Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions

Abstract

A new representation of solutions to the equation $-y"+q(x)y={\omega }^{2}y$ is obtained. For every $x$ the solution is represented as a Neumann series of Bessel functions depending on the spectral parameter $\omega$. Due to the fact that the representation is obtained using the corresponding transmutation operator, a partial sum of the series approximates the solution uniformly with respect to $\omega$ which makes it especially convenient for the approximate solution of spectral problems. The numerical method based on the proposed approach allows one to compute large sets of eigendata with a nondeteriorating accuracy.

26 pages, 7 figures, 3 tables. Added new Proposition 4.4, introduction expanded; some typos correcetd

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