

These equations guarantee that $C_k^{\left(N\right)}=0$ for $k>N$, so the $Hd$ solutions can be written into finite polynomial form. $C_{N+1}^{\left(N\right)}$ can be computed from Equation (66), and it is again an $(N+1)$’th order algebraic expression of q, the roots of which determine the allowed values of q. The equations containing $\alpha$ and q in Equation (31), i.e., the third and the fourth ones, play a special role.

Before discussing the potentials appearing in Table 4, it is worthwhile to analyze their domain of definition. The third potential (with $p_3\ne 0$) is composed of exponential functions, so it is naturally defined on the whole x axis. The remaining four potentials (which correspond to two pairs of potentials with reparametrized coupling coefficients) all contain several potential terms that are singular at $z=0$ (i.e., at $x=0$). It is therefore natural to define them on the positive semi-axis, $x\in [0,\infty )$. Similarly to the potentials generated from the biconfluent Heun equation, the number and nature of the extrema of the potentials discussed here depend on the sign of the coupling coefficients. The ones belonging to the most singular terms are positive (as long as $C/p_i>0$), due to $s_1={\gamma }^2$ and $s_5={ϵ}^2/4$.

Consider first the case $p_3=1$ with $C=d^2$, such that $d=C^{1/2}$, leading to $z\left(x\right)=exp\left(dx\right)$. The potential constructed from this input coincides with the one appearing in Equation (1.3.19) in Ref. [29]

$$V\left(x\right)=a^{2}exp(-2dx)+a[2b-d(2N+1)]exp(-dx)-c(2b+d)exp\left(dx\right)+c^{2}exp\left(2dx\right)$$

(68)

if the parameters are chosen as $\gamma =2a/d$, $\delta =2b/d+1-2N$, $ϵ=-2c/d$, and $s_3=(b^2-2ac)/d^2$. The energy eigenvalues are

$$E=-q{d}^2-{(b-\alpha N)}^2+b^2,$$

(69)

where the possible q values are determined as the roots of the algebraic equation originating from Equation (67).

As it has been mentioned before, the potentials generated from the $p_1=1$ and $p_5=1$ choices contain the same types of terms and differ only in their parametization. It turns out that they coincide with the QES potential displayed in Equation (1.3.72) of Ref. [29], to which we added the centrifugal term corresponding to a radial potential in d dimensions

$$\begin{array}{cc}V\left(x\right)=& {\displaystyle \frac{b^2}{x^4}}+{\displaystyle \frac{b(d+2l+c-3)}{x^3}}-{\displaystyle \frac{a(2N+d+2l-2c-1)}{x}}\hfill \\ & +\left[\left(l+{\displaystyle \frac{d-3}{2}}\right)\left(l+{\displaystyle \frac{d-1}{2}}\right)-c(d+2l-c-2)-2ab-\lambda \right]{\displaystyle \frac{1}{x^2}}.\hfill \end{array}$$

(70)

This potential has repulsive $x^{-4}$ type singularity at the origin and vanishes asymptotically. Depending on the sign of the remaining three coupling coefficients, it can be 0, 1, or 2 minima. The energy expression is

$$E=-a^2$$

(71)

Here, $\lambda$ is obtained from an $(N+1)$’th order algebraic equation. This problem is recovered from the last line of Table 4, with the parameter set $p_5=1$, $C=1$, i.e., with $z\left(x\right)=x$ and $\gamma =2b$, $\delta =d+2l-2c-1$, $ϵ=-2a$, $q=-\lambda$, and $s_5=0$. The alternative form of the same potential follows from the first line of Table 4, with $p_1=1$, $C=1$, i.e., $z\left(x\right)=-1/x$, and the alternative parameter set mentioned before.

This potentials have also been discussed in Ref. [36], where the lowest two bound-state solutions were derived, without reference to recursion relations and special functions in general.

The same duality is present for the potentials originating from $p_2=1$ and $p_4=1$. We present here the form derived from the latter choice with $C=1$, leading to $z\left(x\right)=x^2/4$. The potential is

$$V\left(x\right)={\displaystyle \frac{{ϵ}^2}{4}}{x}^2+\left(4q-2\delta +{\delta }^2+2\gamma ϵ-{\displaystyle \frac{3}{4}}\right){\displaystyle \frac{1}{x^2}}+8\gamma (\delta -2){\displaystyle \frac{1}{x^4}}+16{\gamma }^2{\displaystyle \frac{1}{x^6}},$$

(72)

while the energy expression is

$$E=-ϵ\left({\displaystyle \frac{\delta }{2}}+N\right)$$

(73)

Here, we set the additive constant to $s_4=0$. This potential exhibits singularity at the origin, but due to the positive coupling coefficient of the leading $x^{-6}$ term, it is repulsive there. This is valid also for the alternative parametrization of the same potential, stemming from the $p_2=1$, $C=1$ choice, corresponding to the inverse $z\left(x\right)=4/x^2$ function. This potential tends to plus infinity at both boundaries, so it necessarily has a minimum at a finite x value. It may have another minimum too, if the coupling coefficients of the $x^{-2}$ and the $x^{-4}$ terms have opposite signs.

This potential was discussed in Ref. [37] using the Hill determinant method. A main difference compared to the present approach is that the solutions were written in terms of a Laurent series. Polynomial solutions have also been considered, e.g., in Ref. [38], where the ground and first excited states were determined.

The three potentials displayed in the last three rows of Table 4 reduce to the three well-known potentials discussed in Section 3.1.1 for the parameter reduction that takes the doubly confluent Heun equation to the confluent hypergeometric differential equation.

Similarly to the biconfluent case, potentials generated from the doubly confluent Heun equation using multi-term $\mathsf{\Phi }\left(z\right)$ functions in Equation (29) are not known in the literature.

3.2.4. Potentials from the Triconfluent Heun Equation

The five potentials obtained from the triconfluent Heun equation using five distinct choices of $\varphi \left(z\right)=p_{5-k}{z}^k$, $k=0$, 1, … 5 are displayed in Table 5. Again, there are five parameters and five significant potential terms, one of which turns out to be a constant, similarly to the cases of the biconfluent and doubly confluent equations. However, now the contribution of the Schwartzian derivative term is a potential term distinct from the five significant potential terms. In fact, according to Equation (45), and remembering that in this case, $\mu =k-t=k$, it vanishes (for $k=0$) or corresponds to a weakly attractive inverse square interaction (for the remaining four cases). This also means that these potentials are defined on the positive semi-axis, $x\in [0,\infty )$, with the exception of the first potential ($k=0$), which can also be extended to the full x axis.

The systematic structure of potential terms in Table 5 allows the estimation of the possible potential shapes. All the potential terms are powers of z, and thus of x. $C/p_i>0$ is necessary to generate real potential terms, so $z\left(x\right)$ is a real function in each case, and it grows monotonously with x. The potential terms are thus also monotonous, and they grow or decrease with x, depending on the sign of the $s_i$ coupling coefficients. The powers of x range from $-2$ (the Schwartzian derivative term) except for the first potential, $k=0$, where this term is missing, to 4. The powers of x are all positive integers for the first potential and are all negative for the last one (with $k=4$). It is also seen that the leading term, i.e., the one with the highest power of x, is always the one with the coupling coefficient $C{ϵ}^2/\left(4p_i\right)$, which is positive. This means that the first four are confining potentials. The sign of the coupling coefficient of the terms with lowest power, i.e., $x^{-2}$, is negative (when it exists, i.e., for the last four potentals), so the number of extrema of the potentials is determined by the sign of the $s_i$ coupling coefficients of the terms with intermediate powers. If all are positive, then there can be no local minimum. One negative $s_i$ introduces a local minimum. Two local minima can also occur for a well-chosen combination of positive and negative $s_i$ coupling coefficients.

Similar arguments prove that the solutions (39) can be normalizable only when the leading term in the exponent has a negative sign. Generally, this means $ϵ<0$.

A further feature different from the other examples emerges when one expands the solutions into a power series. It turns out that the recursion relation determining the expansion coefficients contains four terms:

$$(k+1)(k+2)C_{k+2}=-(k-1)\gamma C_{k-1}-(\delta k-q)C_k-[\alpha +ϵ(k-1)]C_{k-1}.$$

(74)

The termination of the power series is thus more complicated and requires the fulfillment of more conditions than in the case of the biconfluent and the doubly confluent Heun equations.

Using the simplified notation $C=1$, $p_i=1$, the five potentials are

$$\begin{array}{ccc}\hfill V\left(x\right)& =& {\displaystyle \frac{{ϵ}^2}{4}}{x}^4+{\displaystyle \frac{\delta ϵ}{2}}{x}^3+\left({\displaystyle \frac{{\delta }^2}{4}}+{\displaystyle \frac{\gamma ϵ}{2}}\right)x^2+\left(ϵ+{\displaystyle \frac{\gamma \delta }{2}}-\alpha \right)x\hfill \end{array}$$

(75)

$$\begin{array}{cc}V\left(x\right)=& {\displaystyle \frac{9{ϵ}^2}{16}}{x}^2+{\displaystyle \frac{\delta ϵ}{2}}{\left({\displaystyle \frac{3}{2}}\right)}^{4/3}{x}^{4/3}+\left({\displaystyle \frac{{\delta }^2}{4}}+{\displaystyle \frac{\gamma ϵ}{2}}\right){\left({\displaystyle \frac{3}{2}}\right)}^{2/3}{x}^{2/3}\hfill \\ & +\left(q+{\displaystyle \frac{\delta }{2}}-{\displaystyle \frac{{\delta }^2}{4}}\right){\left({\displaystyle \frac{2}{3}}\right)}^{2/3}{x}^{-2/3}-{\displaystyle \frac{5}{36}}{\displaystyle \frac{1}{x^2}}\hfill \end{array}$$

(76)

$$\begin{array}{ccc}\hfill V\left(x\right)& =& {\displaystyle \frac{{ϵ}^2}{2}}x+{\displaystyle \frac{\delta ϵ}{2}}{2}^{1/2}{x}^{1/2}+\left(ϵ+{\displaystyle \frac{\gamma \delta }{2}}-\alpha \right)2^{-1/2}{x}^{-1/2}+\left(q+{\displaystyle \frac{\delta }{2}}-{\displaystyle \frac{{\delta }^2}{4}}\right){\displaystyle \frac{1}{2}}{x}^{-1}-{\displaystyle \frac{3}{16}}{\displaystyle \frac{1}{x^2}}\hfill \end{array}$$

(77)

$$\begin{array}{cc}V\left(x\right)=& {\displaystyle \frac{{ϵ}^2}{4}}{\left({\displaystyle \frac{5}{2}}\right)}^{2/5}{x}^{2/5}+\left({\displaystyle \frac{{\delta }^2}{4}}+{\displaystyle \frac{\gamma ϵ}{2}}\right){\left({\displaystyle \frac{2}{5}}\right)}^{2/5}{x}^{-2/5}+\left(ϵ+{\displaystyle \frac{\gamma \delta }{2}}-\alpha \right){\left({\displaystyle \frac{2}{5}}\right)}^{4/5}{x}^{-4/5}\hfill \\ & +\left(q+{\displaystyle \frac{\delta }{2}}-{\displaystyle \frac{{\delta }^2}{4}}\right){\left({\displaystyle \frac{2}{5}}\right)}^{6/5}{x}^{-6/5}-{\displaystyle \frac{21}{100}}{\displaystyle \frac{1}{x^2}}\hfill \end{array}$$

(78)

$$\begin{array}{cc}V\left(x\right)=& {\displaystyle \frac{\delta ϵ}{2}}{3}^{-1/3}{x}^{-1/3}+\left({\displaystyle \frac{{\delta }^2}{4}}+{\displaystyle \frac{\gamma ϵ}{2}}\right)3^{-2/3}{x}^{-2/3}+\left(ϵ+{\displaystyle \frac{\gamma \delta }{2}}-\alpha \right)3^{-1}{x}^{-1}\hfill \\ & +\left(q+{\displaystyle \frac{\delta }{2}}-{\displaystyle \frac{{\delta }^2}{4}}\right)3^{-4/3}{x}^{-4/3}-{\displaystyle \frac{2}{9}}{\displaystyle \frac{1}{x^2}}\hfill \end{array}$$

(79)

Works in this field employed the three-parameter version of the triconfluent Heun equation mentioned earlier [12], which corresponds to the substitution (34). As a consequence of this, the potentials generated in this way are special subcases of the potentials obtained here. The $\delta =0$ substitution, for example, sets $s_2=0$, so the potential terms displayed in the fifth column of Table 5 are canceled, while the $ϵ=-3$ substitution leads to the fixed value of the coupling coefficient $s_1=9/4$ of the terms in the fourth column.

The study in Ref. [39] discusses the special cases of three potentials. Equation (8) of that work corresponds to potential (76) without the term $x^{4/3}$ and with the fixed coupling coefficient of the $x^2$ term. Equation (10) corresponds to potential (77) without the $x^{1/2}$ term and with the fixed coupling coefficient of the x term. Finally, Equation (12) corresponds to potential (79) without the $x^{-1/3}$ term. (Here, $s_1={ϵ}^2/4=9/4$ appears in the constant term). Only the ground-state solutions are discussed in Ref. [39], so there is no reference to the triconfluent Heun equation and its possible polynomial solutions.

The special case of potential (75) is discussed in Refs. [40,41] in the context of the three-parameter version of the triconfluent Heun equation, which is obtained from Equation (33) after the (34) substitution. This eliminates the cubic term due to $\delta =0$; however, in Ref. [41], it appears because a shifted version of the $z\left(x\right)$ function is used there. In Ref. [40], a further restriction corresponding to $\alpha =-3$ is also applied, canceling the linear term and making the potential symmetric under reflection. The first few solutions are calculated and plotted in both works.

The range of solvable potentials is generalized beyond those presented in Table 4 in Ref. [42], where more general, multi-term $\mathsf{\Phi }\left(z\right)$ functions (36) are employed. This results in more complex $z\left(x\right)$ functions and potentials that include extra terms from the Schwartzian derivative. The bound-state solutions are generated by expanding them in a power series, which leads to a four-term recursion relation similar to Equation (74). The expansion coefficients are extracted from the analysis of an $(N+1)\times (N+1)$ matrix resulting from the termination of the recursion.

4. Summary and Conclusions

We applied a transformation method to the biconfluent, doubly confluent, and triconfluent Heun equations in order to generate exactly solvable potentials from the Schrödinger equation. Our aim was to present a synthesis of a number of results concerning both the mathematical and the physical aspects of this field. The transformation method [5,8] has been applied previously to the hypergeometric and confluent hypergeometric differential equations, leading to the Natanzon and the Natanzon confluent potential classes [6]. Since the biconfluent and doubly confluent Heun equations contain the confluent hypergeometric differential equation as a special limit, the potentials generated by this method are expected to be the generalizations of Natanzon confluent potentials. (The Heun equation, which can be considered the generalization of the hypergeometric differential equation, and the confluent Heun equation that contains both differential equations as a special limit will be discussed elsewhere).

Earlier works in this field dealt with the general structure of potentials derived from these Heun-type equations [10,13,17] and the expansion of their solutions in terms of known functions [10,13], e.g., Hermite functions [16], as well as in terms of polynomials [26]. There are also a number of solvable potentials derived by various methods that are clearly beyond the Natanzon (confluent) class but contain potentials from this class as special cases, so integrating them into a common framework seemed desirable. An example is the sextic oscillator, which has been described in terms of methods ranging from the Hill determinant method and continued fractions [31], to the quasi-exactly solvable (QES) framework [28,29] and the Nikiforov–Uvarov method [32]. The relation of the QES approach and the transformation method applied to the biconfluent Heun equation has been demonstrated recently [30]: the two methods coincide when the Hermite functions applied in the expansion of the solutions reduced to Hermite polynomials, which are special forms of the generalized Laguerre polynomials.

We discussed the general form of the transformation method with special attention to the potential terms originating from the Schwartzian derivative. We showed that the solutions of the Schrödinger equation can be translated into finding solutions to a set of algebraic equations (i.e., Equations (19), (25), (31), and (38)) that connect the parameters of the $z\left(x\right)$ transformation function and those appearing in the actual differential equation with the coupling coefficients of the potential and the energy eigenvalues. The number of equations contained in the set was equal to the number of independently tunable potential terms, including a constant that can be absorbed into the energy eigenvalue. This number was 3 for the reference problem, the confluent hypergeometric differential equation, and 5 for the three Heun-type equations. The Schwartzian derivative generally contributed to the potential with two extra terms that contained only the parameters appearing in the transformation function $z\left(x\right)$.

The complexity of the resulting potential was found to depend on the structure of a function, $\mathsf{\Phi }\left(z\right)$, that defined $z\left(x\right)$ through a first-order differential Equation (7) for $z\left(x\right)$. In the simplest case, when $\mathsf{\Phi }\left(z\right)$ contained only an integer power $z^{\mu }$, the potential terms turned out to contain either integer or fractional powers of z (for $\mu \ne -2$) or exponential functions (for $\mu =-2$). The Schwartzian derivative term turned out to be a constant in the latter case, while in the remaining cases, it resulted in a weakly singular attractive inverse square-like term that could be absorbed into one of the main potential terms in each case, except for the triconfluent Heun equation, when it stood alone as a sixth potential term with a fixed coupling coefficient.

The relatively simple potentials originating from single-term $\mathsf{\Phi }\left(z\right)$ functions identified previously [10,13] were analyzed one by one for the confluent hypergeometric, biconfluent, doubly confluent, and triconfluent Heun equations. The reference problem originating from the confluent hypergeometric differential equation recovered well-known potentials, i.e., the radial harmonic oscillator, the radial Coulomb, and the Morse potentials.

The biconfluent Heun equation led to five potentials, three of which contained the three well-known potentials mentioned above as special cases, while two others contained fractional powers of x. The doubly confluent Heun equation led to five potentials, but only three of them were independent due to a symmetry property of this differential equation. The three independent potentials each contained one of the well-known potentials mentioned above. In contrast to this, the five potentials originating from the triconfluent Heun equations contained typically fractional powers of x, and neither of them could be reduced to the three well-known reference potentials.

Potentials generated by using more general $\mathsf{\Phi }\left(z\right)$ functions containing more than one term are known only within the Natanzon confluent potential class [25] and the potential class derived from the triconfluent Heun equations [42].

In the next step, the solutions of potentials obtained from the single-term $\varphi \left(z\right)$ functions were discussed in detail. For some of the potentials, the solutions were written in terms of known special functions, e.g., Hermite functions. However, inspired by the structure of the bound-state solutions of Natanzon (confluent) potentials, for which the bound-state solutions are written in terms of classical orthogonal polynomials, polynomial solutions were also considered for the three Heun-type equations. First applying the results of Ref. [26], the solutions were written in terms of infinite power series, the expansion coefficients of which are determined from recurrence relations. For the biconfluent and doubly confluent Heun equation, the recurrence relation contains three terms, while the triconfluent Heun equation leads to a four-term recurrence relation. (For the confluent hypergeometric function, the recurrence relation is two-term, so the expansion coefficients can be expressed in terms of simple products, as it is well known for the generalized Laguerre polynomials [9]).

In order to terminate the three-term recurrence relations at N, reducing the power series to a polynomial of order N, two conditions had to be satisfied. The first one linked the $\alpha$ parameter to $ϵ$ as $\alpha =-Nϵ$, while the second one prescribed that the expansion coefficient vanishes: $C_{N+1}^{\left(N\right)}=0$. This expression contains the q parameter in the form of an $(N+1)$’th order algebraic equation, so the allowed values of q are obtained as the roots of this equation. These requirements hold in this form for both the biconfluent and the doubly confluent Heun equation. The termination conditions reduce the independent parameters to three ($\gamma$, $\delta$, $ϵ$), plus a non-negative integer N.

The procedure of expressing the bound-state eigenfunctions and the corresponding energy eigenvalues differ for these potentials from that applied to Natanzon-class potentials, including Natanzon confluent potentials that played the role of the reference problem here. In that case, the polynomial solutions directly supply all the bound-state solutions of a potential. The energy eigenvalues and the coupling coefficients of the potential terms are determined from the set of algebraic equations connecting these quantities with the parameters of the actual differential equation (Equations (19), (25), (31), and (38)). For this, a reparametrization may also be necessary in order to express the coupling coefficients of the potential terms ($s_i$) such that N, the order of the polynomials, appears only in the energy expression. This was the case for the Morse and the radial Coulomb potentials, while the reparametrization was not necessary for the radial harmonic oscillator, because then N appeared strictly in the only equation containing E. The situation is different for the potentials obtained from the Heun-type equations. Now, in addition to the set of five algebraic equations, the conditions that guarantee the termination of the power series also have to be observed. In addition to N, the q parameter now also plays an important role. When q appears in the same equation with E, its allowed values obtained as the zeros of the $(N+1)$’th order algebraic equation directly supply the energy eigenvalues, while when this is not the case, they set one of the coupling coefficients of the potential. Furthermore, N also appears in one of the coupling coefficients, so rather than generating all the solutions of a fixed potential, usually, one obtains solutions with one fixed energy eigenvalue belonging to a set of potentials.

The actual potentials generated from the biconfluent (BHE) and doubly confluent (DHE) Heun equations contained several potentials identified previously as quasi-exactly solvable (QES) potentials [29]. In addition to the sextic oscillator (BHE with $z\left(x\right)=x^2/4$), this was the case with the Cornell potential (BHE with $z\left(x\right)=x$), two exponential potentials (BHE and DHE with $z\left(x\right)=exp(-\alpha x)$ and $z\left(x\right)=exp\left(\alpha x\right)$, respectively) and a potential with negative inverse powers of x (DHE with $z\left(x\right)=x$ and $z\left(x\right)=-1/x$). (This latter potential was derived using two parametrizations due to the symmetry of the DHE). Two further potentials (BHE with $z\left(x\right)={(3x/2)}^{2/3}$ and $z\left(x\right)={\left(2x\right)}^{1/2}$) were found as more general forms of potentials derived from an alternative method [35]. Finally, a potential with power-like terms $x^k$, $k=2$, $-2$, $-4$, and $-6$ was also identified with two alternative parametrizations (DHE with $z\left(x\right)=x^2/4$ and $z\left(x\right)=4/x^2$). This potential is missing from the compilation of QES potentials mentioned here [29] but has been found in independent studies [37,38].

The potentials obtained from the triconfluent Heun equation are technically more complicated, as they can be obtained from conditions necessary to terminate a four-term recursion relation. The five potentials in this class contain four typically power-like terms with both integer and non-integer exponents, and four of them also contain a fifth weakly attractive inverse square potential term with fixed coupling coefficients. Special limits of four of the five potentials are known from the literature [39,40,41], where usually a three-parameter version of the triconfluent Heun equation is considered. Furthermore, studies in this field have been extended to potential obtained from non-trivial, multi-term $\mathsf{\Phi }\left(z\right)$ functions (36).

A further possibility is to consider potentials derived with a more general structure of the $\mathsf{\Phi }\left(z\right)$ function defining the $z\left(x\right)$ transformation function. This amounts to choosing several $p_i$ parameters to be non-zero. This choice leads to technically more involved situations: ($\mathit{i}$) the $z\left(x\right)$ function may be obtained only implicitly through $x\left(z\right)$; ($\mathit{ii}$) the potential picks up extra terms with fixed coupling coefficients; ($\mathit{iii}$) the reparametrization of the parameter set in order to obtain the N-independent $s_i$ coupling coefficient in the potential may become more complicated. This procedure has been applied only to the confluent hypergeometric function, resulting in a non-trivial Natanzon confluent potential [25], so its application to biconfluent and doubly confluent Heun equations still remains to be performed. (It should be noted though that a number of potentials of this type are known [8] from the Natanzon class).

The results reviewed here are mainly of mathematical nature, and the question of possible applications of these potentials arises naturally. Their common feature is that they contain four (or in the case of the triconfluent Heun equation (THE), five) terms, and a rich variety of potential shapes can be generated by an appropriate choice of the parameters. In addition to two examples (BHE $p_1$ and DHE $p_3$), the potentials contain power-like terms with integer or fractional exponents ranging from $x^{-6}$ to $x^6$, usually also containing an inverse square-like centrifugal term $x^{-2}$. Due to the singularity at $x=0$, these potentials are naturally defined on the positive semi-axis $x\in [0,\infty )$, although some of them can be extended to the full x axis. Potentials with dominant positive-power terms are confining potentials, which can have single- or double-well structures. These are typically the ganeralizations of the harmonic oscillator potential. An example of this is the sextic oscillator (BHE $p_2$), which has been applied successfully to describe transitions between different collective shape phases of nuclei [27]. There are numerous quantum mechanical systems in which double-well potentials can be found useful. The quartic potential (THE $p_5$) is employed in many physical systems. Another group of potentials contain dominant terms with negative powers (BHE $p_5$, DHE $p_1$, $p_5$, and $p_2$, $p_4$). These potentials may describe atomic systems with screened point charges. There are also potentials with mixed (positive and negative) powers of x (BHE $p_2$, $p_4$, THE $p_2$, $p_3$, $p_4$, $p_5$), which can be useful to describe asymptotically confined systems with charged point-like behavior near the origin. An example is the Cornell potential (BHE $p_2$), which contains both Coulombic and oscillator-like terms and can describe quark systems. Extending the studies of potentials to multi-term $\mathsf{\Phi }\left(z\right)$ functions may open the door to more versatile interaction forms.

A similar study focusing on the Heun and confluent Heun equations also seems worthwhile. Potentials obtained from these differential equations are generalizations of Natanzon-class potentials, i.e., those derived from the hypergeometric differential equation. The main difference between these differential equations and those discussed here is that the former ones have more regular singularities, which requires a different mathematical formulation. Nevertheless, preliminary results are available for the general form of the potentials [14,15] and their solutions expanded in terms of hypergeometric functions. More recently, polynomial solutions of these potentials have also been discussed [26,43]. The expansion coefficients of these polynomials satisfy a three-term recurrence relation, similarly to those discussed in the present work for the biconfluent and doubly confluent Heun equations. It was also shown that closed-form solutions of these recursion relations are possible and recover the $X_1$-type Jacobi and Laguerre polynomials that supply the solutions of the rationally extended versions of the harmonic oscillator, Scarf I, Scarf II, and the generalized Pöschl–Teller potentials. It is also known that the original and the rationally extended versions of these potentials are connected by transformations formulated in terms of supersymmetric quantum mechanics (SUSYQM); see, e.g., Refs. [44,45]. The possibility of applying SUSYQM to potentials derived from Heun-type equations in a wider range may also be studied.