Uniform, on the entire axis, convergence of the spectral expansion for Schrödinger operator with a potential-distribution

Abstract

A uniform, on ℝ, estimate for the increment of the spectral function θ(λ; x, y) at x = y is proved for the self-adjoint Schrödinger operator A defined on the entire axis ℝ by the differential operation (−d/dx)² + q(x) with a potential-distribution q(x) that uniformly locally belongs to the space W₂⁻¹. As a consequence, it is shown that for any function f(x) from the domain of power Aα of the operator with α > 1/4, the spectral expansion of the function that corresponds to the operator A is convergent absolutely and uniformly on the entire axis ℝ.