Bethe-Sommerfeld conjecture in semiclassical settings

Publication date: 1 February 2019

Abstract: Under certain assumptions (including

d g e 2)

we prove that the spectrum of a scalar operator in

m a t h s c r L^{2} (m a t h b b R^{d})

�egin{equation*} A_varepsilon (x,hD)= A^0(hD) + varepsilon B(x,hD), end{equation*} covers interval

(a u - e p s i l o n, a u + e p s i l o n)

, where

A^{0}

is an elliptic operator and

B (x, h D)

is a periodic perturbation,

v a r e p s i l o n = O (h^{v} a r k a p p a)

,

v a r k a p p a > 0

. Further, we consider generalizations.

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