A representation for Jost solutions and an efficient method for solving the spectral problem on the half line

Abstract

For the one‐dimensional Schrödinger equation with short‐range potential on a half‐line x >0, the knowledge of the Jost solution e ( ρ , x )∼ e i ρ x , Im ρ ≥ 0, x → ∞ allows one to solve corresponding spectral problems. In the present work, a new series representation for e ( ρ , x ) is derived with the aid of the Levin formula for the Jost solution and a recently proposed Fourier‐Laguerre series expansion of the integral kernel from the Levin formula. The representation for e ( ρ , x ) has the form , where, for the coefficients b n ( x ), a simple recurrent integration procedure is obtained and the parameter belongs to the unit disk. An analogous representation is derived for the derivative of the Jost solution as well. With the aid of the series representations, numerical solution of the classical spectral problem on the half‐line becomes an easy task. Indeed, computation of the eigenvalues reduces to finding zeros of a polynomial for z ∈(−1,1). For computing corresponding normalizing constants, a simple formula is derived. Moreover, computation of the (usually) most difficult spectral characteristics, the spectral density function (or its derivative) for all positive values of the spectral parameter reduces to calculation of an easily computable function along the unitary upper half‐circle. Numerical illustrations of the efficiency of the computational method based on the obtained results are presented.