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Periodic magnetic Schrödinger equation with symmetry of order six: Spectral measure. II. (Équation de Schrödinger magnétique périodique avec symétrie d’ordre six: Mesure du spectre. II.) (French) Zbl 0826.35147

This is the continuation of the study of the Schrödinger operator with periodic magnetic field and electric potential in the case of a sixfold rotational symmetry and in the semiclassical limit (the Planck constant \(h\) tending to 0). The first part appeared in [the author, Mém. Soc. Math. Fr., Nouv. Sér. 51, 139 p. (1992; Zbl 0769.35075)]. The author treats the so-called triangular case in which the potential reaches its minimum once per periodicity cell.
It was proved in the first part that the lower part of the spectrum coincides with the spectra of pseudo-differential operators quantized with a new explicitly computable Planck constant \(\widehat h\) but this was done outside some critical values of the principal symbol.
The same result is obtained here near the critical values. This permits to show that the measure of this part of the spectrum is estimated by constant\(\times \widehat h\).
Reviewer: B.Helffer (Orsay)

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35J10 Schrödinger operator, Schrödinger equation
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis

Citations:

Zbl 0769.35075
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References:

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