48 research outputs found
Perturbation theory for spectral gap edges of 2D periodic Schr\"odinger operators
We consider a two-dimensional periodic Schr\"odinger operator
with being the lattice of periods. We investigate the structure of the
edges of open gaps in the spectrum of . We show that under arbitrary small
perturbation periodic with respect to where is some
integer, all edges of the gaps in the spectrum of which are perturbation
of the gaps of become non-degenerate, i.e. are attained at finitely many
points by one band function only and have non-degenerate quadratic
minimum/maximum. We also discuss this problem in the discrete setting and show
that changing the lattice of periods may indeed be unavoidable to achieve the
non-degeneracy.Comment: 25 pages; several typos are fixed and comments are added; subsection
3.2 is expanded to include more detailed proof of Theorem 3.
Perturbation theory for almost-periodic potentials I. One-dimensional case
We consider the family of operators
in with
almost-periodic potential . We study the behaviour of the integrated density
of states (IDS) when and
is a fixed energy. When is quasi-periodic (i.e. is a finite sum of complex
exponentials), we prove that for each the IDS has a complete
asymptotic expansion in powers of ; these powers are either integer,
or in some special cases half-integer. These results are new even for periodic
. We also prove that when the potential is neither periodic nor
quasi-periodic, there is an exceptional set of energies (which we
call ) such that for any
there is a complete power asymptotic expansion
of IDS, and when , then even two-terms power
asymptotic expansion does not exist. We also show that the super-resonant set
is uncountable, but has measure zero. Finally, we prove that the
length of any spectral gap of has a complete asymptotic
expansion in natural powers of when .Comment: journal version, some misprints are fixed; 28 pages, 1 figur
Stability for the inverse resonance problem for the CMV operator
For the class of unitary CMV operators with super-exponentially decaying
Verblunsky coefficients we give a new proof of the inverse resonance problem of
reconstructing the operator from its resonances - the zeros of the Jost
function. We establish a stability result for the inverse resonance problem
that shows continuous dependence of the operator coefficients on the location
of the resonances