An Example Of Accurate Microlocal Tunneling In One Dimension

TL;DR

This paper extends tunneling effect formulas to pseudo-differential operators in one dimension, providing spectral analysis and eigenvalue splitting asymptotics under symmetry assumptions.

Contribution

It introduces the first example of tunneling formulas for pseudo-differential operators, expanding the understanding of spectral properties in this context.

Findings

Asymptotic formula for eigenvalue splitting derived

Spectral analysis of pseudo-differential operators conducted

Extension of tunneling effect formulas to new operator class

Abstract

We investigate the spectral analysis of a class of pseudo-differential operators in one dimension. Under symmetry assumptions, we prove an asymptotic formula for the splitting of the first two eigenvalues. This article is a first example of extension to pseudo-differential operators of the tunneling effect formulas known for the symmetric electric Schr{\"o}dinger operator.