Related papers: Gap control by singular Schr\"odinger operators in…
We study the Schr\"odinger operator $-\Delta -\alpha \delta (x-\Gamma)$ in $L^2(\R^3)$ with a $\delta$ interaction supported by an infinite non-planar surface $\Gamma$ which is smooth, admits a global normal parameterization with a…
We consider Schr\"odinger operators with periodic potentials on periodic discrete graphs. The spectrum of the Schr\"odinger operator consists of an absolutely continuous part (a union of a finite number of non-degenerated bands) plus a…
As in arXiv: math. 0809.2365 we consider classical system of interacting particles $\mathcal{P}_1, ..., \mathcal{P}_n$ on the line with only neighboring particles involved in interaction. On the contrast to arXiv: math. 0809.2365 now {\it…
We develop a theory of interaction effects in graphene superlattices, where tunable superlattice periodicity can be used as a knob to control the gap at the Dirac point. Applied to graphene on hexa-boron-nitride (G/h-BN), our theory…
Given a Hermitian vector bundle over an infinite weighted graph, we define the Laplacian associated to a unitary connection on this bundle and study the essential self-adjointness of a perturbation of this Laplacian by an operator-valued…
The paper deals with singular Schr\"odinger operators of the form \begin{gather*} -{\mathrm{d}^2\over \mathrm{d} x^2 } + \sum_{k\in\mathbb{Z} }\gamma_k \delta(\cdot-z_k),\quad \gamma_k\in\mathbb{R}, \end{gather*} in…
The dynamics of a many-particle system are often modeled by mapping the Hamiltonian onto a Schr\"odinger equation. An alternative approach is to solve the Hamiltonian equations directly in a model space of many-body configurations. In a…
We study a Cahn-Hilliard model for phase separation in composite materials with multiple periodic microstructures. These are modeled by considering a highly oscillating potential. The focus of this paper is in the case where the scales of…
We consider Schr\"odinger operators on the real line with limit-periodic potentials and show that, generically, the spectrum is a Cantor set of zero Lebesgue measure and all spectral measures are purely singular continuous. Moreover, we…
We investigate the limiting behavior of solutions with infinitely many peaks to nonlinear Schr\"odinger equations [-epsilon^2 Delta u_epsilon + u_epsilon = u_epsilon^p, u_epsilon > 0 in R^n,] as epsilon -> 0, where p is Sobolev subcritical.…
At the example of two coupled waveguides we construct a periodic second order differential operator acting in a Euclidean domain and having spectral gaps whose edges are attained strictly inside the Brillouin zone. The waveguides are…
Let $(\lambda_-,\lambda_+)$ be a spectral gap of a periodic Schr\"odinger operator $A$ on the lattice ${\mathbb Z}^d$. Consider the operator $A(\alpha)=A-\alpha V$ where $V$ is a decaying positive potential on ${\mathbb Z}^d$. We study the…
This is a survey of the basic results on the behavior of the number of the eigenvalues of a Schr\"odinger operator, lying below its essential spectrum. We discuss both fast decaying potentials, for which this behavior is semiclassical, and…
In [15] we proposed a set of sufficient conditions for the approximate controllability of a discrete-spectrum bilinear Schr\"odinger equation. These conditions are expressed in terms of the controlled potential and of the eigenpairs of the…
We study asymptotically and numerically the fundamental gap -- the difference between the first two smallest (and distinct) eigenvalues -- of the fractional Schr\"{o}dinger operator (FSO) and formulate a gap conjecture on the fundamental…
The goal of this work is to prove global controllability and stabilization properties for the fractional Schr\"odinger equation on $d$-dimensional compact Riemannian manifolds without boundary $(M,g)$. To prove our main results we use…
We prove that the wave operators for Schr\"odinger operators with multi-center local point interactions are the scaling limits of the ones for Schr\"odinger operators with regular potentials. We simultaneously present a proof of the…
We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$ when $H$ has a threshold eigenvalue. We adapt our recent results for $m\geq 1$ when…
We develop direct scattering theory for one-dimensional Schr\"odinger operators with steplike potentials, which are asymptotically close to different Bohr almost periodic infinite-gap potentials on different half-axes.
We consider the Dirichlet Laplacian in a straight planar strip perturbed by a bounded periodic symmetric operator. We prove the classical Bethe-Sommerfeld conjecture for this operator, namely, that this operator has finitely many gaps in…