Ito's diffusion in multidimensional scattering with sign-indefinite potentials

TL;DR

This paper demonstrates the existence of absolutely continuous spectrum for multidimensional Schrödinger operators with certain sign-indefinite potentials, using a combination of recent techniques involving stochastic trajectories defined by Ito's equation.

Contribution

It introduces a novel approach combining two recent methods to establish spectral properties of Schrödinger operators with sign-indefinite potentials.

Findings

Absolutely continuous spectrum exists under specified conditions.

Potential is summable over stochastic trajectories with positive probability.

Trajectories defined by Ito's equation escape to infinity almost surely.

Abstract

This paper blends two techniques recently developed in [2] and [3] to prove the presence of absolutely continuous spectrum for the multidimensional Schrodinger operator provided that the potential is summable over trajectory with positive probability. These trajectories are defined by Ito's equation and escape to infinity almost surely.