Pseudo-differential Operators, Transmission Problems and the Large Coupling Limit

TL;DR

This paper investigates the large coupling limit of Schrödinger operators with piecewise-constant potentials, revealing how the potential influences boundary conditions and spectrum behavior as the coupling constant grows large.

Contribution

It establishes that the potential induces a non-local boundary condition on the interface and provides new estimates and spectral descriptions in the large coupling limit.

Findings

Potential determines a non-local boundary condition on the interface.

Provides convergence rate estimates for the large coupling limit.

Describes spectral behavior as the coupling constant approaches infinity.

Abstract

In this paper we prove some new results and give new proofs of known results related to the large coupling limit for stationary Schr\"odinger operators. The operators we consider are of the form $-\Delta +\lambda V(x)$ where $\Delta$ is the Laplacian, $V(x)$ is a real valued piecewise--constant potential having a jump discontinuity across a smooth interface and $\lambda$ is the coupling constant. Our main result is that the potential determines a non-local boundary condition on the interface and we systematically exploit this fact to derive various results about the large coupling problem. In particular, we obtain estimates for convergence rates and a description of the behavior of the spectrum of $-\Delta +\lambda V(x)$ as $\lambda\nearrow\infty$.