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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. 1