  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.
