Passage through a potential barrier and multiple wells

TL;DR

This paper investigates the semiclassical limit of quantum particles in multiple potential wells, showing that eigenvalues satisfy Bohr-Sommerfeld conditions in at least one well, with wave functions influenced by barriers.

Contribution

It proves that each eigenvalue corresponds to at least one well satisfying Bohr-Sommerfeld quantization in a multi-well potential, analyzing wave functions near barriers.

Findings

Eigenvalues satisfy Bohr-Sommerfeld conditions in at least one well.

Barriers influence wave function phases similarly to infinite barriers.

Eigenvalues are near points satisfying Bohr-Sommerfeld conditions.

Abstract

Consider the semiclassical limit, as the Planck constant $\hbar\ri 0$, of bound states of a one-dimensional quantum particle in multiple potential wells separated by barriers. We show that, for each eigenvalue of the Schr\"odinger operator, the Bohr-Sommerfeld quantization condition is satisfied at least for one potential well. The proof of this result relies on a study of real wave functions in a neighborhood of a potential barrier. We show that, at least from one side, the barrier fixes the phase of wave functions in the same way as a potential barrier of infinite width. On the other hand, it turns out that for each well there exists an eigenvalue in a small neighborhood of every point satisfying the Bohr-Sommerfeld condition.