Classes of uniform convergence of spectral expansions for the one-dimensional Schrödinger operator with a distribution potential

Abstract

For the self-adjoint Schrödinger operator ℒ defined on ℝ by the differential operation −(d/dx)² + q(x) with a distribution potential q(x) uniformly locally belonging to the space W₂⁻¹, we describe classes of functions whose spectral expansions corresponding to the operator ℒ absolutely and uniformly converge on the entire line ℝ. We characterize the sharp convergence rate of the spectral expansion of a function using a two-sided estimate obtained in the paper for its generalized Fourier transforms.