Related papers: Whittaker-Hill equation and semifinite-gap Schroed…

We study ergodic Schr\"odinger operators defined over product dynamical systems in which one factor is periodic and the other factor is either a subshift over a finite alphabet or an irrational rotation of the circle. In the case in which…

In this article we describe the semi-classical spectrum of a Schrodinger operator on $\mathbb{R}$ with a double well potential. We study the shape of spectrum around the local maximum of the potential. In the classification of singularities…

It is well known that the matrix of a metaplectic operator with respect to phase-space shifts is concentrated along the graph of a linear symplectic map. We show that the algebra generated by metaplectic operators and by pseudodifferential…

We consider 2-dimensional Schroedinger operator with the non-degenerating magnetic field in the domain with the boundary and under certain non-degeneracy assumptions we derive spectral asymptotics with the remainder estimate better than…

Singular Gordon potentials are defined to be distributions from the space W^{-1}_{2,unif}(R) that are sufficiently fast approximated by periodic ones. We prove that Schr\"odinger operators with singular Gordon potentials have no point…

We consider the Schr\"odinger operator on the real line with a $N\ts N$ matrix valued periodic potential, N>1. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define the Lyapunov…

The inverse spectral theory for a self-adjoint one-dimensional Dirac operator associated periodic potentials is formulated via a Riemann-Hilbert problem approach. The resulting formalism is also used to solve the initial value problem for…

The Hill operator admits a band gap structure. As a special case, like the Mathieu operator, one has only open gaps, however, the instability intervals of the Whittaker-Hill operator may be open or closed. In 2007, P. Djakov and B. Mityagin…

We study the problem of how the Floquet property manifests for periodic Schr\"{o}dinger operators which are known to have multiple of asymptotic spectral solutions. The main conclusions are made for elliptic potentials, we demonstrate that…

We consider magnetic Schroedinger operators on quantum graphs with identical edges. The spectral problem for the quantum graph is reduced to the discrete magnetic Laplacian on the underlying combinatorial graph and a certain Hill operator.…

We consider semi-infinite Jacobi matrices corresponding to a point interaction for the discrete Schr\"odinger operator. Our goal is to find explicit expressions for the spectral measure, the resolvent and other spectral characteristics of…

Unbounded (and bounded) Toeplitz operators (TO) with rational symbols are analysed in detail showing that they are densely defined closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains,…

We prove that minimal Dirac operators on the half-line are self-modeling, which means that such an operator is determined by its arbitrary unitary copy uniquely up to a transformation (shape equivalence) which changes its potential by a…

We study the effect of non-negative potentials on the spectral gap of one-dimensional Schr\"odinger operators in the limit of large intervals. In particular, we derive upper and lower bounds on the gap for different classes of potentials…

We consider the fourth order Schr\"odinger operator $H=\Delta^2+V(x)$ in three dimensions with real-valued potential $V$. Let $H_0=\Delta^2$, if $V$ decays sufficiently and there are no eigenvalues or resonances in the absolutely continuous…

In this paper we investigate the spectral expansion for the one-dimensional Schrodinger operator with a periodic complex-valued potential. For this we consider in detail the spectral singularities and introduce new concepts as essential…

We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrodinger operators in $L^2(\Omega; d^n x)$, $n=2,3$, where $\Omega$ is an open set with a compact, nonempty boundary satisfying certain regularity…

We study discrete quasiperiodic Schr\"odinger operators on $\ell^2(\zee)$ with potentials defined by $\gamma$-H\"older functions. We prove a general statement that for $\gamma >1/2$ and under the condition of positive Lyapunov exponents,…

We obtain a complete description of semi-Fredholm spectra of operators of the form $(Tf)(z) = w(z)f(B(z)$ acting on the disc algebra in the case when $B$ is either elliptic or double parabolic finite Blaschke product of degree $d \geq 2$…

In this paper we study spectral properties of Schr\"odinger operators with quasi-periodic potentials related to quasi-periodic action minimizing trajectories for analytic twist maps. We prove that the spectrum contains a component of…