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We consider a polyharmonic operator $H=(-\Delta)^l+V(\x)$ in dimension twowith $l\geq 2$, $l$ being an integer, and a quasi-periodic potential $V(\x)$.We prove that the spectrum of $H$ contains a semiaxis and there is a family ofgeneralized eigenfunctions at every point of this semiaxis with the followingproperties. First, the eigenfunctions are close to plane waves $e^{i< \k,\x>}$at the high energy region. Second, the isoenergetic curves in the space ofmomenta $\k$ corresponding to these eigenfunctions have a form of slightlydistorted circles with holes (Cantor type structure). A new method ofmultiscale analysis in the momentum space is developed to prove these results.

Extended States for Polyharmonic Operators with Quasi-periodic Potentials in Dimension Two