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

A new representation of solutions to the equation y+q(x)y=2y is obtained. For every x the solution is represented as a Neumann series of Bessel functions depending on the spectral parameter . 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 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.