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

A representation for a solution $u(\omega ,x)$ of the equation $-u\mathrm{"}+q(x)u={\omega }^{2}u$, satisfying the initial conditions $u(\omega ,0)=1$, $u^{\mathrm{\prime }}(\omega ,0)=i\omega$ is derived in the form $$u(\omega ,x)=e^{i\omega x}(1+\frac{u_1(x)}{\omega }+\frac{u_2(x)}{{\omega }^2})+\frac{e^{-i\omega x}{u}_3(x)}{{\omega }^2}-\frac{1}{{\omega }^2}\sum _{n=0}^{\mathrm{\infty }}{i}^n{\alpha }_n(x)j_n(\omega x),$$ 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 $\mathrm{\mid }u(\omega ,x)-u_N(\omega ,x)\mathrm{\mid }\le \frac{1}{\mathrm{\mid }\omega {\mathrm{\mid }}^2}{\epsilon }_N(x)\sqrt{\frac{\sinh (2\omega x)}{\omega }}$ for any $\omega \in \mathbb{C}\mathrm{\backslash }\{0\}$, 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 $\omega \in \mathbb{R}\mathrm{\backslash }\{0\}$ the estimate has the form $\mathrm{\mid }u(\omega ,x)-u_N(\omega ,x)\mathrm{\mid }\le \frac{1}{\mathrm{\mid }\omega {\mathrm{\mid }}^2}{\epsilon }_N(x)$. 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 $\omega \to \pm \mathrm{\infty }$ is given.