Ballistic transport for Schrödinger operators with quasi-periodic potentials

We prove the existence of ballistic transport for a Schrödinger operator with a generic quasi-periodic potential in any dimension d > 1

Perturbative diagonalization for Maryland-type quasiperiodic operators with flat pieces

We consider quasiperiodic operators on Zd with unbounded monotone sampling functions ("Maryland-type"), which are not required to be strictly monotone and are allowed to have flat segments. Under several geometric conditions on the frequencies, lengths of the segments, and their positions, we show that these operators enjoy Anderson localization at large disorder

Complete asymptotic expansion of the spectral function of multidimensional almost-periodic Schrodinger operators

We prove the existence of a complete asymptotic expansion of the spectral function (the integral kernel of the spectral projection) of a Schrödinger operator H=−Δ+bH=−Δ+b acting in RdRd when the potential bb is real and either smooth periodic, or generic quasiperiodic (finite linear combination of exponentials), or belongs to a wide class of almost-periodic functions

On absolute continuity of the spectrum of a periodic magnetic Schr\"odinger operator

We consider the Schr\"odinger operator in ${\mathbb R}^n$, $n\geq 3$, with the electric potential $V$ and the magnetic potential $A$ being periodic functions (with a common period lattice) and prove absolute continuity of the spectrum of the operator in question under some conditions which, in particular, are satisfied if $V\in L^{n/2}_{{\mathrm {loc}}}({\mathbb R}^n)$ and $A\in H^q_{{\mathrm {loc}}}({\mathbb R}^n;{\mathbb R}^n)$, $q>(n-1)/2$.Comment: 25 page

On the inverse resonance problem for Jacobi operators—uniqueness and stability

We estimate the difference of the coefficients of two Jacobi operators (from a certain class) from knowledge about their eigenvalues and resonances. More specifically, we prove that if eigenvalues and resonances of the two operators in a sufficiently large disk are respectively close, then the coefficients are close too. A uniqueness result for the inverse resonance problem follows as a corollary

Convergence of perturbation series for unbounded monotone quasiperiodic operators

We consider a class of unbounded quasiperiodic Schrödinger-type operators on ℓ2(Zd) with monotone potentials (akin to the Maryland model) and show that the Rayleigh–Schrödinger perturbation series for these operators converges in the regime of small kinetic energies, uniformly in the spectrum. As a consequence, we obtain a new proof of Anderson localization in a more general than before class of such operators, with explicit convergent series expansions for eigenvalues and eigenvectors. This result can be restricted to an energy window if the potential is only locally monotone and one-to-one. A modification of this approach also allows the potential to be non-strictly monotone and have a flat segment, under additional restrictions on the frequencies

Quasi-conformal mappings and periodic spectral problems in dimension two

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