Consider the semiclassical limit, as the Planck constant $\hbar\to 0$, 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.