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

摘要

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) (1 + u(1)(x)/omega + u(2)(x)/omega(2)) + e(-i omega x)u(3)(x)/omega(2) - 1/omega(2) Sigma(infinity)(n=0) 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 vertical bar u(omega, x)-u(N)(omega, x)vertical bar = 1/vertical bar omega vertical bar(2) epsilon(N)(x) root sinh(2 Im omega x)/Im omega for any u is an element of C{0}, where u(N)(omega, x) is an approximate solution given by truncating the series in the proposed representation for u(omega, x) and epsilon(N)(x) is a non-negative function tending to zero for all x belonging to a finite interval of interest. In particular, for omega is an element of R{0}, the estimate has the form vertical bar u(omega, x) - u(N)(omega, x)vertical bar = 1/vertical bar omega vertical bar(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 ! with an accuracy nondeteriorating (and even improving) when omega - +/- infinity is given. Published by AIP Publishing.