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Bethe-Sommerfeld Conjecture in Semiclassical Settings

  • Victor IvriiEmail author
Chapter

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(xhD) is a periodic perturbation, \(\varepsilon =O(h^\varkappa )\), \(\varkappa >0\).

Further, we consider generalizations.

Key words and phrases

Microlocal Analysis sharp spectral asymptotics integrated density of states periodic operators Bethe-Sommerfeld conjecture 

2010 Mathematics Subject Classification:

35P20 

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

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