On the Schrodinger Operator with Limit-periodic Potential in Dimension Two

Thus is an announcement of the following results. We consider the Schrodinger operator H = -Delta + V(x) in dimension two, V(x) being a limit-periodic potential. We prove that; the spectrum of H contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves e(i((k) over right arrow,(x) over right arrow)) at the high energy region. Second, the isoenergetic curves in the space of momenta (k) over right arrow corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous.