Eigenvalue splitting for a system of Schr\"odinger operators with an energy-level crossing

TL;DR

This paper analyzes the asymptotic distribution of eigenvalues for a coupled 1D Schrödinger system with potential wells and energy-level crossing, providing quantization conditions and precise asymptotics in the semiclassical limit.

Contribution

It introduces Bohr-Sommerfeld quantization conditions for such systems and characterizes eigenvalue splitting behavior at energy-level crossings, especially in symmetric cases.

Findings

Eigenvalue splitting is of order h^{3/2} in symmetric cases.

Main asymptotic term is related to the intersection area of classically allowed regions.

Provides precise eigenvalue asymptotics in the semiclassical limit h→0.

Abstract

We study the asymptotic distribution of the eigenvalues of a one-dimensional two-by-two semiclassical system of coupled Schr\"odinger operators in the presence of two potential wells and with an energy-level crossing. We provide Bohr-Sommerfeld quantization condition for the eigenvalues of the system on any energy-interval above the crossing and give precise asymptotics in the semiclassical limit $h\to 0^+$. In particular, in the symmetric case, the eigenvalue splitting occurs and we prove that the splitting is of polynomial order $h^{\frac32}$ and that the main term in the asymptotics is governed by the area of the intersection of the two classically allowed domains.