A quasiclassical limit of the spectrum of a Schrödinger operator with complex periodic potential

The quasiclassical asymptotics of the spectrum of a one-dimensional Schrödinger operator with periodic complex potential that arises in the statistical mechanics of a Coulomb gas are described. The spectrum is shown to concentrate in a neighbourhood of a tree in the complex plane; the vertices of this tree are calculated explicitly, and the position of its edges can be investigated comprehensively. Equations are derived from which the asymptotic eigenvalues are found; these equations are conditions that certain special periods of a holomorphic form on the Riemann surface of constant classical energy are integers.
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