Bethe-Sommerfeld Conjecture in Semiclassical Settings

Victor Ivrii ()
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Victor Ivrii: University of Toronto, Department of Mathematics

Chapter Chapter 36 in Microlocal Analysis, Sharp Spectral Asymptotics and Applications V, 2019, pp 619-639 from Springer

Abstract: 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.

Keywords: Microlocal Analysis; sharp spectral asymptotics; integrated density of states; periodic operators; Bethe-Sommerfeld conjecture; 35P20 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/978-3-030-30561-1_36

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