Quantization Conditions for a Bundle over a Circle and Semiclassical Spectral Series of the One-Dimensional Schrödinger Operator with a Jump-like Two-Scale Potential

We describe semiclassical spectral series of the one-dimensional Schrödinger operator with a potential having a smoothed jump whose width tends to zero more slowly than the semiclassical parameter. We calculate asymptotic eigenvalues from the quantization conditions of a one-dimensional bundle over a circle constructed from the classical Hamiltonian, and describe the asymptotic behavior of eigenfunctions using a modification of Maslov’s canonical operator on this bundle.