Ito diffusions, modified capacity and harmonic measure. Applications to Schrodinger operators

TL;DR

This paper explores the relationship between Ito diffusions, potential theory, and the spectral properties of multidimensional Schrödinger operators, introducing new probabilistic and geometric methods to analyze scattering and spectral types.

Contribution

It develops a novel variant of potential theory linked to Ito diffusions, providing new estimates on modified harmonic measure and connecting geometry to spectral analysis of Schrödinger operators.

Findings

Established a probabilistic framework for scattering analysis

Derived new estimates on modified harmonic measure

Linked potential support geometry to spectral types

Abstract

Using certain Ito's equation, we introduce the probability on the space of paths and show its relevance to the scattering properties of multidimensional Schrodinger operator. To relate the geometry of the support of potential to the spectral type we develop a special variant of Potential theory and prove some estimates on the modified Harmonic measure.