The Bohr-Sommerfeld quantization condition, revisited

Consider the semiclassical limit, as the Planck constant $hbarri 0$, of bound states of a quantum particle in one-dimensional potential wells. We justify the semiclassical asymptotics of eigenfunctions and recover the Bohr-Sommerfeld quantization condition. Our approach yields some new results: the quantization condition for discontinuous potentials, the precise estimate in the Weyl asymptotic formula for eigenvalues. We reconsider also a passage of a particle through a potential barrier finding a condition which fixes phases of the wave function from the left or from the right of the barrier. This leads to the Bohr-Sommerfeld quantization condition for multiple wells.