SPECTRAL PROPERTIES OF A POLYHARMONIC OPERATOR
WITH LIMIT-PERIODIC POTENTIAL IN DIMENSION TWO.
YOUNG-RAN LEE
UNIVERSITY OF ALABAMA AT BIRMINGHAM
We consider the operator:
l
H = (−∆) +
∞
X
Vn (x),
n=1
where Vn (x) is periodic with the periods growing exponentially as 2n and the L∞ -norm
decaying super-exponentially.
We have shown that if l > 5, a generalized version of the Bethe-Sommerfeld conjecture holds for this operator, in other words, its spectrum contains a semi-axis. We
have proved also that the corresponding eigenfunctions are close to plane waves. In
the proof we use the Kolmogorov-Arnold-Moser method.
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