Semiclassical spectral series of a quantum Schrödinger operator corresponding to a singular circle composed of equilibria

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On a two-dimensional surface, a Schrodinger operator is considered with a potential whose critical points form a closed curve. We pose the problem of describing the semiclassical spectral series corresponding to this curve. The standard construction for describing the spectral series corresponding to isolated nondegenerate equilibria or to periodic trajectories of Hamiltonian systems is not applicable in this situation. In the paper, a description of semiclassical solutions of the spectral problem for the quantum Schrodinger operator that correspond to nonisolated equilibria are presented and, to calculate the splitting of the eigenvalues, it turns out to be necessary to find the spectral series with a higher order of accuracy than it is usually required.