Related papers: Non-periodic one-gap potentials in quantum mechani…
In this note we review spectral properties of magnetic random Schroedinger operators H_omega=H_0+V_omega + U_l + U_r defined on L^2(R x [-L/2,L/2],dx dy) with periodic boundary conditions along y. U_l and U_r are two confining potentials…
We prove that the spectrum of an n-dimensional semiclassical radial Schr\"odinger operator determines the potential within a large class of potentials for which we assume no symmetry or analyticity. Our proof is based on the first two…
We prove that, if an isospectral torus contains a discrete Schr\"odinger operator with nonconstant potential, the shift dynamics on that torus cannot be minimal. Consequently, we specify a generic sense in which finite unions of…
Solitary waves in one-dimensional periodic media are discussed employing the nonlinear Schr\"odinger equation with a spatially periodic potential as a model. This equation admits two families of gap solitons that bifurcate from the edges of…
A formulation of non-relativistic quantum mechanics in terms of Newtonian particles is presented in the shape of a set of three postulates. In this new theory, quantum systems are described by ensembles of signed particles which behave as…
The Schr\"odinger equation is shown to be equivalent to a constrained Liouville equation under the assumption that phase space is extended to Grassmann algebra valued variables. For onedimensional systems, the underlying Hamiltonian…
We study multi-frequency quasiperiodic Schr\"{o}dinger operators on $\mathbb{Z} $. We prove that for a large real analytic potential satisfying certain restrictions the spectrum consists of a single interval. The result is a consequence of…
We consider the one-dimensional Schroedinger equation on a ring, with the cubic term, of either self-attractive or repulsive sign, confined to a narrow segment. This setting can be realized in optics and Bose-Einstein condensates. For the…
We investigate spectral properties of quantum graphs in the form of a periodic chain of rings with a connecting link between each adjacent pair, assuming that wave functions at the vertices are matched through conditions manifestly…
In this paper we prove the existence of the Stark-Wannier quantum resonances for one-dimensional Schrodinger operators with smooth periodic potential and small external homogeneous electric field. Such a result extends the existence result…
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…
We investigate spectral properties of the Laplace operator on a class of non-compact Riemannian manifolds. For a given number $N$ we construct periodic (i.e. covering) manifolds such that the essential spectrum of the corresponding…
We consider a two-dimensional integrable Hamiltonian system with a vector and scalar potential in quantum mechanics. Contrary to the case of a pure scalar potential, the existence of a second order integral of motion does not guarantee the…
The importance of the energy spectrum of bound states and their restrictions in quantum mechanics due to the different methods have been used for calculating and determining the limit of them. Comparison of Schrodinger-like equation…
We consider a cubic nonlinear Schroedinger equation with periodic potential. In a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several Bloch bands is studied. The notion of closed mode systems…
This paper addresses the problem of computing the eigenvalues lying in the gaps of the essential spectrum of a periodic Schrodinger operator perturbed by a fast decreasing potential. We use a recently developed technique, the so called…
The behaviour of the lengths of spectral gaps $\{\gamma_{n}(q)\}_{n=1}^{\infty}$ of the Hill-Schr\"odinger operators S(q)u=-u''+q(x)u,\quad u\in \mathrm{Dom}(S(q)) with real-valued 1-periodic distributional potentials $q(x)\in…
For Schr\"odinger operators on an interval with either convex or symmetric single-well potentials, and Robin or Neumann boundary conditions, the gap between the two lowest eigenvalues is minimised when the potential is constant. We also…
We consider the Schr\"odinger operator $Hy=-y"+(p+q)y$ with a periodic potential $p$ plus a compactly supported potential $q$ on the real line. The spectrum of $H$ consists of an absolutely continuous part plus a finite number of simple…
It is shown that quantum mechanics on noncommutative spaces (NQM) can be obtained by the canonical quantization of some underlying second class constrained system formulated in extended configuration space. It leads, in particular, to an…