Passage through a potential barrier and multiple wells

The semiclassical limit as the Planck constant $\hbar$ tends to $0$ is considered for bound states of a one-dimensional quantum particle in multiple potential wells separated by barriers. It is shown that, for each eigenvalue of the Schrödinger operator, the Bohr–Sommerfeld quantization condition is satisfied for at least one potential well. The proof of this result relies on a study of real wave functions in a neighborhood of a potential barrier. It is shown that, at least from one side, the barrier fixes the phase of the 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.