Weighted Mourre's commutator theory, application to Schr\"odinger operators with oscillating potential

TL;DR

This paper introduces a modified Mourre's commutator theory to establish the limiting absorption principle for Schrödinger operators with long-range oscillating potentials, extending previous results and demonstrating limitations of traditional Mourre theory.

Contribution

A novel variant of Mourre's commutator theory is developed and applied to Schrödinger operators with long-range oscillating potentials, allowing new spectral analysis results.

Findings

Proved limiting absorption principle for Schrödinger operators with oscillating potentials.

Showed traditional Mourre theory cannot handle these operators.

Extended spectral analysis to long-range perturbed Wigner-Von Neumann potentials.

Abstract

We present a variant of Mourre's commutator theory. We apply it to prove the limiting absorption principle for Schr\"odinger operators with a perturbed Wigner-Von Neumann potential at suitable energies. To our knowledge, this result is new since we allow a long range pertubation of the Wigner-Von Neumann potential. Furthermore, we can show that the usual Mourre theory, based on differential inequalities and on the generator of dilations, cannot apply to our Schr\"odinger operators.