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 $u(ω,x)$ of the equation $-u"+q(x)u=ω^2 u$, satisfying the initial conditions $u(ω,0)=1$, $u'(ω,0)=iω$ 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 $α_n$ are calculated by a recurrent integration procedure. The following estimate is proved $\vert u(ω,x) -u_N(ω,x)\vert \leq \frac{1}{\vert ω\vert^2}\varepsilon_N(x)\sqrt{\frac{\sinh(2\mathop{\rm Im}ω\,x)}{\mathop{\rm Im}ω}}$ for any $ω\in\mathbb{C}\backslash \{0\}$, where $u_N(ω,x)$ is an approximate solution given by truncating the series in the representation for $u(ω,x)$ and $\varepsilon_N(x)$ is a nonnegative function tending to zero for all $x$ belonging to a finite interval of interest. In particular, for $ω\in\mathbb{R}\backslash \{0\}$ the estimate has the form $\vert u(ω,x)-u_N(ω,x)\vert \leq \frac{1}{\vertω\vert^2}\varepsilon_N(x)$. A numerical illustration of application of the new representation for computing the solution $u(ω,x)$ on large sets of values of the spectral parameter $ω$ with an accuracy nondeteriorating (and even improving) when $ω\rightarrow \pm \infty$ is given.