Spectral properties of a polyharmonic operator with limit-periodic potential in dimension two

This dissertation is devoted to spectral study of a polyharmonic operator with a limit-periodic potential in dimension two. We consider the operator H=-Dl +n=1infinityVn x in L2R2 , where Vn(x) are periodic with the periods growing exponentially as 2n and the Linfinity -norms decaying super-exponentially, when n→infinity .; It is 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. It is also proved that there is an extensive class of generalized eigenfunctions being close to plane waves in the high energy region. In the proof, we use a modification of the Kolmogorov-Arnold-Moser method.