Asymptotics with respect to the spectral parameter and Neumann series of Bessel functions for solutions of the one-dimensional Schr\"odinger equation

Abstract

A representation for a solution $u(\omega,x)$ of the equation $-u"+q(x)u=\omega^2 u$, satisfying the initial conditions $u(\omega,0)=1$, $u'(\omega,0)=i\omega$ is derived in the form \[ u(\omega,x)=e^{i\omega x}\left( 1+\frac{u_1(x)}{\omega}+ \frac{u_2(x)}{\omega^2}\right) +\frac{e^{-i\omega x}u_3(x)}{\omega^2}-\frac{1}{\omega^2}\sum_{n=0}^{\infty} i^{n}\alpha_n(x)j_n(\omega x), \] 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 $\vert u(\omega,x) -u_N(\omega,x)\vert \leq \frac{1}{\vert \omega \vert^2}\varepsilon_N(x)\sqrt{\frac{\sinh(2\mathop{\rm Im}\omega\,x)}{\mathop{\rm Im}\omega}}$ 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 $\varepsilon_N(x)$ 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 $\vert u(\omega,x)-u_N(\omega,x)\vert \leq \frac{1}{\vert\omega\vert^2}\varepsilon_N(x)$. 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 $\omega \rightarrow \pm \infty$ is given.