The Absence of Positive Energy Bound States for a Class of Nonlocal Potentials

TL;DR

This paper extends a classical theorem to establish conditions under which certain nonlocal potentials do not admit positive energy bound states in the radial Schrödinger equation, enhancing understanding of spectral properties.

Contribution

It generalizes Titchmarsh's theorem to nonlocal potentials and provides simple criteria for the absence of positive energy bound states in quantum mechanics.

Findings

Derived conditions for absence of positive energy bound states

Extended classical Fourier integral theorems to nonlocal potentials

Applied results to specific quantum mechanical potentials

Abstract

We generalize in this paper a theorem of Titchmarsh for the positivity of Fourier sine integrals. We apply then the theorem to derive simple conditions for the absence of positive energy bound states (bound states embedded in the continuum) for the radial Schr\"odinger equation with nonlocal potentials which are superposition of a local potential and separable potentials.