相关论文: On the finite-zone periodic PT-symmetric potential…
The Darboux transformation applied recurrently on a Schroedinger operator generates what is called a {\em dressing chain}, or from a different point of view, a set of supersymmetric shape invariant potentials. The finite-gap potential…
We prove that if f is a self-map of an algebraic variety over a field K, then under certain conditions on X, f and K the set of possible periods of K-valued periodic points of f is finite.
The Parity-Time ($\mathcal{PT}$) symmetric potentials are derived by non-Hermitian supersymmetric quantum mechanics for square well and barrier. These $\mathcal{PT}$-supersymmetric square well and barrier. The partners have complex…
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…
The resonant scattering of gap solitons (GS) of the periodic nonlinear Schr\"odinger equation with a localized defect which is symmetric under the parity and the time-reversal (PT) symmetry, is investigated. It is shown that for suitable…
We review some results and proofs on eigenvalue bounds for random Schr\"odinger operators with complex-valued potentials. We also include new Schatten norm estimates for the resolvent and use them to obtain bounds for sums of eigenvalues.
One-dimensional PT-symmetric quantum-mechanical Hamiltonians having continuous spectra are studied. The Hamiltonians considered have the form $H=p^2+V(x)$, where $V(x)$ is odd in $x$, pure imaginary, and vanishes as $|x|\to\infty$. Five…
In the first part of the paper, we discuss eigenvalue problems of the form -w"+Pw=Ew with complex potential P and zero boundary conditions at infinity on two rays in the complex plane. We give sufficient conditions for continuity of the…
The higher order supersymmetric partners of a stationary periodic potential are studied. The transformation functions associated to the band edges do not change the spectral structure. However, when the transformation is implemented for…
In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schr\"odinger operators whose potentials are obtained by real-valued sampling along the orbits of an ergodic invertible transformation. After an…
The paper is a continuation of the study started in \cite{Yorzh1}. Schrodinger operators on finite compact metric graphs are considered under the assumption that the matching conditions at the graph vertices are of $\delta$ type. Either an…
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…
The paper is concerned with the number of open gaps in spectra of periodic quantum graphs. The well-known conjecture by Bethe and Sommerfeld (1933) says that the number of open spectral gaps for a system periodic in more than one direction…
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…
In this paper we consider the spectrum of the self-adjoint differential operator L generated by the differential expression of order n with the m by m periodic matrix coefficients, where n and m are respectively odd and even integers and…
We exhibit a dense set of limit periodic potentials for which the corresponding one-dimensional Schr\"odinger operator has a positive Lyapunov exponent for all energies and a spectrum of zero Lebesgue measure. No example with those…
We show that the measure of the spectrum of Schr\"odinger operator with potential defined by non-constant function over any minimal aperiodic finite subshift tends to zero, as the coupling constant tends to infinity. We also obtained a…
For one-dimensional Schroedinger operators with complex-valued potentials, we construct pseudomodes corresponding to large pseudoeigenvalues. Our (non-semi-classical) approach results in substantial progress in achieving optimal conditions…
We consider branched quantum wires, whose connection rules provide PT-symmetry for the Schrodinger equation on graph. For such PT-symmetric quantum graph we derive general boundary conditions which keep the Hamiltonian as PT-symmetric with…
Consider the discrete 1D Schr\"odinger operator on $\Z$ with an odd $2k$ periodic potential $q$. For small potentials we show that the mapping: $q\to $ heights of vertical slits on the quasi-momentum domain (similar to the…