An Extension of a Formula of F. S. Rofe-Beketov

Abstract

We address the following problem in this study: given a nontrivial solution, $y_1$, of a four-coefficient Sturm–Liouville eigenvalue differential equation, construct a second solution, $y_2$, linearly independent of $y_1$. In the process of describing the solution of this problem, we review the approaches by d’Alembert and Rofe-Beketov in the case of three-coefficient Sturm–Liouville eigenvalue differential equations.

1. Introduction

A classical problem in the theory of Sturm–Liouville eigenvalue differential equations can be stated as follows: given one nontrivial solution, $y_1$, construct a second solution, $y_2$, linearly independent of $y_1$.

A celebrated solution in the context of three-coefficient Sturm–Liouville eigenvalue differential equations of the form $\tau y=zy$, where the differential expression $\tau$ is of the form

$$\tau =r^{-1}[-(d/dx)p(d/dx)+q]\mathrm{on}\mathrm{an}\mathrm{interval}(a,b),$$

(1)

due to Jean le Rond d’Alembert, reads as follows: suppose $y_1$ satisfies $\tau y=zy$, $z\in \mathbb{C}$, in the interval $(a,b)$; then, a second solution, $y_2$, of $\tau y=zy$, linearly independent of $y_1$, is of the form

$$y_2(z,x)=-y_1(z,x){\int }^x\frac{d{x}^{\prime }}{p\left(x^{\prime }\right)y_1{(z,x^{\prime })}^2},x\in (a,b).$$

(2)

In this case, the Wronskian of $y_2$ and $y_1$ is normalized to equal 1.

Clearly, $y_2(z,·)$ in (2) has “issues” at the zeros of $y_1(z,·)$ in the interval $(a,b)$. A solution of these issues was presented by Fedor S. Rofe-Beketov [1,2], ([3], Sect. 6.4) (see also [4]), who found the following formula for $y_2$:

$$\begin{array}{cc}\hfill y_2(z,x)& =y_1(z,x){\int }^{x}d{x}^{\prime }\{\left[p{\left(x^{\prime }\right)}^{-1}+\overline{q\left(x^{\prime }\right)}-\overline{zr\left(x^{\prime }\right)}\right]{\left(\overline{y_1(z,x^{\prime })}\right)}^2\hfill \\ \hfill & -\left[{\overline{p\left(x^{\prime }\right)}}^{-1}+q\left(x^{\prime }\right)-zr\left(x^{\prime }\right)\right]{\left(\overline{y_1^{\left[1\right]}(z,x^{\prime })}\right)}^2\}\hfill \\ \hfill & \times {\left[|y_1(z,x^{\prime }){|}^2+|y_1^{\left[1\right]}(z,x^{\prime }){|}^2\right]}^{-2}\hfill \\ \hfill & -\overline{y_1^{\left[1\right]}(z,x)}{\left[|y_1{(z,x)|}^2+|y_1^{\left[1\right]}(z,x){|}^2\right]}^{-1},x\in (a,b).\hfill \end{array}$$

(3)

Here, $y^{\left[1\right]}$ abbreviates the first quasi-derivative of y adapted to $\tau$ in (1) and is given by $y^{\left[1\right]}=p{y}^{\prime }$, and now the denominators in (3) can no longer vanish. Once again, the Wronskian of $y_2$ and $y_1$ is normalized to equal 1.

The principal result of this study then consists in an extension of Rofe-Beketov’s Formula (2) to the case of four-coefficient Sturm–Liouville eigenvalue equations. More precisely, $\tau$ is then of the form

$$\tau =r^{-1}\{-(d/dx)p[(d/dx)+s]+sp[(d/dx)+s]+q\}\mathrm{on}\mathrm{an}\mathrm{interval}(a,b),$$

(4)

and if $y_1$ satisfies $\tau y=zy$, $z\in \mathbb{C}$, in the interval $(a,b)$, then a second solution, $y_2$, of $\tau y=zy$, linearly independent of $y_1$, is of the form

$$\begin{array}{cc}\hfill & y_2(z,x,x_0)\hfill \\ \hfill & =-y_1(z,x){\int }_{x_0}^{x}d{x}^{\prime }\{\left[p{\left(x^{\prime }\right)}^{-1}+q\left(x^{\prime }\right)-zr\left(x^{\prime }\right)\right]\left[y_1{(z,x^{\prime })}^2-{\left(y_1^{\left[1\right]}(z,x^{\prime })\right)}^2\right]\hfill \\ \hfill & +4s\left(x^{\prime }\right)y_1(z,x^{\prime })y_1^{\left[1\right]}(z,x^{\prime })\}{\left[y_1{(z,x^{\prime })}^2+{\left(y_1^{\left[1\right]}(z,x^{\prime })\right)}^2\right]}^{-2}\hfill \\ \hfill & +{\displaystyle \frac{y_1^{\left[1\right]}(z,x)}{y_1{(z,x)}^2+{\left(y_1^{\left[1\right]}(z,x)\right)}^2}},x,x_0\in (a,b),\hfill \end{array}$$

(5)

Here, the first quasi-derivative of y adapted to $\tau$ in (4) is given by $y^{\left[1\right]}=p[y^{\prime }+sy]$. Once more, the denominators in (5) cannot vanish and the Wronskian of $y_2$ and $y_1$ is normalized to equal 1.

In Section 2, we very succinctly review four-coefficient Sturm–Liouville eigenvalue equations; Section 3 is devoted to various results from F. S. Rofe-Beketov in the three-coefficient context, and finally, Section 4 presents our extension of Rofe-Beketov’s formulas for $y_2$ to the case of four-coefficient Sturm–Liouville eigenvalue equations.

2. Four-Coefficient Sturm–Liouville Equations in a Nutshell

We briefly summarize the essentials of four- (and hence, three-) coefficient Sturm–Liouville eigenvalue equations and for this purpose assume the following hypotheses throughout this paper.

Hypothesis 1.

Let $-\infty ⩽a<b⩽\infty$ and suppose that p, q, r, and s are Lebesgue measurable in $(a,b)$ with $p^{-1},q,r,s\in L_{loc}^1((a,b);dx)$(in particular, p, q, r, and s need not be real-valued ).

Assuming Hypothesis 1, we introduce the set

$${\mathfrak{D}}_{\tau }\left((a,b)\right)=\left\{g\in A{C}_{loc}\left((a,b)\right)|p[g^{\prime }+sg]\in A{C}_{loc}\left((a,b)\right)\right\}$$

(6)

and the four-coefficient Sturm–Liouville differential expression $\tau \equiv \tau (p,q,r,s)$ defined by

$$\tau f={\displaystyle \frac{1}{r}}\left(-{\left(p[f^{\prime }+sf]\right)}^{\prime }+sp[f^{\prime }+sf]+qf\right)\in L_{loc}^1((a,b);rdx),f\in {\mathfrak{D}}_{\tau }\left((a,b)\right).$$

(7)

In addition, if we introduce the first quasi-derivative of f adapted to the four-coefficient differential expression $\tau$ via

$$f^{\left[1\right]}=p[f^{\prime }+sf],f\in {\mathfrak{D}}_{\tau }\left((a,b)\right),$$

(8)

then ${\mathfrak{D}}_{\tau }\left((a,b)\right)=\left\{g\in A{C}_{loc}\left((a,b)\right)|g^{\left[1\right]}\in A{C}_{loc}\left((a,b)\right)\right\}$ and

$$\tau y={\displaystyle \frac{1}{r}}\left(-{\left(y^{\left[1\right]}\right)}^{\prime }+s{y}^{\left[1\right]}+qy\right)=zy\mathrm{a}.\mathrm{e}.\mathrm{on}(a,b),z\in \mathbb{C},y\in {\mathfrak{D}}_{\tau }\left((a,b)\right),$$

(9)

is a four-coefficient Sturm–Liouville differential equation. At times, it will be advantageous to rewrite (9) in an equivalent form,

$${\left(y^{\left[1\right]}\right)}^{\prime }=s{y}^{\left[1\right]}+[q-zr]y\mathrm{a}.\mathrm{e}.\mathrm{on}(a,b),z\in \mathbb{C},y\in {\mathfrak{D}}_{\tau }\left((a,b)\right).$$

(10)

The special case $s=0$ a.e. on $(a,b)$ in (8), (9), or (10), in obvious notation, will then be called a three-coefficient Sturm–Liouville eigenvalue differential equation. For a detailed treatment of three-coefficient (i.e., $s=0$) Sturm–Liouville eigenvalue differential equations and their associated operators, we refer to ([5], Ch. 6), [6,7], ([8], Sect. 8.4), [9,10,11].

We note that $\tau$ is called regular at a if, in addition to Hypothesis 1, $p^{-1},q,r,s\in L^1((a,c);dx)$ for some (and hence for all) $c\in (a,b)$; the regularity of $\tau$ at the endpoint b is defined analogously.

Definition 1.

Assume Hypothesis 1 and let $z\in \mathbb{C}$. A function, y, is a solution of the Sturm–Liouville equation $\tau f=zf$ in $(a,b)$ if $y\in {\mathfrak{D}}_{\tau }\left((a,b)\right)$ and $\tau y=zy$ pointwise a.e. in $(a,b)$. A solution y of $\tau f=zf$ in $(a,b)$ is nontrivial if y is not the zero function in $(a,b)$.

For each $f,g\in {\mathfrak{D}}_{\tau }\left((a,b)\right)$, the (modified) Wronskian of f and g adapted to $\tau$ is defined by

$$W(f,g)\left(x\right)=f\left(x\right)g^{\left[1\right]}\left(x\right)-f^{\left[1\right]}\left(x\right)g\left(x\right),x\in (a,b).$$

(11)

Remark  1.

One infers that for $f,g\in {\mathfrak{D}}_{\tau }\left((a,b)\right)$, $W(f,g)$ is locally absolutely continuous in $(a,b)$ and its derivative is

$${\left[W(f,g)\right]}^{\prime }\left(x\right)=\left[g\left(x\right)\left(\tau f\right)\left(x\right)-f\left(x\right)\left(\tau g\right)\left(x\right)\right]r\left(x\right)\mathit{for}\mathrm{a}.\mathrm{e}.x\in (a,b).$$

(12)

In particular, if $z\in \mathbb{C}$, then the Wronskian of two solutions, $y_j(z,·)\in {\mathfrak{D}}_{\tau }\left((a,b)\right)$ and $j\in \{1,2\}$, of (9) in $(a,b)$ is constant. Moreover, $W(y_1(z,·),y_2(z,·))\ne 0$ if and only if $y_1(z,·)$ and $y_2(z,·)$ are linearly independent.

We recall the following special case of the existence and uniqueness result ([7], Theorem 14.2.2) (the real-valuedness and positivity a.e. assumptions made there are not needed) and ([12], Theorem 1, p. 51) for initial-value problems corresponding to the differential Equation (9).

Theorem 1.

Assume Hypothesis 1. If $z\in \mathbb{C}$, $c\in (a,b)$, and α, $\beta \in \mathbb{C}$, then there is a unique solution, $y\in {\mathfrak{D}}_{\tau }\left((a,b)\right)$, of (9) in $(a,b)$ that satisfies $y\left(c\right)=\alpha$ and $y^{\left[1\right]}\left(c\right)=\beta$. If, in addition, $\alpha ,\beta ,z\in \mathbb{R}$, then the solution y is real-valued.

Remark 2.

To simplify matters, we choose $p=r=1$ in addition to Hypothesis 1. Then, with

$$f\in A{C}_{loc}\left((a,b)\right),f^{\left[1\right]}=[f^{\prime }+sf]\in A{C}_{loc}((a,b),$$

(13)

one infers that

$$\left(\tau f\right)=\{-{[f^{\prime }+sf]}^{\prime }+s[f^{\prime }+sf]+qf\}\in L_{loc}^1((a,b);dx).$$

(14)

Next, we also assume that

$$s\in L_{loc}^2((a,b);dx),$$

(15)

that is, $s^2\in L_{loc}^1((a,b);dx)$. In this case,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & f^{\left[1\right]}=[f^{\prime }+sf]\in A{C}_{loc}\left((a,b)\right)\mathrm{and}sf\in L_{loc}^2((a,b);dx)\hfill \\ \hfill & \mathit{implies}{f}^{\prime }\in L_{loc}^2((a,b);dx),f^{″}\in H_{loc}^{-1}\left((a,b)\right).\hfill \end{array}\end{array}$$

(16)

In addition, $s{f}^{\prime }\in L_{loc}^1((a,b);dx)$, $f^{″},s^{\prime }f\in H_{loc}^{-1}\left((a,b)\right)$, and

$${f^{\left[1\right]}}^{\prime }={[f^{\prime }+sf]}^{\prime }=[f^{″}+s^{\prime }f]+s{f}^{\prime }\in L_{loc}^1((a,b);dx)$$

(17)

as well as $s^{2}f,qf\in L_{loc}^1((a,b);dx)$ yield

$$\left(\tau f\right)=-f^{″}+s^{\prime }f+\left[s^2+q\right]f\in L_{loc}^1((a,b);dx).$$

(18)

This process generates the $H_{loc}^{-1}\left((a,b)\right)$-potential coefficient

$$s^{\prime }\in H_{loc}^{-1}\left((a,b)\right).$$

(19)

See ([7], Remark 14.3.10) for additional details.

For more background in connection with four-coefficient differential expressions and the underlying Sturm–Liouville operators and their Weyl–Titchmarsh and oscillation theory, we refer, for instance, to [13], ([7], Ch. 14), and the extensive literature cited therein.

To close this section, we provide two examples to illustrate how Schrödinger-type equations with distributional potentials may be naturally defined and studied within the framework of four-coefficient Sturm–Liouville differential equations.

Example 1.

(Delta function potential). Let $x_0\in \mathbb{R}$, $\alpha \in \mathbb{R}\setminus \left\{0\right\}$ be fixed, and recall the sign function ${sgn}_{x_0}$ centered at $x_0$:

$${sgn}_{x_0}\left(x\right)=\left\{\begin{array}{cc}1,\hfill & x>x_0,\hfill \\ -1,\hfill & x<x_0.\hfill \end{array}\right.$$

(20)

Taking $(a,b)=(-\infty ,\infty )$ and

$$p=r=1,s=-(\alpha /2){sgn}_{x_0},q=-{\alpha }^2/4,$$

(21)

in Hypothesis 1, one obtains the differential expression $\tau \equiv {\tau }_{x_0}$ with action given according to (7) by

$$\left({\tau }_{x_0}f\right)\left(x\right)=-{[f^{\prime }+sf]}^{\prime }\left(x\right)+s\left(x\right)f^{\prime }\left(x\right)-\left({\alpha }^2/4\right)f\left(x\right)=-f^{″}\left(x\right),x\in \mathbb{R}\setminus \left\{x_0\right\},$$

(22)

for any function, f, that satisfies

$$f\in A{C}_{loc}\left(\mathbb{R}\right),[f^{\prime }+sf]\in A{C}_{loc}\left(\mathbb{R}\right).$$

(23)

Formally carrying out the differentiation in (22) and using $s^{\prime }=-\alpha {\delta }_{x_0}$ (in the distributional sense ), one infers that the action of ${\tau }_{x_0}$ is (again, formally ) given by

$${\tau }_{x_0}f=-f^{″}+\alpha {\delta }_{x_0}f.$$

(24)

Therefore, under the choices in (21), the four-coefficient differential expression (7) formally corresponds to a point interaction with the strength α centered at $x_0$.

Example 2

(Miura-type potential). Choosing $(a,b)=(-\infty ,\infty )$ and

$$p=r=1,q=0,s\in L_{loc}^2(\mathbb{R};dx),$$

(25)

in Hypothesis 1, one obtains the differential expression $\tau \equiv {\tau }_s$ with action given according to (7) by

$${\tau }_{s}f=-{[f^{\prime }+sf]}^{\prime }+s[f^{\prime }+sf],$$

(26)

for any function, f, that satisfies

$$f\in A{C}_{loc}\left(\mathbb{R}\right),[f^{\prime }+sf]\in A{C}_{loc}\left(\mathbb{R}\right).$$

(27)

Formally carrying out the differentiation in (26), one infers that the action of ${\tau }_s$ is (again, formally) given by

$${\tau }_{s}f=-f^{″}+\left(s^2-s^{\prime }\right)f.$$

(28)

Since $s\in L_{loc}^2(\mathbb{R};dx)$, $s^{\prime }$ is interpreted in the distributional sense. Therefore, (28) represents a differential expression of the Schrödinger-type with a distributional potential coefficient,

$$s^2-s^{\prime }\in H_{loc}^{-1}\left(\mathbb{R}\right)=W_{loc}^{-1,2}\left(\mathbb{R}\right),$$

(29)

which is known as a Miura-type potential.

3. Rofe-Beketov’s Formula for the Case of Three-Coefficient Sturm–Liouville Equations

In this section, we recall Rofe-Beketov’s formula for a second linearly independent solution, $y_2$, of a three-coefficient Sturm–Liouville equation, given one solution, $y_1$.

In this three-coefficient context, $s=0$ a.e. in $(a,b)$, and hence $f^{\left[1\right]}$ in (8) simplifies to

$$f^{\left[1\right]}=p{f}^{\prime },f\in {\mathcal{D}}_{\tau }\left((a,b)\right).$$

(30)

Lemma 1.

In addition to Hypothesis 1 with $s=0$ a.e. in $(a,b)$, let $z\in \mathbb{C}$, and assume that $y_1(z,·)\in {\mathcal{D}}_{\tau }\left((a,b)\right)$.

(i)

Suppose that $y_1(z,·)\ne 0$ in $(a,b)$ and $y_2(z,·)\in {\mathcal{D}}_{\tau }\left((a,b)\right)$. Then,

$${\left(\frac{y_2(z,·)}{y_1(z,·)}\right)}^{\prime }=\frac{-1}{p{y}_1{(z,·)}^2}\mathit{is}\mathit{equivalent}\mathit{to}W(y_2(z,·),y_1(z,·))=1.$$

(31)

(ii)

Suppose that $y_2(z,·)\in {\mathcal{D}}_{\tau }\left((a,b)\right)$ and $W(y_2(z,·),y_1(z,·))=1$. Then,

$$\tau y_1(z,·)=z{y}_1(z,·)\mathit{is}\mathit{equivalent}\mathit{to}\tau y_2(z,·)=z{y}_2(z,·),$$

(32)

that is, if $W(y_2(z,·),y_1(z,·))=1$, then

$$\begin{array}{cc}\hfill & {\left(p{y}_1^{\prime }\right)}^{\prime }=[q-zr]y_{1}a.e.\mathit{on}(a,b)\hfill \\ \hfill & \mathit{is}\mathit{equivalent}\mathit{to}\hfill \\ \hfill & {\left(p{y}_2\right)}^{\prime }=[q-zr]y_{2}a.e.\mathit{on}(a,b).\hfill \end{array}$$

(33)

(iii)

Suppose that $y_1(z,·)\ne 0$ in $(a,b)$ and $\tau y_1(z,·)=z{y}_1(z,·)$ (i.e., ${\left(p{y}_1^{\prime }\right)}^{\prime }=[q-zr]y_1$) a.e. in$(a,b)$. Then, d’Alembert’s formula,

$$y_2(z,x,x_0)=-y_1(z,x){\int }_{x_0}^{x}d{x}^{\prime }\frac{1}{p\left(x^{\prime }\right)y_1{(z,x^{\prime })}^2},x,x_0\in (a,b),$$

(34)

yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & y_2(z,·,x_0)\in {\mathcal{D}}_{\tau }\left((a,b)\right),\tau y_2(z,·,x_0)=z{y}_2(z,·,x_0),\hfill \\ \hfill & \mathit{and}W(y_2(z,·,x_0),y_1(z,·))=1.\hfill \end{array}\end{array}$$

(35)

In particular, the general solution $y(z,·)$ of $\tau y(z,·)=zy(z,·)$ a.e. in $(a,b)$ is of the form

$$\begin{array}{cc}\hfill y(z,x)& =C_1{y}_1(z,x)+C_2{y}_2(z,x,x_0)\hfill \\ \hfill & =C_1{y}_1(z,x)-C_2{y}_1(z,x){\int }_{x_0}^{x}d{x}^{\prime }\frac{1}{p\left(x^{\prime }\right)y_1{(z,x^{\prime })}^2},\hfill \\ \hfill & C_1,C_2\in \mathbb{C},x,x_0\in (a,b).\hfill \end{array}$$

(36)

Moreover, if p is real-valued a.e. in $(a,b)$ and also $y_1(z,·)$ is real-valued, so is $y_2(z,·,x_0)$. Finally, if for some (or all ) $x\in (a,b)$, $y_1(z,x)$ is analytic (resp., entire ) with respect to $z\in \mathrm{\Omega }$, $\mathrm{\Omega }\subseteq \mathbb{C}$ open, then so is $y_2(z,x,x_0)$.

Proof. 

Item $\left(i\right)$ is verified in an elementary manner and item $\left(ii\right)$ follows from differentiating $W(y_2,y_1)=1$ with respect to x, implying that

$$y_2^{\prime }{\left(p{y}_1\right)}^{\prime }+y_2{\left(p{y}_1^{\prime }\right)}^{\prime }-{\left(p{y}_2^{\prime }\right)}^{\prime }{y}_1-\left(p{y}_2^{\prime }\right)y_1=y_2{\left(p{y}_1^{\prime }\right)}^{\prime }-{\left(p{y}_2^{\prime }\right)}^{\prime }{y}_1=0\mathrm{for}\mathrm{a}.\mathrm{e}.x\in (a,b).$$

(37)

Item $\left(iii\right)$ follows from

$$y_2^{\left[1\right]}=y_1^{\left[1\right]}{\int }^x{\displaystyle \frac{d{x}^{\prime }}{\left(p{y}_1^2\right)\left(x^{\prime }\right)}}+{\displaystyle \frac{1}{y_1}},$$

(38)

which, after a short computation, yields ${\left(y_2^{\left[1\right]}\right)}^{\prime }=[q-zr]y_2$. The analyticity of $y_2$ is clear from that of $y_1$, utilizing (34). □

Lemma 2

([3], Lemma 6.6, p. 209). Assume Hypothesis 1 with $s=0$ a.e. in $(a,b)$, let $z\in \mathbb{C}$, and suppose that $y_j(z,·)\in {\mathcal{D}}_{\tau }\left((a,b)\right)$, $j=1,2$. Moreover, assume the normalized Wronskian $W(y_2(z,·),y_1(z,·))=1$ and that $y_1{(z,·)}^2+{\left(y_1^{\left[1\right]}(z,·)\right)}^2\ne 0$ in $(a,b)$. If $\tau y_1(z,·)=z{y}_1(z,·)$ (i.e., ${\left(p{y}_1^{\prime }\right)}^{\prime }=[q-zr]y_1(z,·)$) a.e. in $(a,b)$, then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\left({\displaystyle \frac{y_2(z,·)-\frac{p{y}_1^{\prime }(z,·)}{y_1{(z,·)}^2+{\left(y_1^{\left[1\right]}(z,·)\right)}^2}}{y_1(z,·)}}\right)}^{\prime }={\left({\displaystyle \frac{y_1{y}_2+\left(p{y}_1^{\prime }\right)\left(p{y}_2^{\prime }\right)}{y_1{(z,·)}^2+{\left(y_1^{\left[1\right]}(z,·)\right)}^2}}\right)}^{\prime }\hfill \\ \hfill & =-{\displaystyle \frac{\left[p^{-1}+q-zr\right]\left[y_1{(z,·)}^2-{\left(y_1^{\left[1\right]}(z,·)\right)}^2\right]}{y_1{(z,·)}^2+{\left(y_1^{\left[1\right]}(z,·)\right)}^2}}.\hfill \end{array}\end{array}$$

(39)

Proof. 

Regarding (39), one first verifies, upon employing $W(y_2,y_1)=1$ and utilizing a cancellation, that

$$\left({\displaystyle \frac{y_2(z,·)-\frac{p{y}_1^{\prime }(z,·)}{y_1{(z,·)}^2+{\left(y_1^{\left[1\right]}(z,·)\right)}^2}}{y_1(z,·)}}\right)={\displaystyle \frac{y_1{y}_2+\left(p{y}_1^{\prime }\right)\left(p{y}_2^{\prime }\right)}{y_1{(z,·)}^2+{\left(y_1^{\left[1\right]}(z,·)\right)}^2}},$$

(40)

and hence the left-hand side of (40) is well defined.

The second equality in (39), then, is a straightforward, yet nasty, computation that once more employs $W(y_2,y_1)=1$ and repeatedly takes advantage of cancellations. □

Lemma 2 indicates why the following result due to F. S. Rofe-Beketov [1,2], ([3], Sect. 6.4), see also K. M. Schmidt [4], is plausible.

Theorem 2.

Assume Hypothesis 1 with $s=0$ a.e. in $(a,b)$, let $z\in \mathbb{C}$, and assume that $y_1(z,·)\in {\mathcal{D}}_{\tau }\left((a,b)\right)$. In addition, suppose that $y_1{(z,·)}^2+{\left(y_1^{\left[1\right]}(z,·)\right)}^2\ne 0$ in $(a,b)$ and that $\tau y_1(z,·)=z{y}_1(z,·)$ (i.e., ${\left(p{y}_1^{\prime }\right)}^{\prime }=[q-zr]y_1$) a.e. in $(a,b)$. Then, Rofe-Beketov’s formula,

$$\begin{array}{cc}\hfill & y_2(z,x,x_0)\hfill \\ \hfill & =-y_1(z,x){\int }_{x_0}^{x}d{x}^{\prime }{\displaystyle \frac{\left[p{\left(x^{\prime }\right)}^{-1}+q\left(x^{\prime }\right)-zr\left(x^{\prime }\right)\right]\left[y_1{(z,x^{\prime })}^2-{\left(y_1^{\left[1\right]}(z,x^{\prime })\right)}^2\right]}{{\left[y_1{(z,x^{\prime })}^2+{\left(y_1^{\left[1\right]}(z,x^{\prime })\right)}^2\right]}^2}}\hfill \\ \hfill & +{\displaystyle \frac{y_1^{\left[1\right]}(z,x)}{y_1{(z,x)}^2+{\left(y_1^{\left[1\right]}(z,x)\right)}^2}},x,x_0\in (a,b),\hfill \end{array}$$

(41)

yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & y_2(z,·,x_0)\in {\mathcal{D}}_{\tau }\left((a,b)\right),\tau y_2(z,·,x_0)=z{y}_2(z,·,x_0),\hfill \\ \hfill & \mathit{and}W(y_2(z,·,x_0),y_1(z,·))=1.\hfill \end{array}\end{array}$$

(42)

In particular, the general solution $y(z,·)$ of $\tau y(z,·)=zy(z,·)$ is of the form

$$y(z,x)=C_1{y}_1(z,x)+C_2{y}_2(z,x,x_0),C_1,C_2\in \mathbb{C},x,x_0\in (a,b).$$

(43)

Moreover, if $z\in \mathbb{R}$, $p,q,r$ are real-valued a.e. in $(a,b)$, and $y_1(z,·)$ is real-valued, so is $y_2(z,·,x_0)$. Finally, if for some (or all ) $x\in (a,b)$, $y_1(z,x)$ is analytic (resp., entire ) with respect to $z\in \mathrm{\Omega }$, $\mathrm{\Omega }\subseteq \mathbb{C}$ open, then so is $y_2(z,x,x_0)$.

Remark 3.

$\left(i\right)$ In a sense, Theorem 2 kicks the can down the road since not only can $y_1(z,·)$ have zeros in (a,b) (rendering (34) suspicious), but so can $y_1{(z,·)}^2+{\left(y_1^{\left[1\right]}(z,·)\right)}^2$ in the denominators of (41). Of course, if $z\in \mathbb{R}$, p is real-valued a.e. in $(a,b)$, and $y_1(z,·)$ is real-valued (as it typically is in the context of oscillation theory), then $y_1{(z,x^{\prime })}^2+{\left(y_1^{\left[1\right]}(z,x^{\prime })\right)}^2\ne 0$ in $(a,b)$.

• (ii) In the end, d’Alembert’s Formula (36) and Rofe-Beketov’s Formula (41) are equivalent in the sense that they differ only by a constant (possibly z-dependent) multiple of $y_1$. In fact, at a zero of $y_1(z,·)$ in $(a,b)$, a second-order singularity integrates to a first-order singularity that is multiplied by $y_1$ which has a first-order zero. Thus, a certain “renormalization” in d’Alembert’s Formula (36) is made explicit in Rofe-Beketov’s Formula (41). Moreover, (39) implies that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{d}{dx}}\left({\displaystyle \frac{y_2}{y_1}}\right)& ={\displaystyle \frac{d}{dx}}\left({\displaystyle \frac{p{y}_1^{\prime }/y_1}{y_1^2+{\left(p{y}_1^{\prime }\right)}^2}}\right)-{\displaystyle \frac{\left[p^{-1}+q-zr\right]\left[y_1^2-{\left(p{y}_1^{\prime }\right)}^2\right]}{y_1^2+{\left(p{y}_1^{\prime }\right)}^2}}\hfill \\ \hfill & ⋮\hfill \\ \hfill & =-{\displaystyle \frac{1}{p}}{\displaystyle \frac{W(y_2,y_1)}{y_1^2}}=-{\displaystyle \frac{1}{p{y}_1^2}},\hfill \end{array}\end{array}$$

(44)

illustrating the point just made.

• (iii) Dobrokhotov [14] obtains (41) for the Schrödinger equation $-y^{″}+qy=0$ as an application of the general result ([14], Theorem 1) for constructing a fundamental matrix of a $2n\times 2n$ first-order system when n skew-orthogonal solutions are known. In the context of canonical systems, we also refer to Rofe-Beketov [15] and Rofe-Beketov and Kholkin ([3], Sect. 6.2). See also Roitberg and Sakhnovich [16] for a discussion of general first-order systems in the context of d’Alembert’s formula.

The following result of Rofe-Beketov [1] rectifies this situation in general, yielding a nonzero denominator in the analog of (41):

Theorem 3.

Assume Hypothesis 1 with $s=0$ a.e. in $(a,b)$, let $z\in \mathbb{C}$, and assume that $0\ne y_1(z,·)\in {\mathcal{D}}_{\tau }\left((a,b)\right)$. In addition, suppose that $\tau y_1(z,·)=z{y}_1(z,·)$ (i.e., ${\left(p{y}_1^{\prime }\right)}^{\prime }=[q-zr]y_1$) a.e. in $(a,b)$. Then, Rofe-Beketov’s formula,

$$\begin{array}{cc}\hfill y_2(z,x,x_0)& =y_1(z,x){\int }_{x_0}^{x}d{x}^{\prime }\{\left[p{\left(x^{\prime }\right)}^{-1}+\overline{q\left(x^{\prime }\right)}-\overline{zr\left(x^{\prime }\right)}\right]{\left(\overline{y_1(z,x^{\prime })}\right)}^2\hfill \\ \hfill & -\left[{\overline{p\left(x^{\prime }\right)}}^{-1}+q\left(x^{\prime }\right)-zr\left(x^{\prime }\right)\right]{\left(\overline{y_1^{\left[1\right]}(z,x^{\prime })}\right)}^2\}\hfill \\ \hfill & \times {\left[|y_1(z,x^{\prime }){|}^2+|y_1^{\left[1\right]}(z,x^{\prime }){|}^2\right]}^{-2}\hfill \\ \hfill & -\overline{y_1^{\left[1\right]}(z,x)}{\left[|y_1{(z,x)|}^2+|y_1^{\left[1\right]}(z,x){|}^2\right]}^{-1},x,x_0\in (a,b),\hfill \end{array}$$

(45)

yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & y_2(z,·,x_0)\in {\mathcal{D}}_{\tau }\left((a,b)\right),\tau y_2(z,·,x_0)=z{y}_2(z,·,x_0),\hfill \\ \hfill & \mathit{and}W(y_2(z,·,x_0),y_1(z,·))=1.\hfill \end{array}\end{array}$$

(46)

In particular, the general solution $y(z,·)$ of $\tau y(z,·)=zy(z,·)$ is of the form

$$y(z,x)=C_1{y}_1(z,x)+C_2{y}_2(z,x,x_0),C_1,C_2\in \mathbb{C},x,x_0\in (a,b).$$

(47)

We postpone the proofs of Theorems 2 and 3 to Section 4, where more general four-coefficient analogs will be proved in Theorems 4 and 5.

4. The Extension of Rofe-Beketov’s Formula to the Case of Four-Coefficient Sturm–Liouville Equations

In our principal section, we now describe the extension of Rofe-Beketov’s formula to four-coefficient Sturm–Liouville equations.

We start by noting that Lemma 1 extends essentially verbatim to the present four-coefficient situation:

Lemma 3.

In addition to Hypothesis 1, let $z\in \mathbb{C}$, and assume that $y_1(z,·)\in {\mathcal{D}}_{\tau }\left((a,b)\right)$.

(i)

Suppose that $y_1(z,·)\ne 0$ in $(a,b)$ and $y_2(z,·)\in {\mathcal{D}}_{\tau }\left((a,b)\right)$. Then,

$${\left({\displaystyle \frac{y_2(z,·)}{y_1(z,·)}}\right)}^{\prime }={\displaystyle \frac{-1}{p{y}_1{(z,·)}^2}}\mathit{is}\mathit{equivalent}\mathit{to}W(y_2(z,·),y_1(z,·))=1.$$

(48)

(ii)

Suppose that $y_2(z,·)\in {\mathcal{D}}_{\tau }\left((a,b)\right)$ and $W(y_2(z,·),y_1(z,·))=1$. Then,

$$\tau y_1(z,·)=z{y}_1(z,·)\mathit{is}\mathit{equivalent}\mathit{to}\tau y_2(z,·)=z{y}_2(z,·),$$

(49)

that is, if $W(y_2(z,·),y_1(z,·))=1$, then

$$\begin{array}{cc}\hfill & {\left(y_1^{\left[1\right]}\right)}^{\prime }=s{y}_1^{\left[1\right]}+[q-zr]y_1\mathrm{a}.\mathrm{e}.\mathit{on}(a,b)\hfill \\ \hfill & \mathit{is}\mathit{equivalent}\mathit{to}\hfill \\ \hfill & {\left(y_2^{\left[1\right]}\right)}^{\prime }=s{y}_2^{\left[1\right]}+[q-zr]y_2\mathrm{a}.\mathrm{e}.\mathit{on}(a,b).\hfill \end{array}$$

(50)

(iii)

Suppose that $y_1(z,·)\ne 0$ in $(a,b)$ and $\tau y_1(z,·)=z{y}_1(z,·)$ (i.e., $(y_1^{\left[1\right]}{)}^{\prime }=s{y}_1^{\left[1\right]}+[q-zr]y_1$) a.e. in $(a,b)$. Then, d’Alembert’s formula extends to the present four-coefficient setting, that is,

$$y_2(z,x,x_0)=-y_1(z,x){\int }_{x_0}^{x}d{x}^{\prime }\frac{1}{p\left(x^{\prime }\right)y_1{(z,x^{\prime })}^2},x,x_0\in (a,b),$$

(51)

satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & y_2(z,·,x_0)\in {\mathcal{D}}_{\tau }\left((a,b)\right),\tau y_2(z,·,x_0)=z{y}_2(z,·,x_0),\hfill \\ \hfill & \mathit{and}W(y_2(z,·,x_0),y_1(z,·))=1.\hfill \end{array}\end{array}$$

(52)

In particular, the general solution $y(z,·)$ of $\tau y(z,·)=zy(z,·)$ a.e. in $(a,b)$ is of the form

$$\begin{array}{cc}\hfill y(z,x)& =C_1{y}_1(z,x)+C_2{y}_2(z,x,x_0)\hfill \\ \hfill & =C_1{y}_1(z,x)-C_2{y}_1(z,x){\int }_{x_0}^{x}d{x}^{\prime }\frac{1}{p\left(x^{\prime }\right)y_1{(z,x^{\prime })}^2},\hfill \\ \hfill & C_1,C_2\in \mathbb{C},x,x_0\in (a,b).\hfill \end{array}$$

(53)

Moreover, if p is real-valued a.e. in $(a,b)$ and also $y_1(z,·)$ is real-valued, so is $y_2(z,·,x_0)$. Finally, if for some (or all ) $x\in (a,b)$, $y_1(z,x)$ is analytic (resp., entire ) with respect to $z\in \mathrm{\Omega }$, $\mathrm{\Omega }\subseteq \mathbb{C}$ open, then so is $y_2(z,x,x_0)$.

We omit the proof of Lemma 3 as it is essentially that of Lemma 1, replacing ${\left(p{y}_j^{\prime }\right)}^{\prime }=[q-zr]y_j$ in Lemma 1 with ${\left(y_j^{\left[1\right]}\right)}^{\prime }=s{y}_j^{\left[1\right]}+[q-zr]y_j$, $j=1,2$.

The analog of Theorem 2 then reads as follows:

Theorem 4.

Assume Hypothesis 1, let $z\in \mathbb{C}$, and assume that $y_1(z,·)\in {\mathcal{D}}_{\tau }\left((a,b)\right)$. In addition, suppose that $y_1{(z,·)}^2+{\left(y_1^{\left[1\right]}(z,·)\right)}^2\ne 0$ in $(a,b)$ and $\tau y_1(z,·)=z{y}_1(z,·)$ (i.e., ${\left(p{y}_1^{\prime }\right)}^{\prime }=[q-zr]y_1$) a.e. in $(a,b)$. Then,

$$\begin{array}{cc}\hfill & y_2(z,x,x_0)\hfill \\ \hfill & =-y_1(z,x){\int }_{x_0}^{x}d{x}^{\prime }\{\left[p{\left(x^{\prime }\right)}^{-1}+q\left(x^{\prime }\right)-zr\left(x^{\prime }\right)\right]\left[y_1{(z,x^{\prime })}^2-{\left(y_1^{\left[1\right]}(z,x^{\prime })\right)}^2\right]\hfill \\ \hfill & +4s\left(x^{\prime }\right)y_1(z,x^{\prime })y_1^{\left[1\right]}(z,x^{\prime })\}{\left[y_1{(z,x^{\prime })}^2+{\left(y_1^{\left[1\right]}(z,x^{\prime })\right)}^2\right]}^{-2}\hfill \\ \hfill & +{\displaystyle \frac{y_1^{\left[1\right]}(z,x)}{y_1{(z,x)}^2+{\left(y_1^{\left[1\right]}(z,x)\right)}^2}},x,x_0\in (a,b),\hfill \end{array}$$

(54)

satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & y_2(z,·,x_0)\in {\mathcal{D}}_{\tau }\left((a,b)\right),\tau y_2(z,·,x_0)=z{y}_2(z,·,x_0),\hfill \\ \hfill & \mathit{and}W(y_2(z,·,x_0),y_1(z,·))=1.\hfill \end{array}\end{array}$$

(55)

In particular, the general solution $y(z,·)$ of $\tau y(z,·)=zy(z,·)$ is of the form

$$y(z,x)=C_1{y}_1(z,x)+C_2{y}_2(z,x,x_0),C_1,C_2\in \mathbb{C},x,x_0\in (a,b).$$

(56)

Moreover, if $z\in \mathbb{R}$, $p,q,r,s$ are real-valued a.e. in $(a,b)$, and $y_1(z,·)$ is real-valued, so is $y_2(z,·,x_0)$. Finally, if for some (or all ) $x\in (a,b)$, $y_1(z,x)$ is analytic (resp., entire ) with respect to $z\in \mathrm{\Omega }$, $\mathrm{\Omega }\subseteq \mathbb{C}$ open, then so is $y_2(z,x,x_0)$.

Proof. 

For notational convenience, we introduce the abbreviation

$$\begin{array}{cc}\hfill I_{x_0}(z,x)& ={\int }_{x_0}^{x}d{x}^{\prime }\{\left[p{\left(x^{\prime }\right)}^{-1}+q\left(x^{\prime }\right)-zr\left(x^{\prime }\right)\right]\left[y_1{(z,x^{\prime })}^2-{\left(y_1^{\left[1\right]}(z,x^{\prime })\right)}^2\right]\hfill \\ \hfill & +4s\left(x^{\prime }\right)y_1(z,x^{\prime })y_1^{\left[1\right]}(z,x^{\prime })\}{\left[y_1{(z,x^{\prime })}^2+{\left(y_1^{\left[1\right]}(z,x^{\prime })\right)}^2\right]}^{-2},x\in (a,b),\hfill \end{array}$$

(57)

and for brevity, we will suppress the z and x dependencies of $I_{x_0}(z,x)$ and $y_j(z,x)$, $j\in \{1,2\}$ and simply write $I_{x_0}$ and $y_j$, $j\in \{1,2\}$, throughout the proof. Since $y_1$ is a solution of (9) in $(a,b)$,

$${\left(y_1^{\left[1\right]}\right)}^{\prime }=(q-zr)y_1+s{y}_1^{\left[1\right]}\mathrm{a}.\mathrm{e}.\mathrm{on}(a,b).$$

(58)

Differentiating (54) and making use of (58), one computes

$$\begin{array}{cc}\hfill y_2^{\prime }& =-y_1^{\prime }{I}_{x_0}+{\left[y_1^2+{\left(y_1^{\left[1\right]}\right)}^2\right]}^{-2}\{\left[y_1^2-{\left(y_1^{\left[1\right]}\right)}^2\right]\left[s{y}_1^{\left[1\right]}-p^{-1}{y}_1\right]\hfill \\ \hfill & -2y_1{y}_1^{\prime }{y}_1^{\left[1\right]}-4s{y}_1^2{y}_1^{\left[1\right]}\}\hfill \\ \hfill & =-y_1^{\prime }{I}_{x_0}+{\left[y_1^2+{\left(y_1^{\left[1\right]}\right)}^2\right]}^{-2}\left\{-s{y}_1^2{y}_1^{\left[1\right]}-p^{-1}{y}_1^3-s{\left(y_1^{\left[1\right]}\right)}^3-p^{-1}{y}_1{\left(y_1^{\left[1\right]}\right)}^2\right\}\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill & =-y_1^{\prime }{I}_{x_0}-\left[s{y}_1^{\left[1\right]}+p^{-1}{y}_1\right]{\left[y_1^2+{\left(y_1^{\left[1\right]}\right)}^2\right]}^{-1}.\hfill \end{array}$$

(60)

Therefore,

$$y_2^{\left[1\right]}=-y_1^{\left[1\right]}{I}_{x_0}-y_1{\left[y_1^2+{\left(y_1^{\left[1\right]}\right)}^2\right]}^{-1},$$

(61)

which also implies that $y_2^{\left[1\right]}\in A{C}_{loc}\left((a,b)\right)$. Furthermore, differentiating (61) and using (58), one obtains

$$\begin{array}{cc}\hfill {\left(y_2^{\left[1\right]}\right)}^{\prime }& =-\left[(q-zr)y_1+s{y}_1^{\left[1\right]}\right]I_{x_0}\hfill \\ \hfill & +{\left[y_1^2+{\left(y_1^{\left[1\right]}\right)}^2\right]}^{-2}\{-p^{-1}{y}_1^2{y}_1^{\left[1\right]}-(q-zr)y_1^2{y}_1^{\left[1\right]}+\left[p^{-1}+q-zr\right]{\left(y_1^{\left[1\right]}\right)}^3\hfill \\ \hfill & -2s{y}_1{\left(y_1^{\left[1\right]}\right)}^2-y_1^{\prime }{y}_1^2-y_1^{\prime }{\left(y_1^{\left[1\right]}\right)}^2+2y_1^2{y}_1^{\prime }+2(q-zr)y_1^2{y}_1^{\left[1\right]}\}\hfill \\ \hfill & =-\left[(q-zr)y_1+s{y}_1^{\left[1\right]}\right]I_{x_0}\hfill \\ \hfill & +{\left[y_1^2+{\left(y_1^{\left[1\right]}\right)}^2\right]}^{-2}\left\{(q-zr){\left(y_1^{\left[1\right]}\right)}^3-s{y}_1{\left(y_1^{\left[1\right]}\right)}^2-s{y}_1^3+(q-zr)y_1^2{y}_1^{\left[1\right]}\right\}\hfill \\ \hfill & =-\left[(q-zr)y_1+s{y}_1^{\left[1\right]}\right]I_{x_0}+{\left[y_1^2+{\left(y_1^{\left[1\right]}\right)}^2\right]}^{-1}\left\{(q-zr)y_1^{\left[1\right]}-s{y}_1\right\}.\hfill \end{array}$$

(62)

Combining (61) with (62) and taking the obvious cancellations into account, one obtains

$$\begin{array}{cc}\hfill -{\left(y_2^{\left[1\right]}\right)}^{\prime }+s{y}_2^{\left[1\right]}& =-(q-zr)\left\{-y_1{I}_{x_0}+y_1^{\left[1\right]}{\left[y_1^2+{\left(y_1^{\left[1\right]}\right)}^2\right]}^{-1}\right\}\hfill \\ \hfill & =-(q-zr)y_2\mathrm{a}.\mathrm{e}.\mathrm{on}(a,b).\hfill \end{array}$$

(63)

Hence, $y_2$ is a solution of (9) in $(a,b)$. By (54), one computes

$$\begin{array}{cc}\hfill W(y_2,y_1)\left(x_0\right)& =y_2(z,x_0)y_1^{\left[1\right]}(z,x_0)-y_2^{\left[1\right]}(z,x_0)y_1(z,x_0)\hfill \\ \hfill & ={\displaystyle \frac{{\left(y_1^{\left[1\right]}(z,x_0)\right)}^2}{y_1{\left(x_0\right)}^2+{\left(y_1^{\left[1\right]}(z,x_0)\right)}^2}}+{\displaystyle \frac{y_1{(z,x_0)}^2}{y_1{\left(x_0\right)}^2+{\left(y_1^{\left[1\right]}(z,x_0)\right)}^2}}=1.\hfill \end{array}$$

(64)

The constancy of the Wronskian of solutions (cf. Remark 1) implies that $W(y_2,y_1)=1$.

The real-valuedness of $y_2(z,·)$ if $z\in \mathbb{R}$, $p,q,r,s$ are real-valued a.e. in $(a,b)$ and $y_1(z,·)$ is real-valued is clear from (54).

Finally, if $y_1(z,·)$ is analytic (resp., entire ) in some open set $\mathrm{\Omega }\subseteq \mathbb{C}$, then we next show that $y_1^{\left[1\right]}(z,·)$ is also analytic (resp., entire ) in $\mathrm{\Omega }\subseteq \mathbb{C}$, and, hence, so is $y_2(z,·)$ by (54). To show that the analyticity of $y_1(z,·)$ for $z\in \mathrm{\Omega }$ implies that of $y_1^{\left[1\right]}(z,·)$, one can argue as follows: clearly, $\left(\tau y_1(z,·)\right)\left(x\right)=z{y}_1(z,x)$; equivalently,

$${\left(y_1^{\left[1\right]}(z,x)\right)}^{\prime }-s\left(x\right)y_1^{\left[1\right]}(z,x)=[q\left(x\right)-zr\left(x\right)]y_1(z,x),z\in \mathrm{\Omega },\mathrm{a}.\mathrm{e}.x,x_0\in (a,b),$$

(65)

implies that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill y_1^{\left[1\right]}(z,x)& =\left[y_1^{\left[1\right]}(z,x_0)+{\int }_{x_0}^{x}d{x}^{\prime }[q\left(x^{\prime }\right)-zr\left(x^{\prime }\right)]y_1(z,x^{\prime })e^{-{\int }_{x_0}^{x^{\prime }}d{x}^{\prime \prime }s\left(x^{″}\right)}\right]\hfill \\ \hfill & \times e^{{\int }_{x_0}^{x}d{x}^{\prime }s\left(x^{\prime }\right)},z\in \mathrm{\Omega },\mathrm{a}.\mathrm{e}.x,x_0\in (a,b).\hfill \end{array}\end{array}$$

(66)

Hence, if for all $x\in (a,b)$, $y_1(z,x)$ is analytic with respect to $z\in \mathrm{\Omega }$, then the expression

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill Y(z,x,x_0)=y_1^{\prime }(z,x)-[p\left(x_0\right)/p\left(x\right)]y_1^{\prime }(z,x_0)e^{{\int }_{x_0}^{x}d{x}^{\prime }s\left(x^{\prime }\right)}& \\ \hfill \mathrm{for}\mathrm{a}.\mathrm{e}.x,x_0\in (a,b),\end{array}\end{array}$$

(67)

is also analytic with respect to $z\in \mathrm{\Omega }$. Integrating $Y(z,x,x_0)$ with respect to x yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill y_1(z,x)-y_1(z,x_0)-p\left(x_0\right)y_1^{\prime }(z,x_0){\int }_{x_0}^{x}d{x}^{\prime }p{\left(x^{\prime }\right)}^{-1}{e}^{{\int }_{x_0}^{x^{\prime }}d{x}^{″}s\left(x^{″}\right)}& \\ \hfill \mathrm{for}\mathrm{a}.\mathrm{e}.x,x_0\in (a,b),\end{array}\end{array}$$

(68)

and, thus,

$$p\left(x_0\right)y_1^{\prime }(z,x_0)\mathrm{for}\mathrm{a}.\mathrm{e}.x_0\in (a,b),$$

(69)

as well as

$$p\left(x_0\right)[y_1^{\prime }(z,x_0)+s\left(x_0\right)y_1(z,x_0)]\mathrm{for}\mathrm{all}{x}_0\in (a,b),$$

(70)

are also analytic for $z\in \mathrm{\Omega }$. Since $x_0\in (a,b)$ is arbitrary, this implies that for all $x\in (a,b)$, $y_1^{\left[1\right]}(z,x)$ is analytic for $z\in \mathrm{\Omega }$. □

Remark 4.

To the best of our knowledge, (54) is the first instance in which the extension of a result from three-coefficient to four-coefficient Sturm–Liouville equations yields an explicit dependence on the fourth coefficient s (as opposed to the s-dependence simply being encoded only into the quasi-derivative) in the statement of the result.

Finally, the analog of Theorem 3 now reads as follows:

Theorem 5.

Assume Hypothesis 1, let $z\in \mathbb{C}$, and assume that $0\ne y_1(z,·)\in {\mathcal{D}}_{\tau }\left((a,b)\right)$. In addition, suppose that $\tau y_1(z,·)=z{y}_1(z,·)$ (i.e., ${\left(p{y}_1^{\prime }\right)}^{\prime }=[q-zr]y_1$) a.e. in $(a,b)$. Then,

$$\begin{array}{cc}\hfill y_2(z,x,x_0)& =-y_1(z,x){\int }_{x_0}^{x}d{x}^{\prime }\{\left[p{\left(x^{\prime }\right)}^{-1}+\overline{q\left(x^{\prime }\right)}-\overline{z}\overline{r\left(x^{\prime }\right)}\right]{\left(y_1(z,x^{\prime })\right)}^2\hfill \\ \hfill & +2\left[s\left(x^{\prime }\right)+\overline{s\left(x^{\prime }\right)}\right]\overline{y_1(z,x^{\prime })}\overline{y_1^{\left[1\right]}(z,x^{\prime })}\hfill \\ \hfill & -\left[{\overline{p\left(x^{\prime }\right)}}^{-1}+q\left(x^{\prime }\right)-zr\left(x^{\prime }\right)\right]{\left(y_1^{\left[1\right]}(z,x^{\prime })\right)}^2\}\hfill \\ \hfill & \times {\left[|y_1(z,x^{\prime }){|}^2+|y_1^{\left[1\right]}(z,x^{\prime }){|}^2\right]}^{-2}\hfill \\ \hfill & +{\displaystyle \frac{\overline{y_1^{\left[1\right]}(z,x)}}{|y_1{(z,x)|}^2+|y_1^{\left[1\right]}(z,x){|}^2}},x,x_0\in (a,b),\hfill \end{array}$$

(71)

satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & y_2(z,·,x_0)\in {\mathcal{D}}_{\tau }\left((a,b)\right),\tau y_2(z,·,x_0)=z{y}_2(z,·,x_0),\hfill \\ \hfill & \mathit{and}W(y_2(z,·,x_0),y_1(z,·))=1.\hfill \end{array}\end{array}$$

(72)

In particular, the general solution $y(z,·)$ of $\tau y(z,·)=zy(z,·)$ is of the form

$$y(z,x)=C_1{y}_1(z,x)+C_2{y}_2(z,x,x_0),C_1,C_2\in \mathbb{C},x,x_0\in (a,b).$$

(73)

Proof. 

For notational convenience, we introduce the abbreviation

$$\begin{array}{cc}\hfill J_{x_0}(z,x)& ={\int }_{x_0}^{x}d{x}^{\prime }\{\left[p{\left(x^{\prime }\right)}^{-1}+\overline{q\left(x^{\prime }\right)}-\overline{z}\overline{r\left(x^{\prime }\right)}\right]{\left(y_1(z,x^{\prime })\right)}^2\hfill \\ \hfill & +2\left[s\left(x^{\prime }\right)+\overline{s\left(x^{\prime }\right)}\right]\overline{y_1(z,x^{\prime })}\overline{y_1^{\left[1\right]}(z,x^{\prime })}\hfill \\ \hfill & -\left[{\overline{p\left(x^{\prime }\right)}}^{-1}+q\left(x^{\prime }\right)-zr\left(x^{\prime }\right)\right]{\left(y_1^{\left[1\right]}(z,x^{\prime })\right)}^2\}\hfill \\ \hfill & \times {\left[|y_1(z,x^{\prime }){|}^2+|y_1^{\left[1\right]}(z,x^{\prime }){|}^2\right]}^{-2},x,x_0\in (a,b).\hfill \end{array}$$

(74)

In lieu of directly verifying that (71) is a solution of (9) (as is performed with (54) in the proof of Theorem 4), we will instead show how the right-hand side of (71) is constructed so as to be a solution of (9) in $(a,b)$. This approach illustrates how formulas such as (54) and (71) arise. (Directly verifying that the right-hand side of (71) is a solution of (9) is entirely analogous to the proof of Theorem 4.) For brevity, we will suppress the z and x dependencies of $J_{x_0}(z,x)$ and $y_j(z,x)$, $j\in \{1,2\}$ and simply write $J_{x_0}$ and $y_j$, $j\in \{1,2\}$, throughout the proof.

Equation (9) is equivalent to the system

$$\left(\begin{array}{c}f\\ f^{\left[1\right]}\end{array}\right)=\left(\begin{array}{cc}-s& p^{-1}\\ q-zr& s\end{array}\right)\left(\begin{array}{c}f\\ f^{\left[1\right]}\end{array}\right).$$

(75)

We look for a second solution, $Y_2=Y_2(z,·)$ (the reasoning behind the capital “Y” notation will become clear in the last step of the proof), in the form

$$\left(\begin{array}{c}{Y}_2\\ Y_2^{\left[1\right]}\end{array}\right)=c_1\left(\begin{array}{c}{y}_1\\ y_1^{\left[1\right]}\end{array}\right)+c_2\left(\begin{array}{c}-\overline{y_1^{\left[1\right]}}\\ \overline{y_1}\end{array}\right),$$

(76)

where the scalar-valued functions $c_j=c_j(z,·)$, $j\in \{1,2\}$, are to be determined subject to the condition

$$W(Y_2,y_1)\left(x\right)=1,x\in (a,b).$$

(77)

The condition in (77) requires

$$c_2=-{\left[|y_1{|}^2+|y_1^{\left[1\right]}{|}^2\right]}^{-1}.$$

(78)

Furthermore, differentiating (77) and using the fact that $y_1$ is a solution of (9), one obtains

$$\begin{array}{c}\hfill {\left(Y_2^{\left[1\right]}\right)}^{\prime }{y}_1=\left\{[q-zr]Y_2+s{Y}_2^{\left[1\right]}\right\}{y}_1.\end{array}$$

(79)

As a consequence,

$$Y_2^{\left[1\right]}=[q-zr]Y_2+s{Y}_2^{\left[1\right]},$$

(80)

and it follows that $Y_2$ satisfies the equation that results from equating the second components in (76), independent of the choice for $c_1$. Thus, in order for $Y_2$ to solve (76), we need to find $c_1$ so that $Y_2$ satisfies the equation that results from equating the first components in (76). That is, we need to find $c_1$ so that

$$Y_2^{\prime }=p^{-1}{Y}_2^{\left[1\right]}-s{Y}_2.$$

(81)

Using (75), the equation in (81) is equivalent to

$$\begin{array}{c}\hfill Y_2^{\prime }=p^{-1}\left(c_1{y}_1^{\left[1\right]}+c_2\overline{y_1}\right)-c_{1}s{y}_1+c_{2}s\overline{y_1^{\left[1\right]}};\end{array}$$

(82)

that is,

$$c_1^{\prime }{y}_1+c_1{y}_1^{\prime }-c_2^{\prime }\overline{y_1^{\left[1\right]}}-c_2{\left(\overline{y_1^{\left[1\right]}}\right)}^{\prime }=p^{-1}\left(c_1{y}_1^{\left[1\right]}+c_2\overline{y_1}\right)-c_{1}s{y}_1+c_{2}s\overline{y_1^{\left[1\right]}}.$$

(83)

Since

$$y_1^{\prime }=p^{-1}{y}_1^{\left[1\right]}-s{y}_1,$$

(84)

(83) reduces to

$$c_1^{\prime }{y}_1=c_2\left\{p^{-1}\overline{y_1}+{\left(\overline{y_1^{\left[1\right]}}\right)}^{\prime }+s\overline{y_1^{\left[1\right]}}\right\}+c_2^{\prime }\overline{y_1^{\left[1\right]}}.$$

(85)

By (78) and (84),

$$\begin{array}{cc}\hfill c_2^{\prime }& =c_2^2\left[y_1^{\prime }\overline{y_1}+y_1{\left(\overline{y_1}\right)}^{\prime }+{\left(y_1^{\left[1\right]}\right)}^{\prime }\overline{y_1^{\left[1\right]}}+y_1^{\left[1\right]}{\left(\overline{y_1^{\left[1\right]}}\right)}^{\prime }\right]\hfill \\ \hfill & =c_2^2\left[p^{-1}\overline{y_1}+{\left(\overline{y_1^{\left[1\right]}}\right)}^{\prime }\right]y_1^{\left[1\right]}+c_2^2\left[{\overline{p}}^{-1}{y}_1+{\left(y_1^{\left[1\right]}\right)}^{\prime }\right]\overline{y_1^{\left[1\right]}}-c_2^2\left[s+\overline{s}\right]{\left|y_1\right|}^2.\hfill \end{array}$$

(86)

Equations (85) and (86) combine to yield

$$\begin{array}{cc}\hfill c_1^{\prime }{y}_1& =c_2^2\{-\left(|y_1{|}^2+|y_1^{\left[1\right]}{|}^2\right)\left[p^{-1}\overline{y_1}+{\left(\overline{y_1^{\left[1\right]}}\right)}^{\prime }+s\overline{y_1^{\left[1\right]}}\right]+\left[p^{-1}\overline{y_1}+{\left(\overline{y_1^{\left[1\right]}}\right)}^{\prime }\right]|y_1^{\left[1\right]}{|}^2\hfill \\ \hfill & +\left[{\overline{p}}^{-1}{y}_1+{\left(y_1^{\left[1\right]}\right)}^{\prime }\right]{\left(\overline{y_1^{\left[1\right]}}\right)}^2-\left[s+\overline{s}\right]\overline{y_1^{\left[1\right]}}\}\hfill \\ \hfill & =c_2^2\{-\left[p^{-1}\overline{y_1}+{\left(\overline{y_1^{\left[1\right]}}\right)}^{\prime }+s\overline{y_1^{\left[1\right]}}+\left[s+\overline{s}\right]\overline{y_1^{\left[1\right]}}\right]{\left|y_1\right|}^2-s\overline{y_1^{\left[1\right]}}|y_1^{\left[1\right]}{|}^2\hfill \\ \hfill & +\left[{\overline{p}}^{-1}{y}_1+[q-zr]y_1+s{y}_1^{\left[1\right]}\right]{\left(\overline{y_1^{\left[1\right]}}\right)}^2\}\hfill \\ \hfill & =c_2^2{y}_1\left\{-\left[p^{-1}+\overline{q}-\overline{z}\overline{r}\right]{\left(\overline{y_1}\right)}^2-2\left[s+\overline{s}\right]\overline{y_1^{\left[1\right]}}\overline{y_1}+\left[{\overline{p}}^{-1}+q-zr\right]{\left(\overline{y_1^{\left[1\right]}}\right)}^2\right\}.\hfill \end{array}$$

(87)

Therefore,

$$c_1^{\prime }=c_2^2\left\{-\left[p^{-1}+\overline{q}-\overline{z}\overline{r}\right]{\left(\overline{y_1}\right)}^2-2\left[s+\overline{s}\right]\overline{y_1^{\left[1\right]}}\overline{y_1}+\left[{\overline{p}}^{-1}+q-zr\right]{\left(\overline{y_1^{\left[1\right]}}\right)}^2\right\},$$

(88)

and upon integrating from $x_0$ to x, one obtains

$$c_1=c_1\left(x_0\right)+J_{x_0}.$$

(89)

In turn, (76) then implies that

$$Y_2=c_1\left(x_0\right)y_1+y_1{J}_{x_0}-c_2\overline{y_1^{\left[1\right]}}.$$

(90)

Solutions of (9) form a subspace, and since $W(c_1\left(x_0\right)y_1,y_1)\equiv 0$, the $c_1\left(x_0\right)y_1$ term may be removed from (90) without affecting the value of the Wronskian. Hence, one chooses

$$y_2=Y_2-c_1\left(x_0\right)y_1=y_1{J}_{x_0}-c_2\overline{y_1^{\left[1\right]}},$$

(91)

which is (71). □

Under certain additional conditions in $p,q,r,s$ (for instance, if $\tau$ is regular at a and/or b), then $x_0=a$ and/or $x_0=b$ are permissible in (54) and (71).