Yulia Karpeshina

On the density of states for the periodic Schrödinger operator

An asymptotic formula for the density of states of the polyharmonic periodic operator (−δ)l+V inRn,n≥2,l>1/2 is obtained. Special consideration is given to the case of the Schrödinger equationn=3,l=1,V being a periodic potential, where the second term of the asymptotic is found.

Absolutely Continuous Spectrum of a Polyharmonic Operator with a Limit Periodic Potential in Dimension Two

We consider a polyharmonic operator H = (−Δ) l + V(x) in dimension two with l ≥ 6, l being an integer, and a limit-periodic potential V(x). We prove that the spectrum contains a semiaxis of absolutely continuous spectrum.

Ballistic Transport for the Schrödinger Operator with Limit-Periodic or Quasi-Periodic Potential in Dimension Two

We prove the existence of ballistic transport for the Schrödinger operator with limit-periodic or quasi-periodic potential in dimension two. This is done under certain regularity assumptions on the potential which have been used in prior work to establish the existence of an absolutely continuous component and other spectral properties. The latter include detailed information on the structure of generalized eigenvalues and eigenfunctions. These allow one to establish the crucial ballistic lower bound through integration by parts on an appropriate extension of a Cantor set in momentum space, as well as through stationary phase arguments.

Extended states for polyharmonic operators with quasi-periodic potentials in dimension two

We consider a polyharmonic operator H = ( − Δ)l + V(x) in dimension two with l ⩾ 2, l being an integer, and a quasi-periodic potential V(x). We prove that the absolutely continuous 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 ei⟨k, x⟩ at the high energy region. Second, the isoenergetic curves in the space of momenta k corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). A new method of multiscale analysis in the momentum space is developed to prove these results.

On polyharmonic operators with limit-periodic potential in dimension two

This is an announcement of the following results. We consider a polyharmonic operator H = (−∆)l + V (x) in dimension two with l ≥ 6 and 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 at the high-energy region. Second, the isoenergetic curves in the space of momenta corresponding to these eigenfunctions have the form of slightly distorted circles with holes (Cantor-type structure). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous. We study the operator (1) H = (−∆) + V (x) in two dimensions, l ≥ 6, V (x) being a limit-periodic potential:

Multiscale analysis in momentum space for quasi-periodic potential in dimension two

We consider a polyharmonic operator H=(−Δ)l+V(x) in dimension two with l ⩾ 2, l being an integer, and a quasi-periodic potential V(x). We prove that the absolutely continuous 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 ei⟨ϰ,x⟩ at the high energy region. Second, the isoenergetic curves in the space of momenta ϰ corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). A new method of multiscale analysis in the momentum space is developed to prove these results.

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

This is an an nouncement of the following results. We consider the Schrodinger operator H=−Δ+V(x) in dimension two, V(x) being a limitperiodic 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{ei257-1} at the high energy region. Second, the isoenergetic curves in the space of momenta \( \vec k \) 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.