Related papers: Spectral gaps for periodic Schr\"odinger operators…
Schr\"odinger operators with periodic (possibly complex-valued) potentials and discrete periodic operators (possibly with complex-valued entries) are considered, and in both cases the computational spectral problem is investigated: namely,…
The spectrum of a Schr\"odinger operator with periodic potential generally consists of bands and gaps. In this paper, for fixed m, we consider the problem of maximizing the gap-to-midgap ratio for the m-th spectral gap over the class of…
Sharp spectral asymptotics for multidimensional Schroedinger operators with the strong magnetic field are derived under rather weak smoothness conditions. I assume that magnetic intensity matrix has constant defect r>0 at each point. In…
The spectral series of the Schr\"odinger operator with a delta-potential on a three-dimensional compact spherically symmetric manifold in the semiclassical limit as $h\to0$ are described.
In this note we provide an explicit lower bound on the spectral gap of one-dimensional Schr\"odinger operators with non-negative bounded potentials and subject to Neumann boundary conditions.
We study generalised magnetic Schroedinger operators of the form H(A,V)=h(P^A)+V, where h is an elliptic symbol, P^A is the generator of the magnetic translations, with A a vector potential defining a variable magnetic field B, and V is a…
We consider a Schr\"odinger operator $(h\mathbf D -\mathbf A)^2$ with a positive magnetic field $B=\curl\mathbf A$ in a domain $\Omega\subset\R^2$. The imposing of Neumann boundary conditions leads to spectrum below $h\inf B$. This is a…
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…
We provide sufficient conditions for the discreteness of spectrum for magnetic Schr\"odinger operators. They generalize the classical result by K.Friedrichs (1934) and earlier results by J.Avron, I.Herbst and B.Simon (1978), A.Dufresnoy…
We consider the zigzag half-nanotubes (tight-binding approximation) in a uniform magnetic field which is described by the magnetic Schr\"odinger operator with a periodic potential plus a finitely supported perturbation. We describe all…
We consider a semi-periodic two-dimensional Schr\"odinger operator which is cut at an angle. When the cut is commensurate with the periodic lattice, the spectrum of the operator has the band-gap Bloch structure. We prove that in the…
The paper studies the Hill--Schr\"odinger operators with potentials in the space $H^\omega \subset H^{-1}\left(\mathbb{T}, \mathbb{R}\right)$. The main results completely describe the sequences arising as the lengths of spectral gaps of…
We consider the Schr\"odinger operator ${\bf H}=(i\nabla+A)^2 $ in the space $L_2({\mathbb R}^3)$ with a magnetic potential $A $ created by an infinite straight current. We perform a spectral analysis of the operator ${\bf H}$ almost…
We discuss Schr\"odinger operators on a half-line with decaying oscillatory potentials, such as products of an almost periodic function and a decaying function. We provide sufficient conditions for preservation of absolutely continuous…
We consider the Schr\"odinger operator $H(\mu) = \nabla_{\bf A}^*\nabla_{\bf A} + \mu V$ on a Riemannian manifold $M$ of bounded geometry, where $\mu>0$ is a coupling parameter, the magnetic field ${\bf B}=d{\bf A}$ and the electric…
We survey results concerning the spectral properties of limit-periodic operators. The main focus is on discrete one-dimensional Schr\"odinger operators, but other classes of operators, such as Jacobi and CMV matrices, continuum…
I consider magnetic Schr\"odinger operator in dimension $d=2$ assuming that coefficients are smooth and magnetic field is non-degenerating. Then I extend the remainder estimate $O(\mu^{-1}h^{-1}+1)$ derived in \cite{Ivr1} for the case when…
We consider the Schr\"odinger operator on zigzag graphs with a periodic potential. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite…
A periodic one-dimensional Schroedinger operator is called semifinite-gap if every second gap in its spectrum is eventually closed. We construct explicit examples of semifinite-gap Schroedinger operators in trigonometric functions by…
We study in a semiclassical regime a two-dimensional magnetic periodic Schr\"odinger operator. We first review some results for the square (Harper), triangular and hexagonal (case of the graphene) lattices. Then we study the case considered…