The Tunneling Effect for Schr\"odinger operators on a Vector Bundle

TL;DR

This paper investigates the tunneling effect in Schr"odinger operators on vector bundles over manifolds, analyzing eigenvalue splitting in the semiclassical limit using WKB quasimodes and geometric considerations of geodesics.

Contribution

It introduces a detailed analysis of eigenvalue splitting due to tunneling in vector bundle Schr"odinger operators, incorporating geometric and semiclassical techniques.

Findings

Eigenvalue splitting is characterized by minimal geodesics connecting potential wells.

The polynomial prefactor in eigenvalue splitting depends on the dimension of the geodesic submanifold.

A framework for analyzing tunneling effects in complex geometric settings is established.

Abstract

In the semiclassical limit h to 0, we analyze a class of self-adjoint Schr\"odinger operators H_h = h^2 L + h W + V id_E acting on sections of a vector bundle E over an oriented Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has non-degenerate minima at a finite number of points m_1, ... m_r in M, called potential wells. Using quasimodes of WKB-type near m_j for eigenfunctions associated with the low lying eigenvalues of H_h, we analyze the tunneling effect, i.e. the splitting between low lying eigenvalues, which e.g. arises in certain symmetric configurations. Technically, we treat the coupling between different potential wells by an interaction matrix and we consider the case of a single minimal geodesic (with respect to the associated Agmon metric) connecting two potential wells and the case of a submanifold of minimal geodesics of dimension l + 1. This dimension l determines the polynomial prefactor for exponentially small eigenvalue splitting.