Direct and Inverse Sturm-Liouville Problems

Inhaltsverzeichnis

1. Frontmatter

2. 1. Introduction

   Vladislav V. Kravchenko

   Abstract

   Since the pioneering work of D. Bernoulli, J. d’Alembert, L. Euler, J. Fourier and later on of S. D. Poisson, Ch. Sturm and J. Liouville, the theory of Sturm-Liouville problems is an integral part of the professional preparation of mathematicians, physicists and engineers, and at the same time an important and actively developing research field.

3. Typical Problem Statements

   1. Frontmatter

   2. 2. Preliminaries on Sturm-Liouville Equations

      Vladislav V. Kravchenko

      Abstract

      We will consider the second-order linear ordinary differential equation

      $$\displaystyle -y^{\prime \prime }+q(x)y=\lambda y $$

      on a finite or infinite interval. This equation is called the Sturm-Liouville equation, or often the one-dimensional Schrö dinger equation . Conditions satisfied by the coefficient q(x), frequently called the potential, will be specified along the way, depending on the problem under consideration. The complex number λ is called the spectral parameter . Very often it will be convenient to work with its square root \(\rho := \sqrt {\lambda }\), which is usually chosen so that Imρ ≥ 0.

   3. 3. Direct and Inverse Sturm-Liouville Problems on Finite Intervals

      Vladislav V. Kravchenko

      Abstract

      Consider Eq. (2.1) on the interval (0, π). If originally the equation is considered on another finite interval (a, b), then by a simple change of the independent variable \(t=\frac {x-a}{b-a}\pi \) it can always be transferred to (0, π). Let q be a real-valued function, q ∈ L ₂(0, π) and h, H be real numbers. Together with the equation

      $$\displaystyle -y^{\prime \prime }+q(x)y=\lambda y,\quad 0<x<\pi $$

      consider the homogeneous boundary conditions

      $$\displaystyle y^{\prime }(0)-hy(0)=y^{\prime }(\pi )+Hy(\pi )=0. $$

      There exists an infinite sequence of real numbers \(\left \{ \lambda _{n}\right \} _{n=0}^{\infty }\) such that λ _n < λ _m if n < m , λ _n → +∞ when n →∞, and for every λ _n the equation

      $$\displaystyle -y_{n}^{\prime \prime }+q(x)y_{n}=\lambda _{n}y_{n},\quad 0<x<\pi $$

      admits a nontrivial solution y _n satisfying the conditions (3.2).

   4. 4. Direct and Inverse Sturm-Liouville Problems on a Half-Line

      Vladislav V. Kravchenko

      Abstract

      Consider Eq. (2.1) on a half-line:

      $$\displaystyle -y^{\prime \prime }+q(x)y=\lambda y,\quad x>0, $$

      where q(x) is a real-valued function satisfying the condition

      $$\displaystyle \int _{0}^{\infty }\left ( 1+x\right ) \left \vert q(x)\right \vert dx<\infty . $$

      Roughly speaking, this condition means that the potential decays rather rapidly at infinity and is integrable on any finite interval. In physics such potentials are called short-range potentials . Consider an initial condition

      $$\displaystyle y^{\prime }(0)-hy(0)=0,\quad h\in \mathbb {R}. $$

   5. 5. Quantum Scattering Problem on the Half-Line

      Vladislav V. Kravchenko

      Abstract

      Consider Eq. (4.1), where the potential q satisfies the condition (4.2). Let e(ρ, x) be the corresponding Jost solution. The function F(ρ) := e(ρ, 0) is called the Jost function and the quotient

      $$\displaystyle S(\rho ):= \frac {F(-\rho )}{F(\rho )} $$

      is traditionally called the scattering matrix, or simply S-matrix (see, e.g., [51]). Notice that due to (4.7) we have that

      $$\displaystyle F(\rho )=1+ \frac {\omega (x)}{i\rho }+o\left (\frac {1}{\rho }\right ),\quad \left \vert \rho \right \vert \rightarrow \infty . $$

      Instead of the initial condition (4.3), consider the condition

      $$\displaystyle y(0)=0. $$

   6. 6. Scattering Problem on the Line

      Vladislav V. Kravchenko

      Abstract

      Consider now the one-dimensional Schrödinger equation on the whole real line:

      $$\displaystyle -y^{\prime \prime }+q(x)y=\lambda y,\quad x\in (-\infty ,\infty ), $$

      where q(x) is a real-valued function defined on (−∞, ∞) and satisfies the condition

      $$\displaystyle \int _{-\infty }^{\infty }\left ( 1+\left \vert x\right \vert \right ) \left \vert q(x)\right \vert dx<\infty . $$

      Besides the Jost solution at plus infinity, let us introduce the Jost solution at minus infinity, defined by the asymptotic relations,

      $$\displaystyle g^{\left ( \nu \right ) }(\rho ,x)=\left ( -i\rho \right ) ^{\nu }e^{-i\rho x}\left ( 1+o(1)\right ) ,\quad x\rightarrow -\infty ,\quad \nu =0,1, $$

      uniformly in \(\overline {\Omega _{+}}\).

   7. 7. Inverse Scattering Transform Method

      Vladislav V. Kravchenko

      Abstract

      In this chapter we give a brief description of a spectacular and very important application of the direct and inverse scattering problems formulated in the previous chapter. Here the following remark is pertinent. Among specialists in applied fields there can be encountered an opinion that the mathematical theory of the inverse spectral problems has a limited applicability because in practice only a very restricted part of the spectral data can be available from measurements, typically this concerns inverse Sturm-Liouville problems on a finite interval. Needless to say that the powerful mathematical theory specifying conditions for the existence and uniqueness of the solution as well as offering certain methods for its computation is of great value also in practical problems. However, the inverse scattering transform method gives us a beautiful example of application of the direct and inverse spectral problems for which “all the data” are indeed required and can be known in practice. Here the quotes are used of course because when it comes to computation finite instead of infinite intervals can be considered and hence the knowledge of the reflection coefficient on a finite interval only can be used for recovering the potential. Nevertheless, up to this refinement, the complete spectral data can be supposed to be known.

4. Transmutation Operators and Series Representations for Solutions of Sturm-Liouville Equations

   1. Frontmatter

   2. 8. Main Transmutation Operators

      Vladislav V. Kravchenko

      Abstract

      Let us consider a solution u of the equation

      $$\displaystyle -u^{\prime \prime }+q(x)u=\lambda u,\quad -b<x<b $$

      satisfying the initial conditions

      $$\displaystyle u(\rho ,0)=1,\quad u^{\prime }(\rho ,0)=i\rho , $$

      \(\rho =\sqrt {\lambda }\in \mathbb {C}\). Here the underlying interval is supposed to be symmetric and q is a complex-valued function belonging to L ₂(−b, b). The following important result is well known.

   3. 9. Construction of Transmutations and Series Representations for Solutions

      Vladislav V. Kravchenko

      Abstract

      Let f be a solution of the equation f ^′′− q(x)f = 0 on the interval (0, b) and f(0) = 1, f ^′(0) = h. Thus, f(x) = φ(0, x) which is the solution of the Cauchy problem (3.3) and (3.4) with λ = 0.

   4. 10. Series Representations for the Kernel A(x, t) and for the Jost Solution

      Vladislav V. Kravchenko

      Abstract

      In this chapter we derive, following Delgado et al. (Math Methods Appl Sci (2019). https://doi.org/10.1002/mma.5881) and Kravchenko (Math. Methods Appl. Sci. 42, 1321–1327, 2019), a series representation for the Jost solution e(ρ, x) introduced in Chap. 4, explaining all the steps in detail. An analogous representation for the second Jost solution g(ρ, x) (see Chap. 6) is given in Chap. 16, where it is used to solve the inverse scattering problem on the line.

5. Solution of Direct Sturm-Liouville Problems

   1. Frontmatter

   2. 11. Sturm–Liouville Problems on Finite Intervals

      Vladislav V. Kravchenko

      Abstract

      In this chapter, following Kravchenko and Porter (Math Method Appl Sci 33, 459–468, 2010) and Kravchenko et al. (Appl Math Comput 314(1), 173–192, 2017), we briefly explain how the SPPS and NSBF representations from Sect. 2.2 and Chap. 9, respectively, can be used for solving the Sturm–Liouville problem on a finite interval.

   3. 12. Spectral Problems on Infinite Intervals

      Vladislav V. Kravchenko

      Abstract

      In this section we consider the Sturm–Liouville problem on the half-line (see Chap. 4) consisting in computation of spectral data of the problem.

6. Solution of Inverse Sturm-Liouville Problems

   1. Frontmatter

   2. 13. The Inverse Sturm–Liouville Problem on a Finite Interval

      Vladislav V. Kravchenko

      Abstract

      In Sect. 3.2 we introduced the classical inverse Sturm–Liouville problem on a finite interval which is formulated as follows. Given two sequences of real numbers \(\left \{ \lambda _{n}\right \} _{n=0}^{\infty }\) and \(\left \{ \alpha _{n}\right \} _{n=0}^{\infty }\) such that λ _n < λ _m for n < m, α _n > 0, and the relations (3.5) are valid. Find the real-valued potential q(x) and the real numbers h and H, such that \(\left \{ \lambda _{n}\right \} _{n=0}^{\infty }\) is the spectrum of the Sturm–Liouville problem

      $$\displaystyle -y^{\prime \prime }+q(x)y=\lambda y,\quad 0<x<\pi , $$

      $$\displaystyle y^{\prime }(0)-hy(0)=y^{\prime }(\pi )+Hy(\pi )=0, $$

      and α _n, n = 0, 1, … are the corresponding norming constants.

   3. 14. Solution of the Inverse Problem on the Half-Line

      Vladislav V. Kravchenko

      Abstract

      In the present chapter we consider the inverse Sturm-Liouville problem on the half-line introduced in Chap. 4. Thus, we assume that the set of the spectral data

      $$\displaystyle \left \{ V(\lambda ),\,\lambda >0;\quad \left \{ \lambda _{n},\,\alpha _{n}\right \} _{n=\overline {1,N}}\right \} $$

      is known, where V (λ) is continuous and positive for λ > 0, the eigenvalues λ _n, if they exist, are negative, and α _n > 0. The function V (λ) has the asymptotics [69], [172, p. 147]

      $$\displaystyle V(\lambda )=\frac {1}{\pi \rho }\left ( 1+o\left ( \frac {1}{\rho }\right ) \right ) ,\quad \rho >0,\quad \rho \rightarrow \infty . $$

   4. 15. Solution of the Inverse Quantum Scattering Problem on the Half-Line

      Vladislav V. Kravchenko

      Abstract

      Here we consider the inverse problem formulated in Chap. 5. Given the scattering data (5.4), find the corresponding short-range potential q.

   5. 16. Solution of the Inverse Scattering Problem on the Line

      Vladislav V. Kravchenko

      Abstract

      In this chapter we consider the inverse scattering problem on the line introduced in Chap. 6. Given a set of scattering data J ⁺ or J ⁻ as in (6.3), find the potential q(x) satisfying the condition (6.2).

7. Backmatter