The Estimates L₁-L_∞ for the Reduced Radial Equation of Schrödinger

Herminio Blancarte

Facultad de Ingeniera, Universidad Autónoma de Querétaro, Centro Universitario, Cerro de las Campanas s/n, Col. Las Campanas, Santiago de Querétaro, México.

DOI: 10.4236/apm.2019.95023                

Abstract

Estimates of the type L₁-L_∞ for the Schrödinger Equation on the Line and on Half-Line with a regular potential V(x), express the dispersive nature of the Schrödinger Equation and are the essential elements in the study of the problems of initial values, the asymptotic times for large solutions and Scattering Theory for the Schrödinger equation and non-linear in general; for other equations of Non-linear Evolution. In general, the estimates L_p-L_p' express the dispersive nature of this equation. And its study plays an important role in problems of non-linear initial values; likewise, in the study of problems nonlinear initial values; see [1] [2] [3]. On the other hand, following a series of problems proposed by V. Marchenko [4], that we will name Marchenko’s formulation, and relate it to a generalized version of Theorem 1 given in [1], the main theorem (Theorem 1) of this article provides a transformation operator W that transforms the Reduced Radial Schrödinger Equation (RRSE) (whose main characteristic is the addition a singular term of quadratic order to a regular potential V(x)) in the Schrödinger Equation on Half-Line (RSEHL) under W. That is to say; W eliminates the singular term of quadratic order of potential V(x) in the asymptotic development towards zero and adds to the potential V(x) a bounded term and a term exponentially decrease fast enough in the asymptotic development towards infinity, which continues guaranteeing the uniqueness of the potential V(x) in the condition of the infinity boundary. Then the L₁-L_∞ estimates for the (RRSE) are preserved under the transformation operator , as in the case of (RSEHL) where they were established in [3]. Finally, as an open question, the possibility of extending the L₁-L_∞ estimates for the case (RSEHL), where added to the potential V(x) an analytical perturbation is mentioned.

Keywords

The Schrödinger Equation on the Half-Line, Reduced Radial Equation of Schrödinger, Conditions Sufficient to Establish the Uniqueness of the Potential and Boundary Conditions Are Named the Generalized Theorem 1, The Marchenko’s Formulation, Reduction of Estimates L₁-L_∞ for the Reduced Radial Equation of Schrôdinger to Equation on Half-Line

1. Introduction

The inverse problem addressed in [1] , Theorem 1, pages 481-483, where the uniqueness of the potential $q_2\left(x\right)$ and the condition of value boundary:

$y^{\prime }\left(0\right)-h_{2}y\left(0\right)=0,$ (1)

were established, for the second problem of boundary value which will be named simply Problem 2 (similarly the first problem of boundary value will be named Problem 1: with hypotheses: $0<x<a$, the boundary condition:

$y^{\prime }\left(0\right)-h_{1}y\left(0\right)=0,$ (2)

and the potential $q_1\left(x\right)$ is given in the Problem 1.

Similarly now assuming that distribution spectral function R extends when $a=\infty$, see [4] , Chapter 2, Section 3, page 153, Problem 6 A and the boundary condition

$y\left(0\right)=0,$ (3)

in the Problem 1, that is

$-y^{″}\left(x\right)+q_1\left(x\right)y\left(x\right)=s^{2}y\left(x\right),0<x<\infty ,s\in \left\{\lambda ,\mu :\lambda ,\mu \in ℝ\right\}$ (4)

$y\left(0\right)=0.$

Then the uniqueness of the potential $q_2\left(x\right)$ is established for the Problem 2

$-y^{″}\left(x\right)+q_2\left(x\right)y\left(x\right)=s^{2}y\left(x\right),0<x<\infty ,s\in \left\{\lambda ,\mu :\lambda ,\mu \in ℝ\right\}$ (5)

$y\left(0\right)=0.$

Conditions Sufficient to establish the uniqueness of the potential and boundary conditions in Theorem 1, established above will be named Generalized Theorem 1.

Now if we name Problem 1 for $i\in \left\{\mathrm{1,2}\right\}$ to the boundary value problem:

$-y^{″}\left(x\right)+q_i\left(x\right)y\left(x\right)=s^{2}y\left(x\right),0<x<\infty ,$ (6)

$y_i\left(0\right)=0.$

and the solutions ${\omega }_i\left(s\mathrm{,}x\right)$, respective for $s\in \left\{\lambda \mathrm{,}\mu \mathrm{<mo>\; :\; </mo>}\lambda \mathrm{,}\mu \in ℝ\right\}$. We obtain de reformulation of results in terms of the Transformation Operators.

The final conclusion of [1] , formula (42), Thm 1, page 483, Section 4,

$\begin{array}{l}{\omega }_2\left(\lambda ,x\right)={\omega }_1\left(\lambda ,x\right)-\frac{c{\omega }_1\left(\mu ,x\right)}{1+c{\displaystyle {\int }_0^x{\omega }_1{\left(\mu ,t\right)}^2\text{d}t}}{\displaystyle {\int }_0^x{\omega }_1\left(\mu ,t\right){\omega }_1\left(\lambda ,t\right)\text{d}t},\hfill \\ h_2={{\omega }^{\prime }}_2\left(\lambda ,0\right)=h_1-c,\hfill \\ q_2=\frac{1}{2}{q}_1-c{\left\{\frac{{\omega }_1\left(\mu ,x\right)}{1+c{\displaystyle {\int }_0^x{\omega }_1{\left(\mu ,t\right)}^2\text{d}t}}\right\}}^{\prime }.\hfill \end{array}\}$ (7)

and Generalized Theorem 1 can rewrite in a formula for a solution ${\omega }_2\left(\lambda \mathrm{,}x\right)$ of the Problem 2, for $s=\lambda$ of the following manner

${\omega }_2\left(\lambda ,x\right)={\omega }_1\left(\lambda ,x\right)+\frac{c{\omega }_2\left(\mu ,x\right)W\left({\omega }_1\left(\mu ,x\right),{\omega }_1\left(\lambda ,x\right)\right)}{{\mu }^2-{\lambda }^2},$ (8)

where ${\omega }_1\left(\mu \mathrm{,}x\right)\mathrm{,}{\omega }_1\left(\lambda \mathrm{,}x\right)$ are solutions of the Problem 1, for $s\in \left\{\lambda \mathrm{,}\mu \right\}$ respectively. The Wronskian

$W\left({\omega }_1\left(\mu ,x\right),{\omega }_1\left(\lambda ,x\right)\right):={{\omega }^{\prime }}_1\left(\mu ,x\right){\omega }_1\left(\lambda ,x\right)-{\omega }_1\left(\mu ,x\right){{\omega }^{\prime }}_1\left(\lambda ,x\right)\ne 0,\text{'}:=\frac{\text{d}}{\text{d}x}.$

Now in terms of the transformation operators. Let us define

$\mathbb{I}f\mathrm{<mo>\; :\; </mo>}=f\mathrm{,}\text{and}\mathbb{W}f\mathrm{<mo>\; :\; </mo>}=\frac{c{\omega }_2\left(\mu \mathrm{,}x\right)W\left({\omega }_1\left(\mu \mathrm{,}x\right)\mathrm{,}f\right)}{{\mu }^2-{\lambda }^2}\mathrm{,}\mu \ne \lambda \mathrm{.}$ (9)

If ${\omega }_2\left(\mu \mathrm{,}x\right)$ is the solution of the Problem 2, for $s=\mu$. For ${\omega }_2\left(\mu \mathrm{,}x\right)$ solution know of the Problem 2, ${\omega }_1\left(\mu \mathrm{,}x\right)$ denote a fixed solution of Problem 1, in both problems: fixed $s:=\mu$ then the equation above suggests the following transformation operator

${\omega }_2\left(\lambda ,x\right)=\left(\mathbb{I}+\mathbb{W}\right){\omega }_1\left(\lambda ,x\right),$ (10)

with the property that a solution ${\omega }_1\left(\lambda \mathrm{,}x\right)$ for the Problem 1 is transformed in a solution ${\omega }_2\left(\lambda \mathrm{,}x\right)$ for the Problem 2, for $s:=\lambda$.

In a complementary way to our approach and following a series of problems proposed by [4] , Chapter 2, Section 3, Problems: 4, 5, 6. Pages 149-153 that the formulation of Marchenko will be named. One get the Theorem 1, which is formulated in terms of the transformation operators as follows: if we assumed that $y_1\left(x\mathrm{,}\mu \right)$ denote a fixed solution de Problem 1 for $s:=\mu$, ${\phi }_1\left(\lambda \mathrm{,}x\right)$ denota an arbitrary solution of Problem 1 and $y_1\left(x\mathrm{,}\mu \right)\ne 0$ on $\left(\mathrm{0,}\infty \right)$, then defined us the operator

$\mathbb{W}:=\frac{W\left(y_1,\cdot \right)}{y_1\left({\mu }^2-{\lambda }^2\right)}\Rightarrow \mathbb{W}{\phi }_1\left(\lambda ,x\right)=\frac{W\left(y_1\left(x,\mu \right),{\phi }_1\left(\lambda ,x\right)\right)}{y_1\left(x,\mu \right)\left({\mu }^2-{\lambda }^2\right)}:={\phi }_2\left(\lambda ,x\right)$ (11)

where ${\phi }_2\left(\lambda \mathrm{,}x\right)$ is the solution of the equation:

$-{{\phi }^{″}}_2\left(\lambda ,x\right)+q_2\left(x\right){\phi }_2\left(\lambda ,x\right)={\lambda }^2{\phi }_2\left(\lambda ,x\right),0<x<\infty ,$ (12)

${\phi }_2\left(\lambda ,0\right)=0.$

and

$q_2\left(x\right):=-q_1-\frac{{y^{″}}_1}{y_1}+\frac{{{\phi }^{\prime }}_1}{{\phi }_2}+\frac{{y^{\prime }}_1}{y_1^2}.$ (13)

Two Examples for Theorem 1 are proposed, in the context of the previous Marchenko’s formulation. Example 1 for the Problem 1 corresponds to Schrödinger Equation on the Half-Line with a regular potential $q\left(x\right):=V\left(x\right)$, that is,

${\displaystyle {\int }_0^{\infty }}x\left|q\left(x\right)\right|\text{d}x<\infty ,$ (14)

which will be denoted by the abbreviation (RSEHL) to which we have added the singular term $\frac{l\left(l+1\right)}{x^2}$, obtain us the named Reduced Radial Schrödinger Equation (RRSE)

$-\frac{{\text{d}}^2}{\text{d}{r}^2}{\psi }_l\left(k,r\right)+\left[q\left(r\right)+\frac{l\left(l+1\right)}{r^2}\right]{\psi }_l\left(k,r\right)=k^2{\psi }_l\left(k,r\right),r>0,$ (15)

${\psi }_l\left(k,0\right)=0.$

for the partial wave of angular momentum l and wave number k. See [5] , Chapter 1, Sections 1.1-1.3, 1.5, pages 1-10, 13-16. And it will correspond to the Example 2 for the Problem 2 the (RSEHL)

$H=-\frac{{\text{d}}^2}{\text{d}{x}^2}+V\left(x\right),0<x<\infty ,$ (16)

$\varphi \left(0\right)=0.$

The two proposed examples for the above for Theorem 1 are constructed by using the regular solution ${\phi }_l\left(s\mathrm{,}x\right)$ and the Jost solution $f_l\left(s\mathrm{,}x\right)$ for (RRSQ) and its properties, the pair of solutions $y_1\left(\mu \mathrm{,}x\right)$ and ${\phi }_1\left(\lambda \mathrm{,}x\right)$ for the parameters $s=\mu$ and $s=\lambda$ will be built and the asymptotic developments and estimates are made in neighborhoods of zero and infinity respectively. The Wronskian $W\left(y_1\left(\mu \mathrm{,}x\right)\mathrm{,}{\phi }_1\left(\lambda \mathrm{,}x\right)\right)$ and the solution ${\phi }_2\left(\lambda ,x\right):=\mathbb{W}{\phi }_1\left(\lambda ,x\right)$ will be expressed in terms of the function of Jost $F_l\left(k\right),k=\mu ,\lambda$ and its properties. See [5] , Chapter 1, Sections 1.4-1.5, pages 11-16. Subsequently, the asymptotic developments and estimates are made in neighborhoods of zero and infinity for the terms

$\frac{{y^{″}}_1}{y_1}\mathrm{,}$ (17)

$\frac{{{\phi }^{\prime }}_1}{{\phi }_2}\mathrm{,}$ (18)

and

$\frac{{y^{\prime }}_1}{y_1^2}\mathrm{,}$ (19)

of potential $q_2\left(x\right)$. When performing the previous calculation using the sum of the respective asymptotic developments, the elimination of the singular term of the potencial $q_1\left(x\right)$ is obtained in the case of the neighborhood of zero and bounded terms and exponentially decrease fast enought are added for the potential $q_1\left(x\right)$ in a neighborhood of infinity.

The elimination of the singular term $\frac{l\left(l+1\right)}{x^2}$ when $x\to 0$ and the addition of an exponential decay term and a bounded term when $x\to \infty$ of the potential $q_1\left(x\right)$ in the Example 1, when applying the transformation operator $\mathbb{W}$ transforms the potential $q_1\left(x\right)$ into the regular potential $q_2\left(x\right):=q\left(x\right)=\mathbb{W}{q}_1$ in the Problem 2, as a consequence of the potential obtained

$q_2\left(x\right):=-q_1-\frac{{y^{″}}_1}{y_1}+\frac{{{\phi }^{\prime }}_1}{{\phi }_2}+\frac{{y^{\prime }}_1}{y_1^2}$

in Theorem 1. Where it is used strongly the fundamental properties of preserving the initial data $y_1\left(0\right)=0$ in Problem 1, as well as, the preservation of spectral and scattering properties in both Problems 1 and 2, by the operator $\mathbb{W}$. See [1] , Section 6, pages 486-487.

The previous fact, constitutes a fundamental part of the previous Marchenko’s formulation and gives the possibility of extending the $L^1-L^{\infty }$ Estimates for the (RSEHL) established in [6] was proved with the potential $q\left(x\right):=V\left(x\right)$ to be real and regular (RSEHL) now for the Reduced Radial Schrödinger Equation (RRSE). That is, the RRSE is transformed into the equation RSEHL through the transformation operator $\mathbb{W}$. And the following generalization of Theorem 2.1 of [6] is obtained.

Theorem 2. (The $L^1-L^{\infty }$ estimate). Supponse that V is regular and

$H=-\frac{{\text{d}}^2}{\text{d}{x}^2}+V\left(x\right)+\frac{l\left(l+1\right)}{x^2},0<x<\infty ,$ (20)

$\varphi \left(0\right)=0.$

Then

${\Vert {\text{e}}^{-itH}{P}_c\Vert }_{B\left(L^1,L^{\infty }\right)}\le C\frac{1}{\sqrt{t}},$ (21)

of moreover, $V\in L^1$, then

${\Vert {\text{e}}^{-itH}{P}_c\Vert }_{B\left(W_{1,1},W_{1,\infty }\right)}\le C\frac{1}{\sqrt{t}}.$ (22)

The article is organized as follows. In Section 2: Existence of the transformation operator $\mathbb{W}$, Theorem 1 is demonstrated in which for the solution given by $\mathbb{W}{\phi }_1\left(\lambda ,x\right):={\phi }_2\left(\lambda ,x\right)$ one get a formula for the potential

$q_2\left(x\right)=q_1-\frac{{y^{″}}_1}{y_1}+\frac{{{\phi }^{\prime }}_1}{{\phi }_2}+\frac{{y^{\prime }}_1}{y_1^2}$.

In Section 3: the two examples: Reducced Radial Schrödinger Equation (RRSE) and Schrödinger Equation on the Half-Line (RSEHL), the Example for the potential $q_2\left(x\right)$ stated in Theorem 1 are constructed calculating the terms: $\frac{{y^{″}}_1}{y_1},\frac{{{\phi }^{\prime }}_1}{{\phi }_2},\frac{{y^{\prime }}_1}{y_1^2}$ using the regular solution ${\phi }_l\left(s\mathrm{,}x\right)$, the solution of Jost $f_l\left(s\mathrm{,}x\right)$, the function of Jost $F_l\left(k\right)$ and its properties, for (RRSE) in the (Problem 1). In section 4 the demonstration of Theorem 2 is given. In Section 5, we establish the Conclusions and Open Problems. An appendix has been added in which, the asymptotic developments and estimates are made in neighborhoods of zero and infinity of the addends that form the potential $q_2\left(x\right)$ are obtained.

Warning: In order to respect the notations of the texts [3] [4] and [5] throughout the paper we will use the letters

$k:=s,r:=x,V\left(x\right):=q(x)$

the potential regular, indistinctly.

2. Existence of the Transformation Operator $\mathbb{W}$

Theorem 1. Let’s consider two Sturm-Liouville equations

$-{y^{″}}_j+q_j\left(x\right)y_j={\lambda }^2{y}_j,j=1,2,x\in \left(a,b\right),b\ge \infty ,$ (23)

$q_j\left(x\right)$ continuous only in the interior points of $\left(a\mathrm{,}b\right)$. Consider in particular, the following pair of boundary value problems of Sturm-Liouville on the Half Line

$-y^{″}+q_1\left(x\right)y=s^{2}y,y\left(0\right)=0,\text{where}s\in \left\{\lambda ,\mu \right\},x>0,$ (24)

and

$-y^{″}+q_2\left(x\right)y=s^{2}y,y\left(0\right)=0\text{where}s\in \left\{\lambda ,\mu \right\},x>0,$ (25)

Let $q_1\left(x\right)\mathrm{,}{q}_2\left(x\right)$ are continuous on $\left(\mathrm{0,}\infty \right)$. If $y_1\left(x,\mu \right):=y_1$ is a fixed solution of the first Equation (2) for $s=\mu$ and let ${\phi }_1\left(\lambda ,x\right):={\phi }_1$ an arbitrary solution of (2) for $s=\lambda$, then

$-{y^{″}}_1+q_1\left(x\right)y_1={\mu }^2{y}_1,y_1\left(0,\mu \right)=0\text{and}-{{\phi }^{″}}_1+q_1\left(x\right){\phi }_1={\lambda }^2{\phi }_1,{\phi }_1\left(0,\lambda \right)=0.$

Suppose

$y_1\left(x,\mu \right)\ne 0,\forall x>0,$ (26)

If

${\phi }_2\left(\lambda ,x\right):=\frac{W\left(y_1\left(x,\mu \right),{\phi }_1\left(\lambda ,x\right)\right)}{y_1\left(x\right)\left({\mu }^2-{\lambda }^2\right)},\mu \ne \lambda ,$ (27)

is solution of (3) for $s=\lambda$ where the Wronskian

$W\left(y_1\left(x,\mu \right),{\phi }_1\left(\lambda ,x\right)\right):=y_1\left(x,\mu \right){{\phi }^{\prime }}_1\left(\lambda ,x\right)-{y^{\prime }}_1\left(x,\mu \right){\phi }_1\left(\lambda ,x\right)\ne 0,$ (28)

$\text{'}\mathrm{<mo>\; :\; </mo>}=\frac{\text{d}}{\text{d}x}$. Then ${\phi }_2\left(\lambda \mathrm{,}x\right)$ satisfies the equation

$-{{\phi }^{″}}_2+q_2\left(x\right){\phi }_2={\lambda }^2{\phi }_2,{\phi }_2\left(0,\lambda \right)=0,$ (29)

where

$q_2\left(x\right)={\mu }^2-2q_1+\frac{{{\phi }^{\prime }}_1}{{\phi }_2}+\frac{{y^{\prime }}_1}{y_1^2}$ (30)

Remark 2. In order to facilitate the reading we will carry out the abuse of the ${\phi }_1\left(\lambda ,x\right):={\phi }_1:={\phi }_1\left(x\right)$ and $y_1\left(x,\mu \right):=y_1:=y_1\left(x\right)$ notation.

Proof. Since ${\left\{\frac{{y^{\prime }}_1}{y_1}\right\}}^{\prime }=\frac{y_1{y^{″}}_1-{\left[{y^{\prime }}_1\right]}^2}{y_1^2}=\frac{{y^{″}}_1}{y_1}-\frac{{\left[{y^{\prime }}_1\right]}^2}{y_1^2}=\frac{{y^{″}}_1}{y_1}-\frac{{\left[{y^{\prime }}_1\right]}^2}{y_1^2}=\frac{{y^{″}}_1}{y_1}+{y^{\prime }}_1{\left(\frac{1}{y_1}\right)}^{\prime }$, then

${\left\{\frac{{y^{\prime }}_1}{y_1}\right\}}^{\prime }=\frac{{y^{″}}_1}{y_1}+{y^{\prime }}_1{\left(\frac{1}{y_1}\right)}^{\prime },$ (31)

and $-{y^{″}}_1+q_1\left(x\right)y_1={\mu }^2{y}_1$ then

$\frac{{y^{″}}_1}{y_1}=q_1\left(x\right)-{\mu }^2.$ (32)

Also

$\begin{array}{c}{q}_1\left(x\right)-2{\left\{\frac{{y^{\prime }}_1}{y_1}\right\}}^{\prime }=q_1\left(x\right)-2\frac{{y^{″}}_1}{y_1}-2{y^{\prime }}_1{\left(\frac{1}{y_1}\right)}^{\prime }\\ =q_1\left(x\right)-2\left[q_1\left(x\right)-{\mu }^2\right]-2{y^{\prime }}_1{\left(\frac{1}{y_1}\right)}^{\prime }\\ =-q_1\left(x\right)+2{\mu }^2-2{y^{\prime }}_1{\left(\frac{1}{y_1}\right)}^{\prime },\end{array}$

that is,

$q_1\left(x\right)-2{\left\{\frac{{y^{\prime }}_1}{y_1}\right\}}^{\prime }=-q_1\left(x\right)+2{\mu }^2-2{y^{\prime }}_1{\left(\frac{1}{y_1}\right)}^{\prime }.$ (33)

Since ${\phi }_2\left(\lambda ,x\right):=\frac{W\left(y_1\left(x\right),{\phi }_1\left(\lambda ,x\right)\right)}{y_1\left(x\right)\left({\mu }^2-{\lambda }^2\right)}$ then let’s calculate

$\begin{array}{l}{{\phi }^{\prime }}_2\left(\lambda ,x\right):={\left(\frac{W\left(y_1\left(x\right),{\phi }_1\left(\lambda ,x\right)\right)}{y_1\left(x\right)\left({\mu }^2-{\lambda }^2\right)}\right)}^{\prime }=\frac{1}{\left({\mu }^2-{\lambda }^2\right)}{\left(\frac{W\left(y_1\left(x\right),{\phi }_1\left(x\right)\right)}{y_1\left(x\right)}\right)}^{\prime }\\ =\frac{1}{\left({\mu }^2-{\lambda }^2\right)}\frac{y_1\left(x\right)W^{\prime }\left(y_1\left(x\right),{\phi }_1\left(x\right)\right)-W\left(y_1\left(x\right),{\phi }_1\left(x\right)\right){y^{\prime }}_1\left(x\right)}{y_1^2\left(x\right)}\\ =\frac{1}{\left({\mu }^2-{\lambda }^2\right)}\frac{y_1\left(x\right)W^{\prime }\left(y_1\left(x\right),{\phi }_1\left(x\right)\right)-W\left(y_1\left(x\right),{\phi }_1\left(x\right)\right){y^{\prime }}_1\left(x\right)}{y_1^2\left(x\right)}\\ =\frac{1}{\left({\mu }^2-{\lambda }^2\right)}\frac{W^{\prime }\left(y_1\left(x\right),{\phi }_1\left(x\right)\right)}{y_1\left(x\right)}-\frac{1}{\left({\mu }^2-{\lambda }^2\right)}\frac{W\left(y_1\left(x\right),{\phi }_1\left(x\right)\right)}{y_1\left(x\right)}\times \frac{{y^{\prime }}_1\left(x\right)}{y_1\left(x\right)}\\ =\frac{\left({\mu }^2-{\lambda }^2\right)}{\left({\mu }^2-{\lambda }^2\right)}\frac{{\left\{{\phi }_2\left(\lambda ,x\right)y_1\right\}}^{\prime }}{y_1\left(x\right)}-{\phi }_2\left(\lambda ,x\right)\frac{{y^{\prime }}_1\left(x\right)}{y_1\left(x\right)}=\frac{{\left\{{\phi }_2\left(\lambda ,x\right)y_1\right\}}^{\prime }}{y_1\left(x\right)}-{\phi }_2\left(\lambda ,x\right)\frac{{y^{\prime }}_1\left(x\right)}{y_1(x)}\end{array}$

that is

${{\phi }^{\prime }}_2\left(\lambda ,x\right)=\frac{{\left\{{\phi }_2\left(\lambda ,x\right)y_1\right\}}^{\prime }}{y_1\left(x\right)}-\frac{{\phi }_2\left(\lambda ,x\right){y^{\prime }}_1\left(x\right)}{y_1\left(x\right)},$ (34)

Using (34) let’s calculate

$\begin{array}{l}{{\phi }^{″}}_2\left(\lambda \mathrm{,}x\right)={\left[\frac{{\left\{{\phi }_2\left(\lambda \mathrm{,}x\right)y_1\right\}}^{\prime }}{y_1\left(x\right)}\right]}^{\prime }-{\left[\frac{{\phi }_2\left(\lambda \mathrm{,}x\right){y^{\prime }}_1\left(x\right)}{y_1\left(x\right)}\right]}^{\prime }\\ =\frac{y_1{\left\{{\phi }_2\left(\lambda \mathrm{,}x\right)y_1\right\}}^{\prime \text{}\prime }-{\left\{{\phi }_2\left(\lambda \mathrm{,}x\right)y_1\right\}}^{\prime }{y^{\prime }}_1}{y_1^2\left(x\right)}\\ -\frac{y_1\left(x\right){\left[{\phi }_2\left(\lambda \mathrm{,}x\right){y^{\prime }}_1\left(x\right)\right]}^{\prime }-{\phi }_2\left(\lambda \mathrm{,}x\right){y^{\prime }}_1\left(x\right){y^{\prime }}_1\left(x\right)}{y_1^2\left(x\right)}\\ =\frac{y_1{\left\{{\phi }_2\left(\lambda \mathrm{,}x\right){y^{\prime }}_1+{{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right)y_1\right\}}^{\prime }-\left\{{\phi }_2\left(\lambda \mathrm{,}x\right){y^{\prime }}_1+{{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right)y_1\right\}{y^{\prime }}_1}{y_1^2\left(x\right)}\\ -\frac{y_1\left(x\right)\left[{\phi }_2\left(\lambda \mathrm{,}x\right){y^{″}}_1\left(x\right)+{{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right){y^{\prime }}_1\left(x\right)\right]-{\phi }_2\left(\lambda \mathrm{,}x\right){y^{\prime }}_1\left(x\right){y^{\prime }}_1\left(x\right)}{y_1^2(x)}\end{array}$

$\begin{array}{l}=\frac{y_1\left\{{\left[{\phi }_2\left(\lambda \mathrm{,}x\right){y^{\prime }}_1\right]}^{\prime }+{\left[{{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right)y_1\right]}^{\prime }\right\}-{\phi }_2\left(\lambda \mathrm{,}x\right){\left[{y^{\prime }}_1\right]}^2-{{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right)y_1{y^{\prime }}_1}{y_1^2\left(x\right)}\\ -\frac{y_1\left(x\right){\phi }_2\left(\lambda \mathrm{,}x\right){y^{″}}_1\left(x\right)+y_1\left(x\right){{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right){y^{\prime }}_1\left(x\right)-{\phi }_2\left(\lambda \mathrm{,}x\right){\left[{y^{\prime }}_1\left(x\right)\right]}^2}{y_1^2\left(x\right)}\\ =\frac{y_1\{{\phi }_2\left(\lambda \mathrm{,}x\right){y^{″}}_1+{{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right){y^{\prime }}_1+{{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right){y^{\prime }}_1+{{\phi }^{″}}_2\left(\lambda \mathrm{,}x\right)y_1}{y_1^2\left(x\right)}\\ +\frac{{{\phi }^{″}}_2\left(\lambda \mathrm{,}x\right)y_1+{{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right){y^{\prime }}_1}{y_1^2\left(x\right)}\frac{-{\phi }_2\left(\lambda \mathrm{,}x\right){\left[{y^{\prime }}_1\right]}^2-{{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right)y_1{y^{\prime }}_1}{y_1^2\left(x\right)}\\ -\frac{y_1\left(x\right){\phi }_2\left(\lambda \mathrm{,}x\right){y^{″}}_1\left(x\right)+y_1\left(x\right){{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right){y^{\prime }}_1\left(x\right)-{\phi }_2\left(\lambda \mathrm{,}x\right){\left[{y^{\prime }}_1\left(x\right)\right]}^2}{y_1^2(x)}\end{array}$

$\begin{array}{l}=\frac{{\phi }_2\left(\lambda \mathrm{,}x\right)y_1{y^{″}}_1+3{{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right)y_1{y^{\prime }}_1+2{{\phi }^{″}}_2\left(\lambda \mathrm{,}x\right)y_1^2}{y_1^2\left(x\right)}\\ -\frac{{\phi }_2\left(\lambda \mathrm{,}x\right){\left[{y^{\prime }}_1\right]}^2+{{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right)y_1{y^{\prime }}_1}{y_1^2\left(x\right)}\\ -\frac{y_1\left(x\right){\phi }_2\left(\lambda \mathrm{,}x\right){y^{″}}_1\left(x\right)+y_1\left(x\right){{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right){y^{\prime }}_1\left(x\right)-{\phi }_2\left(\lambda \mathrm{,}x\right){\left[{y^{\prime }}_1\left(x\right)\right]}^2}{y_1^2\left(x\right)}\\ =\frac{{\phi }_2\left(\lambda \mathrm{,}x\right)y_1{y^{″}}_1+3{{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right)y_1{y^{\prime }}_1+2{{\phi }^{″}}_2\left(\lambda \mathrm{,}x\right)y_1^2-{\phi }_2\left(\lambda \mathrm{,}x\right){\left[{y^{\prime }}_1\right]}^2-{{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right)y_1{y^{\prime }}_1}{y_1^2\left(x\right)}\\ -\frac{y_1\left(x\right){\phi }_2\left(\lambda \mathrm{,}x\right){y^{″}}_1\left(x\right)+y_1\left(x\right){{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right){y^{\prime }}_1\left(x\right)-{\phi }_2\left(\lambda \mathrm{,}x\right){\left[{y^{\prime }}_1\left(x\right)\right]}^2}{y_1^2(x)}\end{array}$

$\begin{array}{l}=\frac{2{\phi }_2\left(\lambda \mathrm{,}x\right)y_1{y^{″}}_1+{{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right)y_1{y^{\prime }}_1+2{{\phi }^{″}}_2\left(\lambda \mathrm{,}x\right)y_1^2}{y_1^2\left(x\right)}\\ =\frac{2{\phi }_2\left(\lambda \mathrm{,}x\right){y^{″}}_1}{y_1\left(x\right)}+\frac{{{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right){y^{\prime }}_1}{y_1\left(x\right)}+2{{\phi }^{″}}_2\left(\lambda \mathrm{,}x\right)\end{array}$

That is

${{\phi }^{″}}_2\left(\lambda ,x\right)=\frac{2{\phi }_2\left(\lambda ,x\right){y^{″}}_1}{y_1\left(x\right)}+\frac{{{\phi }^{\prime }}_2\left(\lambda ,x\right){y^{\prime }}_1}{y_1\left(x\right)}+2{{\phi }^{″}}_2\left(\lambda ,x\right)$ (35)

then

$-{{\phi }^{″}}_2\left(\lambda ,x\right)=\frac{2{\phi }_2\left(\lambda ,x\right){y^{″}}_1}{y_1\left(x\right)}+\frac{{{\phi }^{\prime }}_2\left(\lambda ,x\right){y^{\prime }}_1}{y_1\left(x\right)}.$ (36)

Now using (31), (32), (33), Equation (36) can be written as

$\begin{array}{c}-{{\phi }^{″}}_2\left(\lambda ,x\right)=\frac{{\phi }_2\left(\lambda ,x\right){y^{″}}_1+{\phi }_2\left(\lambda ,x\right){y^{″}}_1+{{\phi }^{\prime }}_2\left(\lambda ,x\right){y^{\prime }}_1}{y_1\left(x\right)}\\ =\frac{{\phi }_2\left(\lambda ,x\right){y^{″}}_1+{\left\{{\phi }_2\left(\lambda ,x\right){y^{\prime }}_1\right\}}^{\prime }}{y_1\left(x\right)}\\ =\frac{{\phi }_2\left(\lambda ,x\right){y^{″}}_1}{y_1\left(x\right)}+\frac{{\left\{{\phi }_2\left(\lambda ,x\right){y^{\prime }}_1\right\}}^{\prime }}{y_1\left(x\right)}\\ ={\phi }_2\left(\lambda ,x\right)\left\{q_1\left(x\right)-{\mu }^2\right\}+\frac{{\phi }_2\left(\lambda ,x\right){y^{″}}_1+{{\phi }^{\prime }}_2\left(\lambda ,x\right){y^{\prime }}_1}{y_1(x)}\end{array}$

$\begin{array}{c}={\phi }_2\left(\lambda ,x\right)\left\{q_1\left(x\right)-{\mu }^2\right\}+\frac{{y^{″}}_1}{y_1\left(x\right)}{\phi }_2\left(\lambda ,x\right)+{\phi }_2\left(\lambda ,x\right)\left\{q_1\left(x\right)-{\mu }^2\right\}\\ +\left\{q_1\left(x\right)-{\mu }^2\right\}{\phi }_2\left(\lambda ,x\right)+\frac{{{\phi }^{\prime }}_2\left(\lambda ,x\right){y^{\prime }}_1}{y_1\left(x\right)}\\ =2{\phi }_2\left(\lambda ,x\right)\left\{q_1\left(x\right)-{\mu }^2\right\}+\frac{{{\phi }^{\prime }}_2\left(\lambda ,x\right){y^{\prime }}_1}{y_1(x)}\end{array}$

Therefore

$-{{\phi }^{″}}_2\left(\lambda ,x\right)=2{\phi }_2\left(\lambda ,x\right)\left\{q_1\left(x\right)-{\mu }^2\right\}+\frac{{{\phi }^{\prime }}_2\left(\lambda ,x\right){y^{\prime }}_1}{y_1\left(x\right)}.$ (37)

One can expressed ${{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right)$ in terms of Wronskian

${{\phi }^{\prime }}_2\left(\lambda ,x\right)=\frac{1}{{\mu }^2-{\lambda }^2}\frac{W^{\prime }\left(y_1,{\phi }_1\right)}{y_1}-\frac{{\phi }_2}{y_1},$ (38)

so

${{\phi }^{\prime }}_2\left(\lambda \mathrm{,}x\right)=\left(\frac{W^{\prime }\left(y_1\mathrm{,}{\phi }_1\right)}{W\left(y_1\mathrm{,}{\phi }_1\right)}-\frac{1}{y_1}\right){\phi }_2\left(\lambda \mathrm{,}x\right)\mathrm{,}$ (39)

then (26) and (28) imply that

${\phi }_2\left(\lambda \mathrm{,}x\right)\ne \mathrm{0,}\forall x\in \left(\mathrm{0,}\infty \right)\mathrm{,}$ (40)

and

$\frac{{{\phi }^{\prime }}_2\left(\lambda ,x\right)}{{\phi }_2\left(\lambda ,x\right)}=\frac{W^{\prime }\left(y_1,{\phi }_1\right)}{W\left(y_1,{\phi }_1\right)}-\frac{1}{y_1}.$ (41)

From Equation (27)

$\begin{array}{c}{W}^{\prime }\left(y_1,{\phi }_1\right)={y^{″}}_1{\phi }_1-y_1{{\phi }^{″}}_1=y_1\left(\left[\frac{{y^{″}}_1}{y_1}\right]{\phi }_1-{{\phi }^{″}}_1\right)\\ =y_1\left(\left[q_1\left(x\right)-{\mu }^2\right]{\phi }_1-{{\phi }^{″}}_1\right)\\ =y_1\left(\left[q_1\left(x\right)-{\mu }^2\right]{\phi }_1+\left[{\lambda }^2-q_1\left(x\right)\right]{\phi }_1\right)\\ =y_1\left(-{\mu }^2+{\lambda }^2\right){\phi }_1.\end{array}$

Then

$W^{\prime }\left(y_1,{\phi }_1\right)=-y_1\left({\mu }^2-{\lambda }^2\right){\phi }_1.$ (42)

And from (41) it is obtained

$\begin{array}{c}{{\phi }^{\prime }}_2\left(\lambda ,x\right)=\left(\frac{W^{\prime }\left(y_1,{\phi }_1\right)}{W\left(y_1,{\phi }_1\right)}-\frac{1}{y_1}\right){\phi }_2\left(\lambda ,x\right)\\ =\left(-\frac{y_1\left({\mu }^2-{\lambda }^2\right){\phi }_1}{W\left(y_1,{\phi }_1\right)}-\frac{1}{y_1}\right){\phi }_2\left(\lambda ,x\right)\\ =\left(-\frac{{\phi }_1\left(\lambda ,x\right)}{{\phi }_2\left(\lambda ,x\right)}-\frac{1}{y_1}\right){\phi }_2\left(\lambda ,x\right)\\ =-{\phi }_1\left(\lambda ,x\right)-\frac{{\phi }_2\left(\lambda ,x\right)}{y_1},\end{array}$

that is

${{\phi }^{\prime }}_2\left(\lambda ,x\right)=-{\phi }_1\left(\lambda ,x\right)-\frac{{\phi }_2\left(\lambda ,x\right)}{y_1}.$ (43)

Now, replacing (43) in (37) we get

$\begin{array}{c}-{{\phi }^{″}}_2\left(\lambda ,x\right)=2{\phi }_2\left(\lambda ,x\right)\left\{q_1\left(x\right)-{\mu }^2\right\}+\frac{{{\phi }^{\prime }}_2\left(\lambda ,x\right){y^{\prime }}_1}{y_1\left(x\right)}\\ =2{\phi }_2\left(\lambda ,x\right)\left\{q_1\left(x\right)-{\mu }^2\right\}+\frac{{y^{\prime }}_1}{y_1\left(x\right)}\left[-{\phi }_1\left(\lambda ,x\right)-\frac{{\phi }_2\left(\lambda ,x\right)}{y_1}\right]\\ =2{\phi }_2\left(\lambda ,x\right)\left\{q_1\left(x\right)-{\mu }^2\right\}-\frac{{y^{\prime }}_1{\phi }_1\left(\lambda ,x\right)}{y_1\left(x\right)}-\frac{{\phi }_2\left(\lambda ,x\right){y^{\prime }}_1}{y_1^2}\\ ={\phi }_2\left(\lambda ,x\right)\left\{2\left[q_1\left(x\right)-{\mu }^2\right]-\frac{{y^{\prime }}_1}{y_1^2}\right\}-\frac{{y^{\prime }}_1{\phi }_1\left(\lambda ,x\right)}{y_1\left(x\right)}.\end{array}$

Therefore

$-{{\phi }^{″}}_2\left(\lambda ,x\right)={\phi }_2\left(\lambda ,x\right)\left\{2\left[q_1\left(x\right)-{\mu }^2\right]-\frac{{y^{\prime }}_1}{y_1^2}\right\}-\frac{{y^{\prime }}_1{\phi }_1\left(\lambda ,x\right)}{y_1\left(x\right)}.$ (44)

Using the representation for a given solution ${\phi }_2\left(\lambda \mathrm{,}x\right)$ in (28)

${\phi }_2\left(\lambda ,x\right):=\frac{W\left(y_1\left(x\right),{\phi }_1\left(x\right)\right)}{y_1\left(x\right)\left({\mu }^2-{\lambda }^2\right)},$

we get the equation

$\frac{{y^{\prime }}_1{\phi }_1\left(\lambda ,x\right)}{y_1\left(x\right)}=\frac{{\phi }_2\left(\lambda ,x\right)\left({\mu }^2-{\lambda }^2\right){y^{\prime }}_1{\phi }_1\left(\lambda ,x\right)}{W\left(y_1\left(x\right),{\phi }_1\left(x\right)\right)},$ (45)

then the right member of Equation (44) can be written as

$\begin{array}{l}{\phi }_2\left(\lambda ,x\right)\left\{2\left[q_1\left(x\right)-{\mu }^2\right]-\frac{{y^{\prime }}_1}{y_1^2}\right\}-\frac{{y^{\prime }}_1{\phi }_1\left(\lambda ,x\right)}{y_1\left(x\right)}\\ ={\phi }_2\left(\lambda ,x\right)\left\{2\left[q_1\left(x\right)-{\mu }^2\right]-\frac{{y^{\prime }}_1}{y_1^2}\right\}-\frac{{y^{\prime }}_1{\phi }_1\left(\lambda ,x\right)\left({\mu }^2-{\lambda }^2\right){\phi }_2\left(\lambda ,x\right)}{W\left(y_1\left(x\right),{\phi }_1\left(x\right)\right)}\\ ={\phi }_2\left(\lambda ,x\right)\left\{2\left[q_1\left(x\right)-{\mu }^2\right]-\frac{{y^{\prime }}_1}{y_1^2}-\frac{{y^{\prime }}_1{\phi }_1\left(\lambda ,x\right)\left({\mu }^2-{\lambda }^2\right)}{W\left(y_1\left(x\right),{\phi }_1\left(x\right)\right)}\right\}.\end{array}$

That is

$-{{\phi }^{″}}_2\left(\lambda ,x\right)={\phi }_2\left(\lambda ,x\right)\left\{2\left[q_1\left(x\right)-{\mu }^2\right]-\frac{{y^{\prime }}_1}{y_1^2}-\frac{{y^{\prime }}_1{\phi }_1\left(\lambda ,x\right)\left({\mu }^2-{\lambda }^2\right)}{W\left(y_1\left(x\right),{\phi }_1\left(x\right)\right)}\right\}.$ (46)

Let us observe that

${\phi }_2\left(\lambda ,x\right)=\frac{W\left(y_1,{\phi }_1\right)}{\left({\mu }^2-{\lambda }^2\right)y_1\left(x\right)}\ne 0,\forall x\in \left(0,\infty \right),$

from (26) and (28). Then

$\frac{1}{{\phi }_2\left(\lambda ,x\right)}=\frac{\left({\mu }^2-{\lambda }^2\right)y_1\left(x\right)}{W\left(y_1,{\phi }_1\right)}\iff \frac{\left({\mu }^2-{\lambda }^2\right){y^{\prime }}_1\left(x\right){\phi }_1\left(\lambda ,x\right)}{W\left(y_1,{\phi }_1\right)},$ (47)

since ${\phi }_2\left(\lambda \mathrm{,}x\right)$ is solution for the Equation (25) for $s=\lambda$. Rewriting (24) one get

$q_2\left(x\right)-{\lambda }^2-\frac{{{\phi }^{″}}_2}{{\phi }_2}=0\text{and}{q}_1\left(x\right)-{\mu }^2=\frac{{y^{″}}_1}{y_1}$ (48)

Then we can rewrite the right term of the Equation (46) as

$\begin{array}{l}{\phi }_2\left(\lambda ,x\right)\left\{2\left[q_1\left(x\right)-{\mu }^2\right]-\frac{y^{\prime }}{y_1^2}-\frac{{y^{\prime }}_1{\phi }_1\left(\lambda ,x\right)\left({\mu }^2-{\lambda }^2\right)}{W\left(y_1\left(x\right),{\phi }_1\left(x\right)\right)}\right\}\\ ={\phi }_2\left(\lambda ,x\right)\left\{2\left[q_2\left(x\right)-{\lambda }^2-\frac{{{\phi }^{″}}_2}{{\phi }_2}+\frac{{y^{″}}_1}{y_1}\right]-\frac{{y^{\prime }}_1}{y_1^2}-\frac{{y^{\prime }}_1{\phi }_1\left(\lambda ,x\right)}{{\phi }_2\left(\lambda ,x\right)y_1\left(x\right)}\right\},\end{array}$

associating the terms and using the equations given in (24) and (25) one get

$\begin{array}{l}{\phi }_2\left(\lambda \mathrm{,}x\right)\left\{2\left[q_2\left(x\right)-{\lambda }^2-\frac{{{\phi }^{″}}_2}{{\phi }_2}+\frac{{y^{″}}_1}{y_1}\right]-\frac{{y^{\prime }}_1}{y_1^2}-\frac{{y^{\prime }}_1{\phi }_1\left(\lambda \mathrm{,}x\right)}{{\phi }_2\left(\lambda \mathrm{,}x\right)y_1\left(x\right)}\right\}\\ ={\phi }_2\left(\lambda \mathrm{,}x\right)\left\{2q_2\left(x\right)-2{\lambda }^2+2\left(\frac{{y^{″}}_1}{y_1}-\frac{{{\phi }^{″}}_2}{{\phi }_2}\right)-\frac{{y^{\prime }}_1}{y_1^2}-\frac{{y^{\prime }}_1{\phi }_1\left(\lambda \mathrm{,}x\right)}{{\phi }_2\left(\lambda \mathrm{,}x\right)y_1\left(x\right)}\right\}\\ ={\phi }_2\left(\lambda \mathrm{,}x\right)\left\{{\lambda }^2-\left(-2q_2\left(x\right)+3{\lambda }^2-2\left[\frac{{y^{″}}_1}{y_1}-\frac{{{\phi }^{″}}_2}{{\phi }_2}\right]+\frac{{y^{\prime }}_1}{y_1^2}+\frac{{y^{\prime }}_1{\phi }_1\left(\lambda \mathrm{,}x\right)}{{\phi }_2\left(\lambda \mathrm{,}x\right)y_1\left(x\right)}\right)\right\}\\ ={\phi }_2\left(\lambda \mathrm{,}x\right)\{{\lambda }^2-({\lambda }^2-2\left[q_2\left(x\right)-{\lambda }^2\right]-2\left[q_1-q_2-\left({\mu }^2-{\lambda }^2\right)\right]\begin{array}{c}\\ \end{array}\\ +\frac{{y^{\prime }}_1}{y_1^2}+\frac{{y^{\prime }}_1{\phi }_1\left(\lambda \mathrm{,}x\right)}{{\phi }_2\left(\lambda \mathrm{,}x\right)y_1\left(x\right)})\}\end{array}$

$\begin{array}{l}={\phi }_2\left(\lambda \mathrm{,}x\right)\{{\lambda }^2-({\lambda }^2-2q_2\left(x\right)+2{\lambda }^2-2q_1+2q_2+2\left({\mu }^2-{\lambda }^2\right)\begin{array}{c}\\ \end{array}\\ +\frac{{y^{\prime }}_1}{y_1^2}+\frac{{y^{\prime }}_1{\phi }_1\left(\lambda \mathrm{,}x\right)}{{\phi }_2\left(\lambda \mathrm{,}x\right)y_1\left(x\right)})\}\\ ={\phi }_2\left(\lambda \mathrm{,}x\right)\left\{{\lambda }^2-\left({\lambda }^2-2\left[q_1-{\mu }^2\right]+\frac{{y^{\prime }}_1}{y_1^2}+\frac{{y^{\prime }}_1{\phi }_1\left(\lambda \mathrm{,}x\right)}{{\phi }_2\left(\lambda \mathrm{,}x\right)y_1\left(x\right)}\right)\right\}{\phi }_2\left(\lambda \mathrm{,}x\right)\\ ={\phi }_2\left(\lambda \mathrm{,}x\right)\left\{{\lambda }^2-\left(-2q_1+{\mu }^2-\left[{\mu }^2-{\lambda }^2\right]+\frac{{y^{\prime }}_1}{y_1^2}+\frac{{y^{\prime }}_1}{y_1}\frac{{\phi }_1{y}_1\left({\mu }^2-{\lambda }^2\right)}{W\left(y_1\mathrm{,}{\phi }_1\right)}\right)\right\}\\ ={\phi }_2\left(\lambda \mathrm{,}x\right)\left\{{\lambda }^2-\left(-2q_1+{\mu }^2-\left[{\mu }^2-{\lambda }^2\right]+\frac{{y^{\prime }}_1}{y_1^2}+\frac{{y^{\prime }}_1{\phi }_1\left({\mu }^2-{\lambda }^2\right)}{W\left(y_1\mathrm{,}{\phi }_1\right)}\right)\right\}\end{array}$

$\begin{array}{l}={\phi }_2\left(\lambda \mathrm{,}x\right)\left\{{\lambda }^2-\left(-2q_1+{\mu }^2+\left[{\mu }^2-{\lambda }^2\right]\left\{-1+\frac{{y^{\prime }}_1{\phi }_1}{W\left(y_1\mathrm{,}{\phi }_1\right)}\right\}+\frac{{y^{\prime }}_1}{y_1^2}\right)\right\}\\ ={\phi }_2\left(\lambda \mathrm{,}x\right)\left\{{\lambda }^2-\left(-2q_1+{\mu }^2+\left[{\mu }^2-{\lambda }^2\right]\left\{-1+\frac{{y^{\prime }}_1{\phi }_1}{{y^{\prime }}_1{\phi }_1-y_1{{\phi }^{\prime }}_1}\right\}+\frac{{y^{\prime }}_1}{y_1^2}\right)\right\}\\ ={\phi }_2\left(\lambda \mathrm{,}x\right)\left\{{\lambda }^2-\left(-2q_1+{\mu }^2+\left[{\mu }^2-{\lambda }^2\right]\left\{\frac{-{y^{\prime }}_1\phi +y_1{{\phi }^{\prime }}_1+{y^{\prime }}_1{\phi }_1}{{y^{\prime }}_1{\phi }_1-y_1{{\phi }^{\prime }}_1}\right\}+\frac{{y^{\prime }}_1}{y_1^2}\right)\right\}\\ ={\phi }_2\left(\lambda \mathrm{,}x\right)\left\{{\lambda }^2-\left(-2q_1+{\mu }^2+\left[{\mu }^2-{\lambda }^2\right]\left\{\frac{y_1{{\phi }^{\prime }}_1}{{y^{\prime }}_1{\phi }_1-y_1{{\phi }^{\prime }}_1}\right\}+\frac{{y^{\prime }}_1}{y_1^2}\right)\right\}\\ ={\phi }_2\left(\lambda \mathrm{,}x\right)\left\{{\lambda }^2-\left(-2q_1+{\mu }^2+\left[{\mu }^2-{\lambda }^2\right]\left\{\frac{y_1{{\phi }^{\prime }}_1}{W\left(y_1\mathrm{,}{\phi }_1\right)}\right\}+\frac{{y^{\prime }}_1}{y_1^2}\right)\right\}\end{array}$

$\begin{array}{l}={\phi }_2\left(\lambda \mathrm{,}x\right)\left\{{\lambda }^2-\left(-2q_1+{\mu }^2+\frac{\left[{\mu }^2-{\lambda }^2\right]y_1}{W\left(y_1\mathrm{,}{\phi }_1\right)}\left\{{{\phi }^{\prime }}_1\right\}+\frac{{y^{\prime }}_1}{y_1^2}\right)\right\}\\ ={\phi }_2\left(\lambda \mathrm{,}x\right)\left\{{\lambda }^2-\left(-2q_1+{\mu }^2+\frac{1}{{\phi }_2}\left\{{{\phi }^{\prime }}_1\right\}+\frac{{y^{\prime }}_1}{y_1^2}\right)\right\}\\ ={\phi }_2\left(\lambda \mathrm{,}x\right)\left\{{\lambda }^2-\left(-2q_1-\frac{{y^{″}}_1}{y_1}+q_1+\frac{{{\phi }^{\prime }}_1}{{\phi }_2}+\frac{{y^{\prime }}_1}{y_1^2}\right)\right\}\\ ={\phi }_2\left(\lambda \mathrm{,}x\right)\left\{{\lambda }^2-\left(-q_1-\frac{{y^{″}}_1}{y_1}+\frac{{{\phi }^{\prime }}_1}{{\phi }_2}+\frac{{y^{\prime }}_1}{y_1^2}\right)\right\}\end{array}$

Then we yield the equation sought

$-{{\phi }^{″}}_2\left(\lambda ,x\right)={\phi }_2\left(\lambda ,x\right)\left\{{\lambda }^2-\left(-q_1-\frac{{y^{″}}_1}{y_1}+\frac{{{\phi }^{\prime }}_1}{{\phi }_2}+\frac{{y^{\prime }}_1}{y_1^2}\right)\right\},$ (49)

that is

$-{{\phi }^{″}}_2\left(\lambda ,x\right)+\left(-q_1-\frac{{y^{″}}_1}{y_1}+\frac{{{\phi }^{\prime }}_1}{{\phi }_2}+\frac{{y^{\prime }}_1}{y_1^2}\right){\phi }_2\left(\lambda ,x\right)={\lambda }^2{\phi }_2\left(\lambda ,x\right),$ (50)

if

$q_2\left(x\right):=-q_1-\frac{{y^{″}}_1}{y_1}+\frac{{{\phi }^{\prime }}_1}{{\phi }_2}+\frac{{y^{\prime }}_1}{y_1^2},$ (51)

one yield to the initial Equation (29)

$-{{\phi }^{″}}_2\left(\lambda ,x\right)+q_2\left(x\right){\phi }_2\left(\lambda ,x\right)={\lambda }^2{\phi }_2\left(\lambda ,x\right).$

and the potential (30).

Since

$-\frac{{y^{″}}_1}{y_1}+q_1={\mu }^2\iff -\frac{{y^{″}}_1}{y_1}-q_1={\mu }^2-2q_1,$

then the potential $q_2\left(x\right)$ can also be represented as

$q_2\left(x\right)={\mu }^2-2q_1+\frac{{{\phi }^{\prime }}_1}{{\phi }_2}+\frac{{y^{\prime }}_1}{y_1^2}.$ (52)

it is uniquely determined.

Finally, according to (27) we define the corresponding transformation operator as

$\mathbb{W}:=\frac{W\left(y_1\left(x,\mu \right),\cdot \right)}{y_1\left(x\right)\left({\mu }^2-{\lambda }^2\right)}\Rightarrow \mathbb{W}{\phi }_1\left(x\right):=\frac{W\left(y_1\left(x,\mu \right),{\phi }_1\left(x\right)\right)}{y_1\left(x\right)\left({\mu }^2-{\lambda }^2\right)}={\phi }_2\left(x\right)$ (53)

3. The Two Examples: Reducced Radial Schrödinger Equation and Schrödinger Equation on the Half-Line

The named Problem 1 of Theorem 1 of the preceding section is equivalent to the following system of boundary value problems on the Half Line

$\{\begin{array}{l}-{y^{″}}_1\left(x,\mu \right)+\left\{q\left(x\right)+\frac{l\left(l+1\right)}{x^2}\right\}{y}_1\left(x,\mu \right)={\mu }^2{y}_1\left(x,\mu \right),0<x<\infty ,\hfill \\ y_1\left(0,\mu \right)=0.\hfill \\ -{{\phi }^{″}}_1\left(x,\lambda \right)+\left\{q\left(x\right)+\frac{l\left(l+1\right)}{x^2}\right\}{\phi }_1\left(x,\lambda \right)={\lambda }^2{\phi }_1\left(x,\lambda \right),0<x<\infty ,\hfill \\ {\phi }_1\left(0,\lambda \right)=0.\hfill \end{array}$ (54)

The previous system corresponds to the reducced radial Schrödinger equation

$-\frac{{\text{d}}^2}{\text{d}{r}^2}{\psi }_l\left(k,r\right)+\left[q\left(r\right)+\frac{l\left(l+1\right)}{r^2}\right]{\psi }_l\left(k,r\right)=k^2{\psi }_l\left(k,r\right).$ (55)

for the partial wave of angular momentum l and wave number k.

This equation and the solution has the following properties:

1) The potential $q\left(r\right)$ is regular, that is

${\displaystyle {\int }_0^{\infty }}r\left|q\left(r\right)\right|\text{d}r<\infty ,$ (56)

2) The solution ${\psi }_l$ exists and it is the only one which vanishes at the origin.

3) For physical reasons know in scattering theory, the physical wave function ${\psi }_l$, also must vanish at the origin.

4) The asymptotic behavior is of ${\psi }_l$

${\psi }_l\approx \exp \left(i{\delta }_l\right)\sin \left(kr-\frac{\text{π}}{2}\left(\lambda -\frac{1}{2}\right)+{\delta }_l\right),r\to \infty ,$ (57)

where ${\delta }_l$, is the phase shift, is a real quantity which depends of energy k and the momentum l, that is, ${\delta }_l={\delta }_l\left(k,l\right)$.

5) The momentum angular fixed $\lambda =l_0+\frac{1}{2}$ and all (positive) values of $k^2:={\lambda }^2$.

6) The Jost solution becomes:

$f_l\left(k,r\right)\underset{\underset{k\ne 0}{r\to \infty }}{=}{i}^l{\text{e}}^{ikr}+{\text{e}}^{-\mathrm{Im}kr}o\left(1\right)$ (58)

then

${f^{\prime }}_l\left(k,r\right)\underset{\underset{k\ne 0}{r\to \infty }}{=}{i}^{l+1}k{\text{e}}^{ikr}+{\text{e}}^{-\mathrm{Im}kr}o\left(1\right)$ (59)

7) In the case $\mathrm{Im}k\ge 0$, the Jost solution is given by

$f_l\left(k,r\right)\underset{\underset{k\ne 0}{r\to \infty }}{=}{i}^l{\text{e}}^{ikr}\text{and}{f^{\prime }}_l\left(k,r\right)\underset{\underset{k\ne 0}{r\to \infty }}{=}{i}^{l+1}k{\text{e}}^{ikr}.$ (60)

8) The regular solution ${\phi }_l\left(k\mathrm{,}r\right)$, near $r=0$, is given by the boundary condition

${\phi }_l\left(k,r\right)\underset{r\to 0}{=}\frac{r^{l+1}}{\left(2l+1\right)!!}$ (61)

then

${{\phi }^{\prime }}_l\left(k,r\right)\underset{\underset{\lambda \ne 0}{r\to 0}}{=}\frac{\left(l+1\right)r^l}{\left(2l+1\right)!!}.$ (62)

See [Ch-S] [5] , Chapter 1, Sections: 1.1-1.5, 1.7 pages 1-16, 19. If

$r:=x,k:=s,$ (63)

the solution ${\psi }_l\left(k\mathrm{,}r\right)$ for

$-{{\psi }^{″}}_l\left(k,x\right)+q_1\left(x\right){\psi }_l\left(k,x\right)={\lambda }^2{\psi }_l\left(k,x\right),x>0,$ (64)

${\psi }_l\left(k,0\right)=0.$

where

$q_1\left(x\right):=q\left(x\right)+\frac{l\left(l+1\right)}{x^2},$ (65)

is proposed as

${\psi }_l\left(\lambda ,x\right):=\{\begin{array}{l}{\phi }_l\left(\lambda ,x\right),\text{if}0<x<1\hfill \\ f_l\left(\lambda ,x\right),\text{if}x>1.\hfill \end{array}$ (66)

or

${\psi }_l\left(\lambda ,x\right):=\{\begin{array}{l}\frac{x^{l+1}}{\left(2l+1\right)!!},\text{if}0<x<1\hfill \\ i^l{\text{e}}^{i\lambda x}+{\text{e}}^{-\mathrm{Im}\lambda x},\text{if}x>1.\hfill \end{array}$ (67)

Since

$\frac{{{\psi }^{″}}_l\left(\lambda ,x\right)}{{\psi }_l\left(\lambda ,x\right)}=q\left(x\right)+\frac{l\left(l+1\right)}{x^2}-{\lambda }^2,\text{for}x>0,$ (68)

it follows that

$\frac{{{\psi }^{″}}_l\left(\lambda ,x\right)}{{\psi }_l\left(\lambda ,x\right)}=\{\begin{array}{l}q\left(x\right)+\frac{l\left(l+1\right)}{x^2}-{\lambda }^2,0<x<1\hfill \\ q\left(x\right)+\frac{l\left(l+1\right)}{x^2}-{\lambda }^2,x>1.\hfill \end{array}$ (69)

Next let $y_j\left(x\mathrm{,}\mu \right)$ denote a fixed solution of the j-th equation for $k=\mu$, and let ${\varphi }_1\left(k\mathrm{,}x\right)$ denote an arbitrary solution of the first equation. Then for $\mathrm{Im}k\ge 0$

$\begin{array}{c}{\varphi }_1\left(k,x\right)=y_1\left(k,x\right):=\{\begin{array}{l}{\phi }_l\left(k,x\right),\text{if}0<x<1,\hfill \\ f_l\left(k,x\right),\text{if}x>1.\hfill \end{array}\\ =\{\begin{array}{l}\frac{x^{l+1}}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^l{\text{e}}^{ikx}+{\text{e}}^{-\mathrm{Im}kx}o\left(1\right),\text{if}x>1.\hfill \end{array}\end{array}$ (70)

and

$\begin{array}{c}{{\varphi }^{\prime }}_1\left(k,x\right)={y^{\prime }}_1\left(k,x\right):=\{\begin{array}{l}{{\phi }^{\prime }}_l\left(k,x\right),\text{if}0<x<1,\hfill \\ {f^{\prime }}_l\left(k,x\right),\text{if}x>1.\hfill \end{array}\\ =\{\begin{array}{l}\frac{\left(l+1\right)x^l}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^{l+1}k{\text{e}}^{ikr}-\mathrm{Im}k{\text{e}}^{-\mathrm{Im}kx}o\left(1\right),\text{if}x>1.\hfill \end{array}\end{array}$ (71)

where ${\varphi }_1\left(k\mathrm{,}x\right)\mathrm{,}{y}_1\left(k\mathrm{,}x\right)$ are solutions for the equation $j=1$ for $k=\mu$ and $k=\lambda$ respectively.

Then

${\varphi }_1\left(\lambda ,x\right)=\{\begin{array}{l}\frac{x^{l+1}}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^l{\text{e}}^{i\lambda x},\text{if}x>1.\hfill \end{array}$ (72)

and

${\varphi }_1\left(\mu ,x\right)=\{\begin{array}{l}\frac{x^{l+1}}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^l{\text{e}}^{i\mu x},\text{if}x>1.\hfill \end{array}$ (73)

then

${{\varphi }^{\prime }}_1\left(\lambda ,x\right)=\{\begin{array}{l}\frac{\left(l+1\right)x^l}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^{l+1}\lambda {\text{e}}^{i\lambda x},\text{if}x\ge 1.\hfill \end{array}$ (74)

${{\varphi }^{\prime }}_1\left(\mu ,x\right)=\{\begin{array}{l}\frac{\left(l+1\right)x^l}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^{l+1}\mu {\text{e}}^{i\mu x},\text{if}x\ge 1.\hfill \end{array}$ (75)

for

$y_1\left(x,\mu \right)=\{\begin{array}{l}\frac{x^{l+1}}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^l{\text{e}}^{i\mu x},\text{if}x\ge 1.\hfill \end{array}$ (76)

and

$y_1\left(x,\lambda \right)=\{\begin{array}{l}\frac{x^{l+1}}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^l{\text{e}}^{i\lambda x},\text{if}x\ge 1.\hfill \end{array}$ (77)

then

${y^{\prime }}_1\left(x,\mu \right)=\{\begin{array}{l}\frac{\left(l+1\right)x^l}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^{l+1}\mu {\text{e}}^{i\mu x},\text{if}x>1.\hfill \end{array}$ (78)

and

${y^{\prime }}_1\left(x,\lambda \right)=\{\begin{array}{l}\frac{\left(l+1\right)x^{l+1}}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^{l+1}\lambda {\text{e}}^{i\lambda x},\text{if}x>1.\hfill \end{array}$ (79)

And

${\varphi }_1\left(x,\lambda \right)=\{\begin{array}{l}\frac{x^{l+1}}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^l{\text{e}}^{i\lambda x},\text{if}x\ge 1.\hfill \end{array},$ (80)

$\frac{1}{{\varphi }_1\left(\lambda ,x\right)}=\{\begin{array}{l}\frac{\left(2l+1\right)!!}{x^{l+1}},\text{if}0<x<1,\hfill \\ \frac{1}{i^l{\text{e}}^{i\lambda x}},\text{if}x\ge 1.\hfill \end{array},$ (81)

Now, from Equation (74),

$\begin{array}{c}{{\varphi }^{″}}_1\left(\lambda ,x\right)=\{\begin{array}{l}\frac{\left(l+1\right)l}{\left(2l+1\right)!!}{x}^{l-1},\text{if}0<x<1,\hfill \\ i^{l+1}\lambda i\lambda {\text{e}}^{i\lambda x},\text{if}x\ge 1.\hfill \end{array}\\ =\{\begin{array}{l}\frac{\left(l+1\right)l}{\left(2l+1\right)!!}{x}^{l-1},\text{if}0<x<1,\hfill \\ i^l{i}^2{\lambda }^2{\text{e}}^{i\lambda x},\text{if}x\ge 1.\hfill \end{array}\\ =\{\begin{array}{l}\frac{\left(l+1\right)l}{\left(2l+1\right)!!}{x}^{l-1},\text{if}0<x<1,\hfill \\ -i^l{\lambda }^2{\text{e}}^{i\lambda x},\text{if}x\ge 1.\hfill \end{array}\end{array}$

That is,

${{\varphi }^{″}}_1\left(\lambda ,x\right)=\{\begin{array}{l}\frac{\left(l+1\right)l}{\left(2l+1\right)!!}{x}^{l-1},\text{if}0<x<1,\hfill \\ -i^l{\lambda }^2{\text{e}}^{i\lambda x},\text{if}x\ge 1.\hfill \end{array}$ (82)

Then

$\begin{array}{c}\frac{{{\varphi }^{″}}_1\left(\lambda ,x\right)}{{\varphi }_1\left(\lambda ,x\right)}=\{\begin{array}{l}\frac{\left(l+1\right)l}{\left(2l+1\right)!!}{x}^{l-1},\text{if}0<x<1,\hfill \\ -i^l{\lambda }^2{\text{e}}^{i\lambda x},\text{if}x\ge 1.\hfill \end{array}\times \{\begin{array}{l}\frac{\left(2l+1\right)!!}{x^{l+1}},\text{if}0<x<1,\hfill \\ \frac{1}{i^l{\text{e}}^{i\lambda x}},\text{if}x\ge 1.\hfill \end{array}\\ =\{\begin{array}{l}\frac{\left(l+1\right)l}{\left(2l+1\right)!!}{x}^{l-1}\frac{\left(2l+1\right)!!}{x^{l+1}},\text{if}0<x<1,\hfill \\ -i^l{\lambda }^2{\text{e}}^{i\lambda x}\left(\frac{1}{i^l{\text{e}}^{i\lambda x}}\right),\text{if}x\ge 1.\hfill \end{array}\\ =\{\begin{array}{l}\frac{l\left(l+1\right)}{x^2},\text{if}0<x<1,\hfill \\ -{\lambda }^2,\text{if}x\ge 1.\hfill \end{array}\end{array}$

That is

$\frac{{{\varphi }^{″}}_1\left(\lambda ,x\right)}{{\varphi }_1\left(\lambda \mathrm{,}x\right)}=\{\begin{array}{l}\frac{l\left(l+1\right)}{x^2},\text{if}0<x<1,\hfill \\ -{\lambda }^2,\text{if}x\ge 1.\hfill \end{array}$ (83)

Now since

$q_2:=-q_1-\frac{{y^{″}}_1}{y_1}+\frac{{{\phi }^{\prime }}_1}{{\phi }_2}+\frac{{y^{\prime }}_1}{y_1^2},$

the corresponding summations $\frac{{y^{″}}_1}{y_1},\frac{{{\phi }^{\prime }}_1}{{\phi }_2},\frac{{y^{\prime }}_1}{y_1^2}$ of the potential $q_2\left(x\right)$ (calculated in Appendix), Are given by

$\frac{{y^{″}}_1}{y_1}=\{\begin{array}{l}\frac{l\left(l+1\right)}{x^2},\text{if}0<x<1,\hfill \\ -{\mu }^2,\text{if}x\ge 1.\hfill \end{array}$ (84)

and

$\frac{{y^{\prime }}_1}{y_1^2}=\{\begin{array}{l}\frac{\left(2l+1\right)!!\left(l+1\right)}{x^2},\text{if}0<x<1,\hfill \\ \frac{\mu }{i^{l-1}{\text{e}}^{i\mu x}},\text{if}x\ge 1.\hfill \end{array}$ (85)

Next we defined the solution

${\varphi }_2\left(\lambda ,x\right):=\frac{W\left(y_1\left(x,\mu \right),{\varphi }_1\left(\lambda ,x\right)\right)}{y_1\left(x,\mu \right)\left({\mu }^2-{\lambda }^2\right)}.$

In terms of the Jost function

$F_l\left(k\right)={\left(-k\right)}^{l}W\left(f_l\left(k,x\right),{\phi }_l\left(k,x\right)\right),$ (86)

for

$k\in \left\{\lambda \mathrm{,}\mu \right\}\mathrm{,}$ (87)

Let’s start with the Wroskin

$W\left(y_1\left(x,\mu \right),{\varphi }_1\left(\lambda ,x\right)\right)=y_1\left(x,\mu \right){{\varphi }^{\prime }}_1\left(\lambda ,x\right)-{y^{\prime }}_1\left(x,\mu \right){\varphi }_1\left(\lambda ,x\right)$ (88)

then

${\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)=W\left(y_1\left(x,\lambda \right),{\varphi }_1\left(\lambda ,x\right)\right)=y_1\left(x,\lambda \right){{\varphi }^{\prime }}_1\left(\lambda ,x\right)-{y^{\prime }}_1\left(x,\lambda \right){\varphi }_1\left(\lambda ,x\right)$

and

${\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)=W\left(y_1\left(x,\mu \right),{\varphi }_1\left(\mu ,x\right)\right)=y_1\left(x,\mu \right){{\varphi }^{\prime }}_1\left(\mu ,x\right)-{y^{\prime }}_1\left(x,\mu \right){\varphi }_1\left(\mu ,x\right),$

that is

$\{\begin{array}{l}{y}_1\left(x,\lambda \right){{\varphi }^{\prime }}_1\left(\lambda ,x\right)-{y^{\prime }}_1\left(x,\lambda \right){\varphi }_1\left(\lambda ,x\right)={\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right),\\ y_1\left(x,\mu \right){{\varphi }^{\prime }}_1\left(\mu ,x\right)-{y^{\prime }}_1\left(x,\mu \right){\varphi }_1\left(\mu ,x\right)={\left(-\mu \right)}^{-l}{F}_l\left(\mu \right).\end{array}$ (89)

And

$y_1\left(x,\lambda \right){{\varphi }^{\prime }}_1\left(\lambda ,x\right)-{y^{\prime }}_1\left(x,\lambda \right){\varphi }_1\left(\lambda ,x\right)={\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right),$ (90)

implies

${{\varphi }^{\prime }}_1\left(\lambda ,x\right)=\frac{{\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)+{y^{\prime }}_1\left(x,\lambda \right){\varphi }_1\left(\lambda ,x\right)}{y_1\left(x,\lambda \right)},$ (91)

and

$y_1\left(x,\mu \right){{\varphi }^{\prime }}_1\left(\lambda ,x\right)=\frac{y_1\left(x,\mu \right)}{y_1\left(x,\lambda \right)}\left[{\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)+{y^{\prime }}_1\left(x,\lambda \right){\varphi }_1\left(\lambda ,x\right)\right].$ (92)

Similary

$W\left(y_1\left(x,\mu \right),{\varphi }_1\left(\mu ,x\right)\right):=y_1\left(x,\mu \right){{\varphi }^{\prime }}_1\left(\mu ,x\right)-{y^{\prime }}_1\left(x,\mu \right){\varphi }_1\left(\mu ,x\right)={\left(-\mu \right)}^{-l}{F}_l\left(\mu \right),$ (93)

that is

${y^{\prime }}_1\left(x,\mu \right)=\frac{y_1\left(x,\mu \right){{\varphi }^{\prime }}_1\left(\mu ,x\right)-{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)}{{\varphi }_1\left(\mu ,x\right)}.$ (94)

so

${\varphi }_1\left(\lambda ,x\right){y^{\prime }}_1\left(x,\mu \right)=\frac{{\varphi }_1\left(\lambda ,x\right)}{{\varphi }_1\left(\mu ,x\right)}\left[y_1\left(x,\mu \right),{{\varphi }^{\prime }}_1\left(\mu ,x\right)-{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)\right]$ (95)

The

$\begin{array}{l}W\left(y_1\left(x\mathrm{,}\mu \right)\mathrm{,}{\varphi }_1\left(\lambda \mathrm{,}x\right)\right)\\ =y_1\left(x\mathrm{,}\mu \right){{\varphi }^{\prime }}_1\left(\lambda \mathrm{,}x\right)-{y^{\prime }}_1\left(x\mathrm{,}\mu \right){\varphi }_1\left(\lambda \mathrm{,}x\right)\\ =\frac{y_1\left(x\mathrm{,}\mu \right)}{y_1\left(x\mathrm{,}\lambda \right)}\left[{\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)+{y^{\prime }}_1\left(x\mathrm{,}\lambda \right){\varphi }_1\left(\lambda \mathrm{,}x\right)\right]\\ -\frac{{\varphi }_1\left(\lambda \mathrm{,}x\right)}{{\varphi }_1\left(\mu \mathrm{,}x\right)}\left[y_1\left(x\mathrm{,}\mu \right){{\varphi }^{\prime }}_1\left(\mu \mathrm{,}x\right)-{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)\right]\\ ={\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)\frac{y_1\left(x\mathrm{,}\mu \right)}{y_1\left(x\mathrm{,}\lambda \right)}+{y^{\prime }}_1\left(x\mathrm{,}\lambda \right){\varphi }_1\left(\lambda \mathrm{,}x\right)\frac{y_1\left(x\mathrm{,}\mu \right)}{y_1\left(x\mathrm{,}\lambda \right)}\end{array}$

$\begin{array}{l}-y_1\left(x\mathrm{,}\mu \right){{\varphi }^{\prime }}_1\left(\mu \mathrm{,}x\right)\frac{{\varphi }_1\left(\lambda \mathrm{,}x\right)}{{\varphi }_1\left(\mu \mathrm{,}x\right)}+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)\frac{{\varphi }_1\left(\lambda \mathrm{,}x\right)}{{\varphi }_1\left(\mu \mathrm{,}x\right)}\\ ={\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)\frac{y_1\left(x\mathrm{,}\mu \right)}{y_1\left(x\mathrm{,}\lambda \right)}+\frac{{y^{\prime }}_1\left(x\mathrm{,}\lambda \right){\varphi }_1\left(\lambda \mathrm{,}x\right)y_1\left(x\mathrm{,}\mu \right)}{y_1\left(x\mathrm{,}\lambda \right)}\\ -\frac{y_1\left(x\mathrm{,}\mu \right){{\varphi }^{\prime }}_1\left(\mu \mathrm{,}x\right){\varphi }_1\left(\lambda \mathrm{,}x\right)}{{\varphi }_1\left(\mu \mathrm{,}x\right)}+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)\frac{{\varphi }_1\left(\lambda \mathrm{,}x\right)}{{\varphi }_1\left(\mu \mathrm{,}x\right)}\end{array}$

$\begin{array}{l}={\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)\frac{y_1\left(x\mathrm{,}\mu \right)}{y_1\left(x\mathrm{,}\lambda \right)}+\frac{y_1\left(x\mathrm{,}\mu \right)}{y_1\left(x\mathrm{,}\lambda \right)}{y^{\prime }}_1\left(x\mathrm{,}\lambda \right){\varphi }_1\left(\lambda \mathrm{,}x\right)\\ -\frac{y_1\left(x\mathrm{,}\mu \right)}{{\varphi }_1\left(\mu \mathrm{,}x\right)}{\varphi }_1\left(\lambda \mathrm{,}x\right){{\varphi }^{\prime }}_1\left(\mu \mathrm{,}x\right)+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)\frac{{\varphi }_1\left(\lambda \mathrm{,}x\right)}{{\varphi }_1\left(\mu \mathrm{,}x\right)}\\ ={\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)\frac{y_1\left(x\mathrm{,}\mu \right)}{y_1\left(x\mathrm{,}\lambda \right)}+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)\frac{{\varphi }_1\left(\lambda \mathrm{,}x\right)}{{\varphi }_1\left(\mu \mathrm{,}x\right)}\\ +\frac{y_1\left(x\mathrm{,}\mu \right)}{y_1\left(x\mathrm{,}\lambda \right)}{y^{\prime }}_1\left(x\mathrm{,}\lambda \right){\varphi }_1\left(\lambda \mathrm{,}x\right)-\frac{y_1\left(x\mathrm{,}\mu \right)}{{\varphi }_1\left(\mu \mathrm{,}x\right)}{\varphi }_1\left(\lambda \mathrm{,}x\right){{\varphi }^{\prime }}_1\left(\mu \mathrm{,}x\right)\end{array}$

That is

$\begin{array}{l}W\left(y_1\left(x,\mu \right),{\varphi }_1\left(\lambda ,x\right)\right)\\ ={\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)\frac{y_1\left(x,\mu \right)}{y_1\left(x,\lambda \right)}+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)\frac{{\varphi }_1\left(\lambda ,x\right)}{{\varphi }_1\left(\mu ,x\right)}\\ +\frac{y_1\left(x,\mu \right)}{y_1\left(x,\lambda \right)}{y^{\prime }}_1\left(x,\lambda \right){\varphi }_1\left(\lambda ,x\right)-\frac{y_1\left(x,\mu \right)}{{\varphi }_1\left(\mu ,x\right)}{\varphi }_1\left(\lambda ,x\right){{\varphi }^{\prime }}_1\left(\mu ,x\right).\end{array}$ (96)

The calculation of $W\left(y_1\left(x\mathrm{,}\mu \right)\mathrm{,}{\varphi }_1\left(\lambda \mathrm{,}x\right)\right)$ is shown in Appendix. The result obtained is:

$\begin{array}{l}W\left(y_1\left(x\mathrm{,}\mu \right)\mathrm{,}{\varphi }_1\left(\lambda \mathrm{,}x\right)\right)\\ =\{\begin{array}{l}{\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right),\text{if}0<x<1,\\ {\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right){\text{e}}^{i\left(\mu -\lambda \right)x}+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right){\text{e}}^{i\left(\lambda -\mu \right)x}+i^{l+1}{\text{e}}^{i\left(\lambda +\mu \right)x}\left(i\lambda -\mu \right),\text{if}x>1.\end{array}\end{array}$ (97)

Now let’s write the solution ${\varphi }_2\left(\lambda \mathrm{,}x\right)$ obtained in Appendix:

${\varphi }_2\left(\lambda ,x\right)=\frac{1}{\left({\mu }^2-{\lambda }^2\right)}\{\begin{array}{l}\left[{\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)\right]\frac{\left(2l+1\right)!!}{x^{l+1}},\text{if}0<x<1,\\ {\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)i^{-l}{\text{e}}^{-i\lambda x}+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)i^{-l}{\text{e}}^{i\left(\lambda -2\mu \right)x}\\ +\left({\mu }^2-{\lambda }^2\right)\left(i^{2l+1}\mu {\text{e}}^{ix\mu }-i^{l+1}\lambda {\text{e}}^{\left(2\lambda -\mu \right)ix}\right),\text{if}x>1,\mu \ne \lambda \end{array}$

That is,

$\begin{array}{l}\frac{1}{{\varphi }_2\left(\lambda ,x\right)}\\ =\{\begin{array}{l}\frac{\left({\mu }^2-{\lambda }^2\right)x^{l+1}}{\left[{\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)\right]\left(2l+1\right)!!},\text{if}0<x<1,\\ \frac{\left({\mu }^2-{\lambda }^2\right)}{{\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)i^{-l}{\text{e}}^{-i\lambda x}+\left({\mu }^2-{\lambda }^2\right)i^{2l+1}\mu {\text{e}}^{ix\mu }+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)i^{-l}{\text{e}}^{i\left(\lambda -2\mu \right)x}+\left({\lambda }^2-{\mu }^2\right)i^{l+1}\lambda {\text{e}}^{i\left(2\lambda -\mu \right)x}}\text{if}x>1,\mu \ne \lambda .\end{array}\end{array}$ (98)

Since

${{\phi }^{\prime }}_1=\{\begin{array}{l}\frac{\left(l+1\right)l{x}^l}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^{l+1}\lambda {\text{e}}^{i\lambda x},\text{if}x\ge 1.\hfill \end{array}$

it is obtained (see Appendix)

$\frac{{{\phi }^{\prime }}_1}{{\phi }_2}=\{\begin{array}{l}\frac{\left({\mu }^2-{\lambda }^2\right)l\left(l+1\right)x^{2l+1}}{\left(2l+1\right)!!\left[{\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)\right]\left(2l+1\right)!!},\text{if}0<x<1,\\ \frac{\left({\mu }^2-{\lambda }^2\right)i^{l+1}\lambda {\text{e}}^{i\lambda x}}{{\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)i^{-l}{\text{e}}^{-i\lambda x}+\left({\mu }^2-{\lambda }^2\right)i^{2l+1}\mu {\text{e}}^{ix\mu }+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)i^{-l}{\text{e}}^{i\left(\lambda -2\mu \right)x}+\left({\lambda }^2-{\mu }^2\right)i^{l+1}\lambda {\text{e}}^{i\left(2\lambda -\mu \right)x}}\text{if}x>1,\mu \ne \lambda \end{array}$ (99)

According to Appendix one gets the next asymptotic relation

$\frac{{{\phi }^{\prime }}_1}{{\phi }_2}=\{\begin{array}{l}C{x}^{2l+1},\text{if}0<x<1,\hfill \\ O\left(1\right),\text{if}x>1,\forall l\text{even}.\hfill \end{array}$ (100)

Since

$\frac{{y^{\prime }}_1}{y_1^2}=\{\begin{array}{l}\frac{\left(2l+1\right)!!\left(l+1\right)}{x^{l+2}},\text{if}0<x<1,\hfill \\ \frac{\mu }{i^{l-1}}{\text{e}}^{-i\mu x},\text{if}x>1.\hfill \end{array}$

Then

$\begin{array}{l}\frac{{{\phi }^{\prime }}_1}{{\phi }_2}+\frac{{y^{\prime }}_1}{y_1^2}\\ =\{\begin{array}{l}C{x}^{2l+1}+\frac{\left(2l+1\right)!!\left(l+1\right)}{x^{l+2}},\text{if}0<x<1,\hfill \\ O\left(1\right)+\frac{\mu }{i^{l-1}}{\text{e}}^{-i\mu x},\text{if}x>1,\forall l\text{even}.\hfill \end{array}\\ =\{\begin{array}{l}\frac{\left(2l+1\right)!!\left(l+1\right)}{x^{l+2}}+C{x}^{2l+1},\text{if}0<x<1,\hfill \\ \frac{\mu }{i^{l-1}}{\text{e}}^{-i\mu x}+O\left(1\right),\text{if}x>1,\forall l\text{even}.\hfill \end{array}\end{array}$

Therefore

$\frac{{{\phi }^{\prime }}_1}{{\phi }_2}+\frac{{y^{\prime }}_1}{y_1^2}=\{\begin{array}{l}\frac{\left(2l+1\right)!!\left(l+1\right)}{x^{l+2}}+C{x}^{2l+1},\text{if}0<x<1,\hfill \\ \frac{\mu }{i^{l-1}}{\text{e}}^{-i\mu x}+O\left(1\right),\text{if}x>1,\forall l\text{even}.\hfill \end{array}$ (101)

Now

$q_1\left(x\right)=q\left(x\right)+\frac{l\left(l+1\right)}{x^2}:=\{\begin{array}{l}q\left(x\right)+\frac{l\left(l+1\right)}{x^2},0<x<1,\hfill \\ q\left(x\right),x>1.\hfill \end{array}$ (102)

then

${\mu }^2-2q_1=\{\begin{array}{l}{\mu }^2-2q\left(x\right)-\frac{2l\left(l+1\right)}{x^2},0<x<1,\hfill \\ {\mu }^2-2q\left(x\right),x>1.\hfill \end{array}$ (103)

and

$q_2={\mu }^2-2q_1+\frac{{{\phi }^{\prime }}_1}{{\phi }_2}+\frac{{y^{\prime }}_1}{y_1^2},$

then

$\begin{array}{l}{q}_2\left(x\right)\\ =\{\begin{array}{l}{\mu }^2-2q\left(x\right)-\frac{2l\left(l+1\right)}{x^2},\text{if}0<x<1,\hfill \\ {\mu }^2-2q\left(x\right),\text{if}x>1.\hfill \end{array}+\{\begin{array}{l}\frac{\left(2l+1\right)!!\left(l+1\right)}{x^{l+2}}+C{x}^{2l+1},\text{if}0<x<1,\hfill \\ \frac{\mu }{i^{l-1}}{\text{e}}^{-i\mu x}+O\left(1\right),\text{if}x>1,\forall l\text{even}.\hfill \end{array}\\ =\{\begin{array}{l}{\mu }^2-2q\left(x\right)-\frac{2l\left(l+1\right)}{x^2}+\frac{\left(2l+1\right)!!\left(l+1\right)}{x^{l+2}}+C{x}^{2l+1},\text{if}0<x<1,\hfill \\ {\mu }^2-2q\left(x\right)+\frac{\mu }{i^{l-1}}{\text{e}}^{-i\mu x}+O\left(1\right),\text{if}x>1,\forall l\text{even}.\hfill \end{array}\end{array}$

Therefore

$q_2\left(x\right)=\{\begin{array}{l}{\mu }^2-2q\left(x\right)-\frac{2l\left(l+1\right)}{x^2}+\frac{\left(2l+1\right)!!\left(l+1\right)}{x^{l+2}}+C{x}^{2l+1},\text{if}0<x<1,\hfill \\ {\mu }^2-2q\left(x\right)+\frac{\mu }{i^{l-1}}{\text{e}}^{-i\mu x}+O\left(1\right),\text{if}x>1,\forall l\text{even}.\hfill \end{array}$ (104)

For the case

$0<x<1,$ (105)

Let’s consider the term

$-\frac{2l\left(l+1\right)}{x^2}+\frac{\left(2l+1\right)\mathrm{<mo>\; !\; </mo>\; <mo>\; !\; </mo>}\left(l+1\right)}{x^{l+2}}\mathrm{.}$

If we define

$a:=2l\left(2l+1\right)!!,b:=\left(2l+1\right)!!\left(l+1\right)\Rightarrow a>b>1\text{and}0<x\le \sqrt[l]{\frac{l+1}{2l}}<1$ (106)

then

$-\frac{2l\left(l+1\right)}{x^2}+\frac{\left(2l+1\right)!!\left(l+1\right)}{x^{l+2}}:=-\frac{a}{x^2}+\frac{b}{x^{l+2}}$, $x\le \sqrt[l]{\frac{l+1}{2l}}$

$\Rightarrow x^l\le \frac{l+1}{2l}=\frac{b}{a}\iff a{x}^l<b\Rightarrow a{x}^{l+2}<b{x}^2$

$\Rightarrow \frac{1}{a{x}^{l+2}}>\frac{1}{b{x}^2}\iff \frac{b}{x^{l+2}}>\frac{a}{x^2}\iff \frac{b}{x^{l+2}}-\frac{a}{x^2}>0.$ (107)

Let the infimum

$\alpha :=\inf \left\{\frac{b}{x^{l+2}}-\frac{a}{x^2}:0<x<\sqrt[l]{\frac{l+1}{2l}}\right\}$ (108)

then

$\alpha \ge \mathrm{0,}$ (109)

If

$\alpha >0,$ (110)

let $\epsilon >0,\exists x\in \left(0,\sqrt[l]{\frac{l+1}{2l}}\right)$ such that $\alpha \le \frac{b}{x^{l+2}}-\frac{a}{x^2}<\alpha +\epsilon ,\forall \epsilon >0.$ Then

$\frac{b}{x^{l+2}}-\frac{a}{x^2}=0$, (111)

!which is impossible!

Hence

$\alpha =0.$ (112)

If

$0<x\le \sqrt[l]{\frac{l+1}{2l}},$ (113)

it is obtained that

$-\frac{2l\left(l+1\right)}{x^2}+\frac{\left(2l+1\right)!!\left(l+1\right)}{x^{l+2}}\underset{x\to 0}{=}0,$ (114)

since

$C{\left(\frac{l\left(l+1\right)}{\left(2l+1\right)!!}\right)}^2{x}^{2l+1}\underset{x\to 0}{=}0.$ (115)

for

$0<x\le \sqrt[l]{\frac{l+1}{2l}}<1,$ (116)

so

$\begin{array}{l}{q}_2\left(x\right)={\mu }^2-2q\left(x\right)-\frac{2l\left(l+1\right)}{x^2}+\frac{\left(2l+1\right)!!\left(l+1\right)}{x^{l+2}}+C{\left(\frac{l\left(l+1\right)}{\left(2l+1\right)!!}\right)}^2{x}^{2l+1}\\ \underset{x\to 0}{=}{\mu }^2-2q\left(x\right).\end{array}$ (117)

That is

$q_2\left(x\right)\underset{x\to 0}{=}{\mu }^2-2q\left(x\right),\text{for}0<x\le \sqrt[l]{\frac{l+1}{2l}}<1.$ (118)

Therefore

$q_2\left(x\right)=\{\begin{array}{l}{\mu }^2-2q\left(x\right),\text{for}0<x\le \sqrt[l]{\frac{l+1}{2l}}<1\hfill \\ {\mu }^2-2q\left(x\right)+\frac{\mu }{i^{l-1}}{\text{e}}^{-i\mu x}+O\left(1\right),\text{if}x>1,l\text{even}.\hfill \end{array}$ (119)

So our final conclusion is, that we have obtained the uniqueness of this potential which is regular when $x\to 0$ and, bounded with exponentially decreases fast enough when $x\to \infty$.

4. The Estimates $L_1-L_{\infty }$ for the Reduced Radial Equation of Schrödinger

Theorem 2. (The $L^1-L^{\infty }$ estimates). Suppose that $V$ is regular and

$H=-\frac{{\text{d}}^2}{\text{d}{x}^2}+V\left(x\right)+\frac{l\left(l+1\right)}{x^2},0<x<\infty ,$ (120)

$\varphi \left(0\right)=0.$

Then

${\Vert {\text{e}}^{-itH}{P}_c\Vert }_{B\left(L^1,L^{\infty }\right)}\le C\frac{1}{\sqrt{t}},$ (121)

of moreover, $V\in L^1$, then

${\Vert {\text{e}}^{-itH}{P}_c\Vert }_{B\left(W_{1,1},W_{1,\infty }\right)}\le C\frac{1}{\sqrt{t}}.$ (122)

Proof. For Theorem 1, the operator $\mathbb{W}$ transforms the potential $V\left(x\right)+\frac{l\left(l+1\right)}{x^2}$ in univocally determinated potential

$q_2\left(x\right):=\{\begin{array}{l}{\mu }^2-2V\left(x\right),\text{for}0<x\le \sqrt[l]{\frac{l+1}{2l}}<1\hfill \\ {\mu }^2-2V\left(x\right)+\frac{\mu }{i^{l-1}}{\text{e}}^{-i\mu x}+O\left(1\right),\text{if}x>1,l\text{even}.\hfill \end{array}$

And if ${\phi }_1\left(x\right)$ is solution of (120) then

${\phi }_2\left(x\right)=\mathbb{W}{\phi }_1\left(x\right),$ (123)

is solution of

$-{{\phi }^{″}}_2\left(x\right)+q_2\left(x\right){\phi }_2\left(x\right)={\mu }^2{\phi }_2\left(x\right),$ (124)

For

$0<x<1$ (125)

we have $q_2\left(x\right)={\mu }^2-2V\left(x\right)\iff q_2\left(x\right)+V\left(x\right)={\mu }^2-V(x)$

$\iff \left[q_2\left(x\right)+V\left(x\right)\right]{\phi }_2\left(x\right)=\left[{\mu }^2-V\left(x\right)\right]{\phi }_2\left(x\right)=-{{\phi }^{″}}_2\left(x\right).$

That is

$-{{\phi }^{″}}_2\left(x\right)+V\left(x\right){\phi }_2\left(x\right)={\mu }^2{\phi }_2\left(x\right),$ (126)

which is Schrödinger Equation on the Half-Line (RSEHL) for $0<x<1$.

For the case

$x>1,$ (127)

The sum of the terms added to the potential

$\frac{\mu }{i^{l-1}}{\text{e}}^{-i\mu x}+O\left(1\right)\mathrm{,}$ (128)

form a term bound with exponentially decreases fast enough which allow that

${\displaystyle {\int }_b^{\infty }}x\left|q_2\left(x\right)\right|\text{d}x<\infty ,\text{for}b>0.$ (129)

See [5] , Chapter 1, Secc 1.1, formula (1.1.3), page 2 which guarantees that the equation (RSEHL) is also fulfilled in this case. Therefore (121) and (122) are satisfied. $\square$

5. Conclusions and Open Problems

As a comment by J. Bourgain in [7] , page 27, the importance of the $L^1-L^{\infty }$ estimates is mentioned, in the context of search of global solutions for the equations of non-linear Schrödinger. In particular, the obtained ones in [8] perform the form

${\Vert {\text{e}}^{it\left(\Delta -V\right)}\varphi \Vert }_{\infty }\le C{\left|t\right|}^{-d/2}{\Vert \varphi \Vert }_1,\text{for}d\ge 3.$ (130)

Also, in [7] , page 27, it is mentioned that “it would be most interesting to prove the analogue (19) in low dimension $d=1,2$. This is certainly a project of independent importance”. In [3] and [6] these estimations were demonstrated for the case $d=1$ on the line and half-line respectively. See [1] , page 487. Now we have established the $L^1-L^{\infty }$ estimates for the case Reduced Radial Schrödinger Equation (RRSE).

Finally as an open problem the possibility of extending the $L^1-L^{\infty }$ estimates for the (RSEHL) but now adding an analytical perturbation to the potential $q\left(x\right)$ ; as mentioned in [2] , 13 Open Problems, page 55, where is considered the differential operator

$Lu=-u^{″}+q\left(x\right)u,0<x<\infty .$ (131)

Since according to the Analytic Perturbation Theory the potential $q\left(x\right)$ can be written

$q\left(x\right)=q_1\left(x\right)+q_2\left(x\right)+q_3\left(x\right),$ (132)

where all the $q_i\left(x\right)$ are real valued, $q_1\left(x\right)\ge 0$ is locally integrable, $q_2\left(x\right)$ is uniformly locally integrable and $q_3\left(x\right)$ is majorized by $1/4x^2$. Then a closed form $b\left[u\right]={\displaystyle {\int }_0^{\infty }}\left({\left|u^{\prime }\right|}^2+q\left(x\right){\left|u\right|}^2\right)\text{d}x$ can be constructed and the associated selfadjoint operator (self-adjoint extension) H of L in $L^2\left(\mathrm{0,}\infty \right)$ with an appropriate domain (including the boundary condition $u\left(0\right)=0$ ). See [9] , Chapter 7, Section 8, pages 408-409.

Dedication

Dedicated to Professor Vladimir Alexandrovich Marchenko to celebrate his ninety-seventh birthday: born July 7, 1922. Sponsor: Fellow Sistema Nacional de Investigadores.

Acknowledgements

The author is grateful to the anonymous referees for their valuable comments. Also, to Miguel Navarro Saad and Gustavo Gómez Ros for their valuable help in technical editing and drafting the manuscript respectively. Especially to Jasmyn Chen, Editorial Assistant of APM for her generous willingness and professionalism.

Appendix

Next, the calculations of the addends

$\frac{{y^{″}}_1}{y_1}\mathrm{,}\frac{{{\phi }^{\prime }}_1}{{\phi }_2}\mathrm{,}\frac{{y^{\prime }}_1}{y_1^2}$ (133)

that form the potential

$q_2\left(x\right)=q_1-\frac{{y^{″}}_1}{y_1}+\frac{{{\phi }^{\prime }}_1}{{\phi }_2}+\frac{{y^{\prime }}_1}{y_1^2}$

will be shown.

It starts with

$y_1\left(x,\mu \right):=\{\begin{array}{l}\frac{x^{l+1}}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^l{\text{e}}^{i\mu x},\text{if}x\ge 1.\hfill \end{array}$

then

${y^{\prime }}_1\left(x,\mu \right):=\{\begin{array}{l}\frac{l+1}{\left(2l+1\right)!!}{x}^l,\text{if}0<x<1,\hfill \\ i^{l+1}\mu {\text{e}}^{i\mu x},\text{if}x\ge 1.\hfill \end{array}$

and since

${\varphi }_1\left(\lambda ,x\right)=\{\begin{array}{l}\frac{x^{l+1}}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^l{\text{e}}^{i\lambda x},\text{if}x\ge 1.\hfill \end{array}$

${{\varphi }^{\prime }}_1\left(x,\lambda \right):=\{\begin{array}{l}\frac{l+1}{\left(2l+1\right)!!}{x}^l,\text{if}0<x<1,\hfill \\ i^{l+1}\lambda {\text{e}}^{i\lambda x},\text{if}x\ge 1.\hfill \end{array}$

Since

$y_1\left(x,\mu \right):=\{\begin{array}{l}\frac{x^{l+1}}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^l{\text{e}}^{i\mu x},\text{if}x\ge 1.\hfill \end{array}$

then

$\frac{1}{y_1\left(x,\mu \right)}=\{\begin{array}{l}\frac{\left(2l+1\right)!!}{x^{l+1}},\text{if}0<x<1,\hfill \\ \frac{1}{i^l{\text{e}}^{i\mu x}},\text{if}x\ge 1.\hfill \end{array}$

$\frac{1}{y_1{\left(x,\mu \right)}^2}=\{\begin{array}{l}\frac{\left(2l+1\right)!{!}^2}{x^{2\left(l+1\right)}},\text{if}0<x<1,\hfill \\ \frac{1}{i^{2l}{\text{e}}^{2i\mu x}},\text{if}x\ge 1.\hfill \end{array}$

and

${y^{\prime }}_1\left(x,\mu \right):=\{\begin{array}{l}\frac{l+1}{\left(2l+1\right)!!}{x}^l,\text{if}0<x<1,\hfill \\ i^{l+1}\mu {\text{e}}^{i\mu x},\text{if}x\ge 1.\hfill \end{array}$

and

${y^{″}}_1\left(x,\mu \right):=\{\begin{array}{l}\frac{l\left(l+1\right)}{\left(2l+1\right)!!}{x}^{l-1},\text{if}0<x<1,\hfill \\ i^{l+2}{\mu }^2{\text{e}}^{i\mu x},\text{if}x\ge 1.\hfill \end{array}$

Therefore

$\begin{array}{c}\frac{{y^{″}}_1}{y_1}=\{\begin{array}{l}\frac{l\left(l+1\right)}{\left(2l+1\right)!!}{x}^{l-1},\text{if}0<x<1,\hfill \\ i^{l+2}{\mu }^2{\text{e}}^{i\mu x},\text{if}x\ge 1.\hfill \end{array}\times \{\begin{array}{l}\frac{\left(2l+1\right)!!}{x^{l+1}},\text{if}0<x<1,\hfill \\ \frac{1}{i^l{\text{e}}^{i\mu x}},\text{if}x\ge 1.\hfill \end{array}\\ =\{\begin{array}{l}\frac{l\left(l+1\right)}{\left(2l+1\right)!!}{x}^{l-1}\times \frac{\left(2l+1\right)!!}{x^{l+1}},\text{if}0<x<1,\hfill \\ i^{l+2}{\mu }^2{\text{e}}^{i\mu x}\frac{1}{i^l{\text{e}}^{i\mu x}},\text{if}x\ge 1.\hfill \end{array}\\ =\{\begin{array}{l}\frac{l\left(l+1\right)}{x^2},\text{if}0<x<1,\hfill \\ -{\mu }^2,\text{if}x\ge 1.\hfill \end{array}\end{array}$

That is

$\frac{{y^{″}}_1}{y_1}=\{\begin{array}{l}\frac{l\left(l+1\right)}{x^2},\text{if}0<x<1,\hfill \\ -{\mu }^2,\text{if}x\ge 1.\hfill \end{array}$ (134)

Now

$\begin{array}{c}\frac{{y^{\prime }}_1}{y_1^2}={y^{\prime }}_1\left(x,\mu \right):=\{\begin{array}{l}\frac{l+1}{\left(2l+1\right)!!}{x}^l,\text{if}0<x<1,\hfill \\ i^{l+1}\mu {\text{e}}^{i\mu x},\text{if}x\ge 1.\hfill \end{array}\times \{\begin{array}{l}\frac{\left(2l+1\right)!{!}^2}{x^{2\left(l+1\right)}},\text{if}0<x<1,\hfill \\ \frac{1}{i^{2l}{\text{e}}^{2i\mu x}},\text{if}x\ge 1.\hfill \end{array}\\ =\{\begin{array}{l}\frac{l+1}{\left(2l+1\right)!!}{x}^l\times \frac{\left(2l+1\right)!{!}^2}{x^{2\left(l+1\right)}},\text{if}0<x<1,\hfill \\ i^{l+1}\mu {\text{e}}^{i\mu x}\frac{1}{i^{2l}{\text{e}}^{2i\mu x}},\text{if}x\ge 1.\hfill \end{array}\\ =\{\begin{array}{l}\frac{\left(2l+1\right)!!\left(l+1\right)}{x^2},\text{if}0<x<1,\hfill \\ \frac{\mu }{i^{l-1}{\text{e}}^{i\mu x}},\text{if}x\ge 1.\hfill \end{array}\end{array}$

That is

$\frac{{y^{\prime }}_1}{y_1^2}=\{\begin{array}{l}\frac{\left(2l+1\right)!!\left(l+1\right)}{x^2},\text{if}0<x<1,\hfill \\ \frac{\mu }{i^{l-1}{\text{e}}^{i\mu x}},\text{if}x\ge 1.\hfill \end{array}$ (135)

Now since

$\begin{array}{l}W\left(y_1\left(x\mathrm{,}\mu \right)\mathrm{,}{\varphi }_1\left(\lambda \mathrm{,}x\right)\right)\\ ={\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)\frac{y_1\left(x\mathrm{,}\mu \right)}{y_1\left(x\mathrm{,}\lambda \right)}+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)\frac{{\varphi }_1\left(\lambda \mathrm{,}x\right)}{{\varphi }_1\left(\mu \mathrm{,}x\right)}\\ +\frac{y_1\left(x\mathrm{,}\mu \right)}{y_1\left(x\mathrm{,}\lambda \right)}{y^{\prime }}_1\left(x\mathrm{,}\lambda \right){\varphi }_1\left(\lambda \mathrm{,}x\right)-\frac{y_1\left(x\mathrm{,}\mu \right)}{{\varphi }_1\left(\mu \mathrm{,}x\right)}{\varphi }_1\left(\lambda \mathrm{,}x\right){{\varphi }^{\prime }}_1\left(\mu \mathrm{,}x\right)\mathrm{.}\end{array}$

let us calculate each of the $W\left(y_1\left(x\mathrm{,}\mu \right)\mathrm{,}{\varphi }_1\left(\lambda \mathrm{,}x\right)\right)$ terms. Since

$\frac{y_1\left(x,\mu \right)}{y_1\left(x,\lambda \right)}=\{\begin{array}{l}1,\text{if}0<x<1,\hfill \\ {\text{e}}^{i\left(\mu -\lambda \right)x},\text{if}x>1.\hfill \end{array}$

$\frac{{\varphi }_1\left(\lambda ,x\right)}{{\varphi }_1\left(\mu ,x\right)}=\{\begin{array}{l}1,\text{if}0<x<1,\hfill \\ {\text{e}}^{i\left(\lambda -\mu \right)x},\text{if}x>1.\hfill \end{array}$

$y_1\left(x,\mu \right):=\{\begin{array}{l}\frac{x^{l+1}}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^l{\text{e}}^{i\mu x},\text{if}x>1.\hfill \end{array}$

${y^{\prime }}_1\left(x,\mu \right):=\{\begin{array}{l}\frac{l+1}{\left(2l+1\right)!!}{x}^l,\text{if}0<x<1,\hfill \\ i^{l+1}\mu {\text{e}}^{i\mu x},\text{if}x>1.\hfill \end{array}$

${\varphi }_1\left(\lambda ,x\right)=\{\begin{array}{l}\frac{x^{l+1}}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^l{\text{e}}^{i\lambda x},\text{if}x>1.\hfill \end{array}$

${{\varphi }^{\prime }}_1\left(\lambda ,x\right)=\{\begin{array}{l}\frac{\left(l+1\right)x^l}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^{l+1}\lambda {\text{e}}^{i\lambda x},\text{if}x>1.\hfill \end{array}$

first let’s calculate the terms

$\begin{array}{l}\frac{y_1\left(x,\mu \right)}{y_1\left(x,\lambda \right)}{y^{\prime }}_1\left(x,\lambda \right){\varphi }_1\left(\lambda ,x\right)\\ =\{\begin{array}{l}\frac{l+1}{\left(2l+1\right)!!}{x}^l,\text{if}0<x<1,\hfill \\ i^{l+1}\mu {\text{e}}^{i\mu x},\text{if}x>1.\hfill \end{array}\times \{\begin{array}{l}\frac{x^{l+1}}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^l{\text{e}}^{i\lambda x},\text{if}x>1.\hfill \end{array}\times \{\begin{array}{l}1,\text{if}0<x<1,\hfill \\ {\text{e}}^{i\left(\mu -\lambda \right)x},\text{if}x>1.\hfill \end{array}\\ =\{\begin{array}{l}\frac{\left(l+1\right)}{\left(2l+1\right)!{!}^2}{x}^{2l+1},\text{if}0<x<1,\hfill \\ i^{2l+1}\lambda {\text{e}}^{i\left(\lambda +\mu \right)x},\text{if}x>1.\hfill \end{array}\end{array}$

That is

$\frac{y_1\left(x,\mu \right)}{y_1\left(x,\lambda \right)}{y^{\prime }}_1\left(x,\lambda \right){\varphi }_1\left(\lambda ,x\right)=\{\begin{array}{l}\frac{\left(l+1\right)}{\left(2l+1\right)!{!}^2}{x}^{2l+1},\text{if}0<x<1,\hfill \\ i^{2l+1}\lambda {\text{e}}^{i\left(\lambda +\mu \right)x},\text{if}x>1.\hfill \end{array}$

Next

$\begin{array}{l}\frac{y_1\left(x,\mu \right)}{{\varphi }_1\left(\mu ,x\right)}{\varphi }_1\left(\lambda ,x\right){{\varphi }^{\prime }}_1\left(\mu ,x\right)=\frac{{\varphi }_1\left(\lambda ,x\right)}{{\varphi }_1\left(\mu ,x\right)}{y}_1\left(x,\mu \right){{\varphi }^{\prime }}_1\left(\mu ,x\right)\\ =\{\begin{array}{l}1,\text{if}0<x<1,\hfill \\ {\text{e}}^{i\left(\lambda -\mu \right)x},\text{if}x>1.\hfill \end{array}\times \{\begin{array}{l}\frac{x^{l+1}}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^l{\text{e}}^{i\mu x},\text{if}x>1.\hfill \end{array}\times \{\begin{array}{l}\frac{\left(l+1\right)x^l}{\left(2l+1\right)!!},\text{if}0<x<1,\hfill \\ i^{l+1}\lambda {\text{e}}^{i\lambda x},\text{if}x>1.\hfill \end{array}\end{array}$

That is

$\frac{y_1\left(x,\mu \right)}{{\varphi }_1\left(\mu ,x\right)}{\varphi }_1\left(\lambda ,x\right){{\varphi }^{\prime }}_1\left(\mu ,x\right)=\{\begin{array}{l}\frac{l+1}{\left(2l+1\right)!{!}^2}{x}^{2l+1},\text{if}0<x<1,\hfill \\ i^{l+1}\mu {\text{e}}^{i\left(\lambda +\mu \right)x},\text{if}x>1.\hfill \end{array}$

Then

$\begin{array}{l}\frac{y_1\left(x,\mu \right)}{y_1\left(x,\lambda \right)}{y^{\prime }}_1\left(x,\lambda \right){\varphi }_1\left(\lambda ,x\right)-\frac{y_1\left(x,\mu \right)}{{\varphi }_1\left(\mu ,x\right)}{\varphi }_1\left(\lambda ,x\right){{\varphi }^{\prime }}_1\left(\mu ,x\right)\\ =\{\begin{array}{l}\frac{\left(l+1\right)}{\left(2l+1\right)!{!}^2}{x}^{2l+1},\text{if}0<x<1,\hfill \\ i^{2l+1}\lambda {\text{e}}^{i\left(\lambda +\mu \right)x},\text{if}x>1.\hfill \end{array}-\{\begin{array}{l}\frac{l+1}{\left(2l+1\right)!{!}^2}{x}^{2l+1},\text{if}0<x<1,\hfill \\ i^{l+1}\mu {\text{e}}^{i\left(\lambda +\mu \right)x},\text{if}x>1.\hfill \end{array}\\ =\{\begin{array}{l}0,\text{if}0<x<1,\hfill \\ \left(i^l\lambda -\mu \right)i^{l+1}{\text{e}}^{i\left(\lambda +\mu \right)x},\text{if}x>1.\hfill \end{array}\end{array}$

Therefore

$\begin{array}{l}\frac{y_1\left(x,\mu \right)}{y_1\left(x,\lambda \right)}{y^{\prime }}_1\left(x,\lambda \right){\varphi }_1\left(\lambda ,x\right)-\frac{y_1\left(x,\mu \right)}{{\varphi }_1\left(\mu ,x\right)}{\varphi }_1\left(\lambda ,x\right){{\varphi }^{\prime }}_1\left(\mu ,x\right)\\ =\{\begin{array}{l}0,\text{if}0<x<1,\hfill \\ \left(i^l\lambda -\mu \right)i^{l+1}{\text{e}}^{i\left(\lambda +\mu \right)x},\text{if}x>1.\hfill \end{array}\end{array}$

Now let’s calculate the term:

${\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)\frac{y_1\left(x,\mu \right)}{y_1\left(x,\lambda \right)}+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)\frac{{\varphi }_1\left(\lambda ,x\right)}{{\varphi }_1\left(\mu ,x\right)}$

${\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)\{\begin{array}{l}1,\text{if}0<x<1,\hfill \\ {\text{e}}^{i\left(\mu -\lambda \right)x}\text{if}x\ge 1.\hfill \end{array}+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)\{\begin{array}{l}1,\text{if}0<x<1,\hfill \\ {\text{e}}^{i\left(\lambda -\mu \right)x},\text{if}x\ge 1.\hfill \end{array}$

$\begin{array}{l}\{\begin{array}{l}{\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right),\text{if}0<x<1,\hfill \\ {\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right){\text{e}}^{i\left(\mu -\lambda \right)x},\text{if}x>1.\hfill \end{array}+\{\begin{array}{l}{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right),\text{if}0<x<1,\hfill \\ {\left(-\mu \right)}^{-l}{F}_l\left(\mu \right){\text{e}}^{i\left(\lambda -\mu \right)x},\text{if}x>1.\hfill \end{array}\\ +\{\begin{array}{l}{\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right),\text{if}0<x<1,\hfill \\ {\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right){\text{e}}^{i\left(\mu -\lambda \right)x}+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right){\text{e}}^{i\left(\lambda -\mu \right)x},\text{if}x>1.\hfill \end{array}\end{array}$

That is

$\begin{array}{l}{\left(-1\right)}^{-l}{F}_l\left(\lambda \right)\frac{y_1\left(x,\mu \right)}{y_1\left(x,\lambda \right)}+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)\frac{{\varphi }_1\left(\lambda ,x\right)}{{\varphi }_1\left(\mu ,x\right)}\\ =\{\begin{array}{l}{\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right),\text{if}0<x<1,\hfill \\ {\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right){\text{e}}^{i\left(\mu -\lambda \right)x}+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right){\text{e}}^{i\left(\lambda -\mu \right)x}\text{if}x>1.\hfill \end{array}\end{array}$

Then

$\begin{array}{l}W\left(y_1\left(x,\mu \right),{\varphi }_1\left(\lambda ,x\right)\right)\\ ={\left(-\lambda \right)}^{-l}{F}_l\left(\lambda \right)\frac{y_1\left(x,\mu \right)}{y_1\left(x,\lambda \right)}+{\left(-\mu \right)}^{-l}{F}_l\left(\mu \right)\frac{{\varphi }_1\left(\lambda ,x\right)}{{\varphi }_1\left(\mu ,x\right)}\\ +\frac{y_1\left(x,\mu \right)}{y_1\left(x,\lambda \right)}{y^{\prime }}_1\left(x,\lambda \right){\varphi }_1\left(\lambda ,x\right)-\frac{y_1\left(x,\mu \right)}{{\varphi }_1(\mu }\end{array}$