Ground state decay for Schr\"odinger operators with confining potentials

TL;DR

This paper provides precise estimates of the ground state for Schrödinger operators with confining potentials using a novel semigroup approach based on resolvent and Feynman--Kac formula, especially effective for slowly varying, radial, increasing potentials.

Contribution

It introduces a new, concise proof method for ground state estimates of Schrödinger operators with confining potentials, improving understanding for specific potential classes.

Findings

Two-sided estimates of ground states are established.

The semigroup approach simplifies the proof process.

Results are sharp for slowly varying, radial, increasing potentials.

Abstract

We give two-sided estimates of a ground state for Schr\"odinger operators with confining potentials. We propose a semigroup approach, based on resolvent and the Feynman--Kac formula, which leads to a new, rather short and direct proof. Our results take the sharpest form for slowly varying, radial and increasing potentials.