Spectral asymptotics for Hill's equation near the potential maximum

We study the spectrum of a one-dimensional semiclassical periodic Schrödinger operator near the potential maximum. The potential is supposed to be analytic and to have only one point of maximum over the period, which in addition is assumed to be non-degenerate. The spectrum is the union of bands, and some energy value is belonging to the spectrum, if and only if the trace of the operator of translation by the period acting on the two-dimensional space of solutions of the corresponding stationary Schrödinger equation lies in [−2,2].

By application of a result by Helffer and Sjöstrand we can take the dilation generator as a model near the potential maximum, and by application of a Fourier integral operator we find local solutions that have a standard WKB-form away from the point of maximum. So we may extend them over the whole period in order to compute the asymptotics of the trace of the translation matrix such that we finally obtain the size of the bands and the gaps separating them.