Asymptotics with respect to the spectral parameter and Neumann series of Bessel functions for solutions of the one-dimensional Schrödinger equation

Abstract

A representation for a solution $u(\omega ,x)$ of the equation $-u"+q(x)u={\omega }^{2}u$, satisfying the initial conditions $u(\omega ,0)=1$, $u^{\prime }(\omega ,0)=i\omega$ is derived in the form \[ u(ω,x)=e^{iωx}\left( 1+\frac{u_1(x)}ω+ \frac{u_2(x)}{ω^2}\right) +\frac{e^{-iωx}u_3(x)}{ω^2}-\frac{1}{ω^2}\sum_{n=0}^{\infty} i^{n}α_n(x)j_n(ωx), \] where $u_m(x)$, $m=1,2,3$ are given in a closed form, $j_n$ stands for a spherical Bessel function of order $n$ and the coefficients ${\alpha }_n$ are calculated by a recurrent integration procedure. The following estimate is proved $|u(\omega ,x)-u_N(\omega ,x)|\le \frac{1}{|\omega {|}^2}{\epsilon }_N(x)\sqrt{\frac{\sinh (2\mathrm{Im}\omega x)}{\mathrm{Im}\omega }}$ for any $\omega \in \mathbb{C}\backslash \{0\}$, where $u_N(\omega ,x)$ is an approximate solution given by truncating the series in the representation for $u(\omega ,x)$ and ${\epsilon }_N(x)$ is a nonnegative function tending to zero for all $x$ belonging to a finite interval of interest. In particular, for $\omega \in \mathbb{R}\backslash \{0\}$ the estimate has the form $|u(\omega ,x)-u_N(\omega ,x)|\le \frac{1}{|\omega {|}^2}{\epsilon }_N(x)$. A numerical illustration of application of the new representation for computing the solution $u(\omega ,x)$ on large sets of values of the spectral parameter $\omega$ with an accuracy nondeteriorating (and even improving) when $\omega \to \pm \infty$ is given.

12pages, 1 figure

References in corpus (2)

• Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions
• Analytic approximation of transmutation operators and related systems of functions

Cited by in corpus (1)

• Reconstruction techniques for complex potentials