In this Section, we expand the solutions in terms of a power series, and identify conditions under which the infinite series reduces to a finite polynomial form. The coefficients of the polynomial generally satisfy a (typically three-term) recurrence relation, so the solution of the problem amounts to finding solutions of the recurrence relation. In some cases, expansion coefficients can be obtained in explicit form, while generally, they can be determined by diagonalizing an matrix. This usually leads to an th-order algebraic equation in one of the parameters, and solving it, the coefficients of solutions, i.e., the polynomial solutions themselves, and the corresponding eigenvalues can be generated with . Even in this case, exact analytic results can be obtained up to .
Each subsection is devoted to one of the Heun-type equations. After the presentation of the general formalism, illustrative examples are given. Previously published works with Schrödinger potentials and polynomial solutions are also cited, pointing out how they fit into the general scheme reviewed in the present work. We generally do not mention works discussing specific potentials related to Heun-type equations, unless they analyze the solutions too.
3.1. The Confluent Heun Equation
The non-symmetrical canonical form of the confluent Heun equation is
Following the discussion in Ref. [
34], we expand the
solutions into a power series around the singular point
as
We note that the more general Frobenius expansion could also be used here, with a pre-factor of the type
. However, from the point of view of our goal, i.e., to find solutions of the Schrödinger Equation (
4), this does not imply a generalization, because with the exception of the triconfluent Heun equation, such a factor (originating from the
term in
) appears in the solutions (
7), as can be seen in
Table 1. From the substitution of Equation (
9) into Equation (
8), a three-term recursion relation follows for the
coefficients [
22], as follows:
The boundary conditions
and
are prescribed here.
Assuming that
and
, the termination of the recursion is achieved at
provided that the following conditions apply:
The first Equation (
11) defines a relation between
and
that sets
. With this, and prescribing
, Equation (
12) is also satisfied, i.e.,
, so the termination of the series is reached. Provided that
, the conditions can be summarized as follows:
Under these conditions, the confluent Heun function reduces to the following polynomial form:
It has to be stressed that a constant function, i.e., a polynomial of the order
, cannot be the solution of Equation (
8) unless
holds for
. Contrary to the case of the classical orthogonal polynomials, the Jacobi, generalized Laguerre, and Hermite polynomials [
2], this requirement is not fulfilled automatically. It holds only if
is valid for
.
A constructive method has been introduced in Ref. [
34] to determine
coefficients that satisfy the recursion relation (
10), as well as the conditions for the termination. Four different solutions have been found. One of them corresponded to the following relation for the parameters:
and resulted in the explicit form
This
is normalized such that
holds.
It can be seen that, for positive values, determines the sign of . This expression is for , while it is for , while it is a monotonously decreasing function of k. This implies that changes sign exactly once as k proceeds from 0 to N, so the polynomial has one zero on the positive real z axis.
The actual form of the confluent Heun equation with the parameter set (
15) is
This equation depends on a single parameter,
, and it also depends on
N, which is the order of the polynomial solution.
Applying the variable transformation
, one is led to the differential equation
which can be recognized as the differential equation of the
-type exceptional Laguerre polynomials
[
19,
35]. This means that, up to a normalization constant, these exceptional polynomials represent a special case of the confluent Heun function [
34]
It is known that the sequence of the
-type exceptional Laguerre polynomials starts with
[
19,
35], a finding that has a natural explanation in the framework based on the confluent Heun equation. It is also known that one of its zeros falls into the negative domain
. This result is also explained by the present scheme: the variable transformation modifies the coefficients of the polynomials by a factor of
, implying that the coefficients follow an alternating sequence, except for one step, i.e., when the expression
changes signs. The coefficients of the
-type exceptional Laguerre polynomials are related to Equation (
16) as
The extra terms in (
20) are due to the factor introduced in the variable transformation and to the different normalization used in the two cases. The (
20) coefficients can also be obtained [
34] from those of the generalized Laguerre polynomials [
2], taking into account the relation between these polynomials and the
-type exceptional Laguerre polynomials [
17], as follows:
Polynomial solutions of the confluent Heun equation have been discussed in Ref. [
22]; however, they are based on the solution of
th-order algebraic equations, which follow from the diagonalization of an
tridiagonal matrix derived from the recursion relation. The connection to the generalized Laguerre polynomials through the
-type exceptional Laguerre polynomials, which allow explicit construction of the
coefficients, was not known at that time.
It can be noted that the systematic constructive approach in Ref. [
34] also recovered the generalized Laguerre polynomials as a special polynomial solution of the confluent Heun equation, as well as two further polynomial systems. The latter two examples had features typical for semi-classical orthogonal polynomials [
36]: their polynomial order
N also appeared in the first-order derivative term of the differential equation (
).
The study in Ref. [
34] was also extended to the investigation of variable transformations by which the confluent Heun equation can be transformed into the Schrödinger equation with exactly solvable quantum potentials. It was found that, among the possible variable transformations, the most natural choice is considering
which recovers the rationally extended harmonic oscillator [
19,
35], as follows:
Here,
N corresponds to
in the notation of Ref. [
19], where
labels the actual degree of the exceptional Laguerre polynomial. This potential can also be obtained from the conventional harmonic oscillator by a supersymmetric transformation with broken supersymmetry, and also fulfills the criteria for shape invariance [
11]. For a pedagogical review, see Ref. [
37]. It may be mentioned that potential (
23) was derived [
38] by SUSY transformations from the radial harmonic oscillator a decade before the concept of rationally extended potentials was introduced. However, its importance as a new shape-invariant potential class was not realized at that time.
In another study [
39], potentials were derived from the symmetrical canonical form of the confluent Heun equation. This choice was more suited to the formalism of
-symmetric quantum mechanics, because the
functions had definite parity with respect to the space reflection operator
, which facilitated identifying the parity properties of the potentials and the solutions. General expressions have been derived for five different variable transformations to generate five different potentials, and conditions under which the potentials admitted
symmetry have been identified. However, the solutions were left in their general (non-polynomial) form.
Finally, we note that a systematic study of generating solvable Schrödinger potentials from the confluent Heun equation has been presented in Ref. [
27], although without discussing the bound-state solutions in detail. Nine independent potentials arising from nine variable transformation functions
have been identified, among them those discussed in the earlier work [
24]. Six of the nine potentials were found to be generalizations of Natanzon (confluent) class potentials. It can be shown that the one denoted by
can be identified with the rationally extended harmonic oscillator discussed above. The potential with the terms
,
, and
appearing in Equation (1.3.33) in Ref. [
32] and its analogous trigonometric version with the terms
,
,
and
from Equation (3) of Ref. [
40] correspond to the potential denoted with
in [
27]. (The first potential misses the fourth
term of the general expression, while the last two terms in the latter potential can be written into the expected
and
terms plus a constant). Similarly, potential (1.3.41) in Ref. [
32] and potential (4) in Ref. [
40] with the terms
,
, and
correspond to a limited three-term hyperbolic version of the potential denoted with
in [
27]. These potentials, similar to two other ones appearing in Equations (2) and (5) in Ref. [
40], which are trivial transforms or special cases of the two mentioned potentials, have been obtained within the quasi-exactly solvable (QES) setting [
31]. Furthermore, the periodic
-symmetric potential
in Ref. [
41] can also be rewritten into another form containing the
and
terms.
3.2. The Heun Equation
The Heun equation is written as [
22,
33]
where we changed the notation used in
Table 1 (
and
) in order to avoid confusion in the later stages of the discussion.
Let us expand the Heun function into a power series around the singular point
, as follows:
where the factor of
was introduced for practical reasons. Substituting this expansion into the Heun equation and eliminating
using Equation (
3), once again, obtains a three-term recursion relation, as follows:
The boundary conditions
and
are prescribed here too. Assuming again that
and
, the conditions for termination at
are as follows:
and
If the right-hand side of Equation (
27) is zero, i.e., if
is proportional to
, then
has to hold. Furthermore, if the coefficient of
is zero in Equation (
28), then
also holds; i.e., the termination of the recursion is achieved. The latter condition is
This relation is valid for either
or
. Note that the situation is analogous to the condition under which the hypergeometric function
2F1(
a,
b;
c;
z) reduces to a polynomial: there also,
a and
b are interchangeable, and either of them can be chosen as
. The conditions for termination are the following:
The polynomial solutions of the Heun equation are then written as
As it has been mentioned earlier, these equations are equivalent with those obtained from the
replacement.
The Heun equation can be matched with the differential equation of the
-type exceptional Jacobi polynomials
[
19,
35]. These polynomials are defined for
, and they satisfy the differential equation
where
It can be shown that with the substitutions
the Heun equation takes the form (
32). This also means that up to a constant normalization factor,
holds, where
b and
q are related to
and
as in Equation (
34). The relation between the solutions of the Heun equation, the
-type exceptional Jacobi polynomials, and the generalized hypergeometric functions has been discussed in Ref. [
42].
Since the
-type exceptional Jacobi polynomials can be expressed in terms of classical Jacobi polynomials, the
expansion coefficient appearing in Equation (
31) can also be expressed in terms of the expansion coefficients of the classical Jacobi polynomials [
17].
The classical Jacobi polynomials are expressed in terms of the expansion
Since the solutions of the Heun equation have been expanded in the same form in Equation (
25), the expansion coefficients will be obtained from the combination of the expansion coefficients in Equation (
37) with
,
and
in Equation (
36).
Polynomial solutions of the Heun equation have also been discussed in Ref. [
22], where conditions for the termination of the infinite power series have also been analyzed. However, there, the connection with the Jacobi polynomials via the
-type exceptional Jacobi polynomials was not mentioned, which is understandable, as the latter were introduced later [
17].
The zeros of the -type exceptional Jacobi polynomials are known to fall within the domain with the exception of one. The first member of the sequence is a first-order polynomial, similar to the case of the -type exceptional Laguerre polynomials.
There are several exactly solvable potentials with bound-state solutions containing -type exceptional Jacobi polynomials. These can be obtained from variable transformations governed by some functions. All of them are rational extensions of some well-known shape-invariant potentials.
One is the rational extension of the Scarf I potential [
19]
which is obtained for
. (Note that, here,
a is a simple parameter that scales the coordinate, and is different from the
a appearing in Equation (
33).) The bound-state energy eigenvalues are
while the corresponding wavefunctions are
Another choice is
, which results in the rational extension of the generalized Pöschl–Teller potential [
43]
The bound-state eigenfunctions are
while the energy eigenvalues take the form
In both examples, the parametrization guarantees that the singularity of the potential appears outside the domain of definition of the potentials (
and
in the two cases. There is also a third example with
, resulting in the rationally extended Scarf II Potential [
43], which is normally defined on the full real
x axis. In order to avoid singularities, this potential is defined only in the
-symmetric setting, where singularities on the real
x axis can be avoided by applying an imaginary coordinate shift
. All three examples are discussed in Ref. [
44] in the context of supersymmetric transformations connecting the Scarf I, generalized Pöschl–Teller, and Scarf II potentials to their rational extensions. See Ref. [
45] for a review on the subtleties of infinite and finite orthogonal polynomial systems related to Jacobi polynomials and their connection to Romanovski polynomials.
In a systematic study, solvable Schrödinger potentials have been constructed from the Heun equation [
26]. Eleven independent potentials have each corresponded to a specific coordinate transformation. Four of them have already been identified in the earlier study in Ref. [
24]. Nine potentials can be considered generalizations of Natanzon-class potentials. In the solutions, the Heun function is expanded in terms of hypergeometric functions. The rational extensions of Scarf I and the generalized Pöschl–Teller potentials discussed here are not mentioned, but they can be identified as that denoted with
in
Table 1 in Ref. [
26].
3.3. The Biconfluent Heun Equation
The biconfluent Heun equation is written as
Expanding the
function into a power series around the singular point
and substituting it into Equation (
44), again, a three-term recursion relation follows:
is prescribed in the calculations.
Assuming again that
and
, the the recursion terminates at
under the following conditions:
These equations are satisfied if the following conditions apply:
The first equation cancels the coefficient of
in Equation (
48), while the second one (combining it with Equation (
49) sets
in Equation (
47). These two results lead to
, and to the termination of the recursion meaning that
can be written into a finite, polynomial form as
This equation has to be considered together with Equation (
49). In the case of the confluent Heun and the Heun equations, the analogous condition was automatically satisfied, because the solutions of the confluent Heun and the Heun equations reduced to
-type Laguerre and Jacobi polynomials, with coefficients satisfying the required relation, including the one analogous to Equation (
49). This is not the case now, so this condition has to be enforced.
There are five different variable transformations that lead to (in principle) solvable quantum potentials from the biconfluent Heun equation [
14]. When
is applied, the sextic oscillator
is recovered if the parameters are chosen as
while the energy eigenvalues are related to
q through
The wave functions are written as
where
is the polynomial solution (
50). The coefficients of the polynomial are determined by the recursion (
46), while the termination condition (
49) amounts to solving an
th-order algebraic equation for
q, which arises from diagonalizing an
tridiagonal matrix, which, in turn, follows from the recursion (
46). This potential and its solutions have been applied to describe certain collective excitations of nuclei. Closed expressions for the wavefunctions and the corresponding energy eigenvalues have been determined first for
and 1 [
46] and then also for
, i.e., for a cubic algebraic equation for
q [
47].
The sextic oscillator either as a one-dimensional problem or as a radial one, as it stands in Equation (
51), is the archetype of quasi-exactly solvable (QES) potentials [
31]. At the same time, it has been known for a long time [
24] that it can be obtained from the biconfluent Heun equation. The equivalence of the two approaches has been presented in Ref. [
48], where the solutions of the biconfluent Heun equation have been expressed in terms of Hermite functions. That approach recovered the QES results when the Hermite functions reduced to Hermite polynomials
. In the present approach, the solutions are expanded in terms of polynomials of
; nevertheless, the final results are the same, as expected.
Before closing this subsection, we note that polynomial solutions of the biconfluent Heun equation (with different parametrization) have been discussed in Ref. [
22] to some detail. It was established that these solutions arise if one of the parameters is equal to a non-positive integer, while another one follows from the roots of a polynomial. This is rather similar to the results obtained in Ref. [
22] for the Heun and confluent Heun equations, and are practically equivalent with the formalism of quasi-exactly solvable potential models [
31]. It is remarkable that these two approaches have been connected only recently [
48]. Somewhat later, a further independent approach to the sextic oscillator has been presented in terms of the extended Nikiforov–Uvarov method [
49].
A systematic study of solvable potentials arising from the biconfluent Heun equation has been carried out in Ref. [
28]; however, the solutions there are expanded in terms of Hermite functions, rather than in power series. That expansion also terminates under certain conditions, leading to a finite sum of Hermite functions in the solutions. Several potentials are identified, depending on the number of Hermite functions involved in the sum. The five potentials found in Ref. [
24] have also been recovered, including the sextic oscillator and some potentials found earlier independently [
50,
51,
52], which include terms with fractional powers of
x. These potentials are identified in Ref. [
28] with the variable transformations
,
, and
. The one with
, containing the terms
,
,
x, and
and polynomial solutions has been discussed in Ref. [
53] and in Ref. [
32] (see Equation (1.3.66)). The latter work applied the quasi-exactly solvable (QES) formalism [
31], with no reference to the biconfluent Heun equation, and also mentioned the fifth potential with
and with the terms
,
, 2, 3, and 4 (see Equations (1.3.29) and (1.3.38) with the
choice).
3.4. The Double Confluent Heun Equation
The double confluent Heun equation can be written in the general form
Following the steps analogous to those applied to the other equations, the solutions of Equation (
55) can be expanded into a power series around the singular point
as
Substituting it into Equation (
55) leads to a three-term recursion relation, as follows:
The conditions for termination at
do not require any assumptions on
in this case:
The conditions for termination are satisfied if the following equations hold:
These equations guarantee that
for
, so
can be written into the finite, polynomial form as
This equation is formally similar to Equation (
50), but they are essentially different. However, there is a similarity with the case of the biconfluent Heun equation in that the explicit forms of the
coefficients cannot be expressed in closed form; rather, they have to be determined from the roots of an
th-order algebraic equation.
It is known [
14] that there are three different variable transformations leading to relatively simple solvable quantum potentials. Selecting
as an example, the potential
is obtained, and the energy eigenvalues are written as
The wave functions take the form
where
is the polynomial solution (
61). The coefficients of the polynomial are, again, determined by the recursion (
67), with a termination condition (
60), leading to an
th-order algebraic equation for
q.
The potential (
62) is a “mirror image” of the sextic oscillator in the sense that its potential terms (
) are the inverse of those (
) appearing in the sextic oscillator (
51). Another of the three potentials related to the double confluent Heun equation with the transformation function
and the potential terms
,
,
,
, and
has been discussed in Ref. [
32] in the quasi-exactly solvable (QES) setting [
31], without referring to the double confluent Heun equation (see Equation (1.3.79)). In the same work, the third potential with
and with the terms
,
,
, 1, and 2 is also mentioned (see Equation (1.3.19)).
Before closing this subsection, we mention that quasi-polynomial solutions of another parametrization of the double confluent Heun function have been studied in Ref. [
22]. A more detailed comparison of the two approaches seems worthwhile, but we do not consider it here.