18 research outputs found
Absolutely Continuous Spectrum of a Polyharmonic Operator with a Limit Periodic Potential in Dimension Two
We consider a polyharmonic operator in dimension two
with , being an integer, and a limit-periodic potential . We
prove that the spectrum contains a semiaxis of absolutely continuous spectrum.Comment: 33 pages, 8 figure
Spectral properties of a limit-periodic Schr\"odinger operator in dimension two
We study Schr\"{o}dinger operator in dimension two,
being a limit-periodic potential. We prove that the spectrum of 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 a form of slightly distorted circles
with holes (Cantor type structure). Third, the spectrum corresponding to the
eigenfunctions (the semiaxis) is absolutely continuous.Comment: 89 pages, 6 figure
Multiscale Analysis in Momentum Space for Quasi-periodic Potential in Dimension Two
We consider a polyharmonic operator H=(-\Delta)^l+V(\x) in dimension two
with , being an integer, and a quasi-periodic potential V(\x).
We prove that the absolutely continuous spectrum of 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 \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.Comment: 125 pages, 4 figures. arXiv admin note: incorporates arXiv:1205.118
Extended States for Polyharmonic Operators with Quasi-periodic Potentials in Dimension Two
We consider a polyharmonic operator H=(-\Delta)^l+V(\x) in dimension two
with , being an integer, and a quasi-periodic potential V(\x).
We prove that the spectrum of 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 \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.Comment: This is an announcement only. Text with the detailed proof is under
preparation. 11 pages, 4 figures. arXiv admin note: text overlap with
arXiv:math-ph/0601008, arXiv:0711.4404, arXiv:1008.463
Perturbation theory for the Schrödinger operator with a periodic potential
The book is devoted to perturbation theory for the Schrödinger operator with a periodic potential, describing motion of a particle in bulk matter. The Bloch eigenvalues of the operator are densely situated in a high energy region, so regular perturbation theory is ineffective. The mathematical difficulties have a physical nature - a complicated picture of diffraction inside the crystal. The author develops a new mathematical approach to this problem. It provides mathematical physicists with important results for this operator and a new technique that can be effective for other problems. The semiperiodic Schrödinger operator, describing a crystal with a surface, is studied. Solid-body theory specialists can find asymptotic formulae, which are necessary for calculating many physical values