Remarks on criticality theory for Schr\"odinger operators and its application to wave equations with potentials

TL;DR

This paper offers a new perspective on the criticality theory for Schr"odinger operators, linking it to Hilbert lattice structures and exploring implications for wave equations with potentials.

Contribution

It introduces a novel quantity based on Hilbert lattice structure that preserves criticality properties and applies this to analyze large-time behavior of wave equations.

Findings

New quantity characterizes criticality in Schr"odinger operators

Links criticality to Hilbert lattice properties

Analyzes large-time behavior of wave equations with potentials

Abstract

In this paper, we give an alternative perspective of the criticality theory for (nonnegative) Schr\"odinger operators. Schr\"odinger operator $S=-\Delta+V$ is classified as subcritical/critical in terms of the existence/nonexistence of a positive Green function for the associated elliptic equation $Su=f$. Such a property strongly affects to the large-time behavior of solutions to the parabolic equation $\partial_tv+Sv=0$. In this paper, we propose a remarkable quantity in terms of the structure of Hilbert lattices, which keeps some important properties including the notion of criticality theory. As an application, we study the large-time behavior of solutions to the hyperbolic equation $\partial_t^2w+Sw=0$.