Perturbation theory for spectral gap edges of 2D periodic Schr\"odinger operators

We consider a two-dimensional periodic Schr\"odinger operator $H=-\Delta+W$ with $\Gamma$ being the lattice of periods. We investigate the structure of the edges of open gaps in the spectrum of $H$. We show that under arbitrary small perturbation $V$ periodic with respect to $N\Gamma$ where $N=N(W)$ is some integer, all edges of the gaps in the spectrum of $H+V$ which are perturbation of the gaps of $H$ 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 $H^{(\epsilon)}:=-\frac{d^2}{dx^2}+\epsilon V$ in ${\mathbb R}$ with almost-periodic potential $V$. We study the behaviour of the integrated density of states (IDS) $N(H^{(\epsilon)};\lambda)$ when $\epsilon\to 0$ and $\lambda$ is a fixed energy. When $V$ is quasi-periodic (i.e. is a finite sum of complex exponentials), we prove that for each $\lambda$ the IDS has a complete asymptotic expansion in powers of $\epsilon$; these powers are either integer, or in some special cases half-integer. These results are new even for periodic $V$. We also prove that when the potential is neither periodic nor quasi-periodic, there is an exceptional set $\mathcal S$ of energies (which we call $\hbox{the super-resonance set}$) such that for any $\sqrt\lambda\not\in\mathcal S$ there is a complete power asymptotic expansion of IDS, and when $\sqrt\lambda\in\mathcal S$, then even two-terms power asymptotic expansion does not exist. We also show that the super-resonant set $\mathcal S$ is uncountable, but has measure zero. Finally, we prove that the length of any spectral gap of $H^{(\epsilon)}$ has a complete asymptotic expansion in natural powers of $\epsilon$ when $\epsilon\to 0$.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