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Bethe-Sommerfeld Conjecture in Semiclassical Settings

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Abstract

Under certain assumptions (including \(d\ge 2)\) we prove that the spectrum of a scalar operator in \(\mathscr {L}^2({\mathbb {R}}^d)\)

$$\begin{aligned} A_\varepsilon (x, hD)= A^0(hD) + \varepsilon B(x, hD), \end{aligned}$$

covers interval \((\tau -\epsilon ,\tau +\epsilon )\), where \(A^0\) is an elliptic operator and B(x, hD) is a periodic perturbation, \(\varepsilon =O(h^\varkappa )\), \(\varkappa >0\).

Further, we consider generalizations.

This research was supported in part by National Science and Engineering Research Council (Canada) Discovery Grant RGPIN 13827.

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  1. Department of Mathematics, University of Toronto, Toronto, ON, Canada

    Prof. Victor Ivrii

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Correspondence to Victor Ivrii .

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Ivrii, V. (2019). Bethe-Sommerfeld Conjecture in Semiclassical Settings. In: Microlocal Analysis, Sharp Spectral Asymptotics and Applications V. Springer, Cham. https://doi.org/10.1007/978-3-030-30561-1_36

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