scholarly journals Schrödinger operator with a nonlocal potential whose absolutely continuous and point spectra coexist

1991 ◽  
Vol 139 (2) ◽  
pp. 357-380
Author(s):  
A. L. Figotin ◽  
L. A. Pastur
Keyword(s):  
Schrödinger Operator ◽  
Schrodinger Operator ◽  
Nonlocal Potential ◽  
Absolutely Continuous

scholarly journals Absolutely continuous spectrum of multifrequency quasiperiodic Schrödinger operator

2020 ◽  
Vol 279 (6) ◽  
pp. 108632
Author(s):  
Xuanji Hou ◽  
Jing Wang ◽  
Qi Zhou
Keyword(s):  
Continuous Spectrum ◽  
Schrödinger Operator ◽  
Schrodinger Operator ◽  
Absolutely Continuous Spectrum ◽  
Absolutely Continuous

ON THE SPECTRAL AND SCATTERING THEORY OF THE SCHRÖDINGER OPERATOR WITH SURFACE POTENTIAL

2000 ◽  
Vol 12 (04) ◽  
pp. 561-573 ◽  
Author(s):  
AYHAM CHAHROUR ◽  
JAOUAD SAHBANI
Keyword(s):  
Continuous Spectrum ◽  
Surface Potential ◽  
Schrödinger Operator ◽  
Scattering Theory ◽  
Schrodinger Operator ◽  
Absolutely Continuous Spectrum ◽  
Discrete Schrödinger Operator ◽  
Absolutely Continuous ◽  
Spectral And Scattering Theory

We consider a discrete Schrödinger operator H=-Δ+V acting in ℓ2 (ℤd+1), with potential V supported by the subspace ℤd×{0}. We prove that σ (-Δ)=[-2 (d+1), 2(d+1)] is contained in the absolutely continuous spectrum of H. For this we develop a scattering theory for H. We emphasize the fact that this result applies to arbitrary potentials, so it depends on the structure of the problem rather than on a particular choice of the potential.

Schrödinger Operator Levels for a Crystal Film with a Nonlocal Potential

2004 ◽  
Vol 140 (2) ◽  
pp. 1146-1150
Author(s):  
M. S. Smetanina ◽  
Yu. P. Chuburin
Keyword(s):  
Schrödinger Operator ◽  
Schrodinger Operator ◽  
Nonlocal Potential ◽  
Crystal Film

Zeroes of the spectral density of the periodic Schrödinger operator with Wigner–von Neumann potential

2011 ◽  
Vol 153 (1) ◽  
pp. 33-58 ◽  
Author(s):  
SERGUEI NABOKO ◽  
SERGEY SIMONOV
Keyword(s):  
Spectral Density ◽  
Continuous Spectrum ◽  
Schrödinger Operator ◽  
Bound State ◽  
Periodic Operator ◽  
Schrodinger Operator ◽  
Absolutely Continuous Spectrum ◽  
Absolutely Continuous ◽  
Von Neumann ◽  
Band Gap Structure

AbstractWe consider the Schrödinger operator α on the half-line with a periodic background potential and the Wigner–von Neumann potential of Coulomb type: csin(2ωx + δ)/(x + 1). It is known that the continuous spectrum of the operator α has the same band-gap structure as the free periodic operator, whereas in each band of the absolutely continuous spectrum there exist two points (so-called critical or resonance) where the operator α has a subordinate solution, which can be either an eigenvalue or a “half-bound” state. The phenomenon of an embedded eigenvalue is unstable under the change of the boundary condition as well as under the local change of the potential, in other words, it is not generic. We prove that in the general case the spectral density of the operator α has power-like zeroes at critical points (i.e., the absolutely continuous spectrum has pseudogaps). This phenomenon is stable in the above-mentioned sense.

scholarly journals On singular spectrum of finite dimensional perturbations (to the Aronszajn-Donoghue-Kac theory)

2019 ◽  
Vol 487 (4) ◽  
pp. 365-369
Author(s):  
M. M. Malamud
Keyword(s):  
Continuous Spectrum ◽  
Schrödinger Operator ◽  
Symmetric Operator ◽  
Weyl Function ◽  
Schrodinger Operator ◽  
Absolutely Continuous Spectrum ◽  
Absolutely Continuous ◽  
Direct Sums ◽  
Singular Spectrum ◽  
Finite Dimensional

The main results of the Aronszajn-Donoghue-Kac theory are extended to the case of n-dimensional (in the resolvent sense) perturbations à of an operator A0 = A0* defined on a Hilbert space H. Applying technique of boundary triplets we describe singular continuous and point spectra of extensions AB of a symmetric operator A acting in H in terms of the Weyl function M(·) of the pair {A, A0} and boundary n-dimensional operator B = B*. Assuming that the multiplicity of singular spectrum of A0 is maximal it is established orthogonality of singular parts EsAв and EsAo of the spectral measures EAв and EAo of the operators AB and A0, respectively. It is shown that the multiplicity of singular spectrum of special extensions of direct sums A = A(1) ⊕ A(2) cannot be maximal as distinguished from multiplicity of the absolutely continuous spectrum. In particular, it is obtained a generalization of the Kac theorem on multiplicity of singular spectrum of Schrodinger operator on the line as well as its clarification. The multiplicity of singular spectrum of special extensions of direct sums A = A(1) ⊕ A(2) are investigated. In particular, it is shown that it cannot be maximal as distinguished from multiplicity of the absolutely continuous spectrum. This result generalizes the Kac theorem on multiplicity of singular spectrum of Schrodinger operator on the line and clarifies it.

scholarly journals Weyl—Titchmarsh-type formula for periodic Schrödinger operator with Wigner—von Neumann potential

2013 ◽  
Vol 143 (2) ◽  
pp. 401-425 ◽  
Author(s):  
Pavel Kurasov ◽  
Sergey Simonov
Keyword(s):  
Continuous Spectrum ◽  
Schrödinger Operator ◽  
Schrodinger Operator ◽  
Absolutely Continuous Spectrum ◽  
Absolutely Continuous ◽  
Von Neumann ◽  
Type Formula ◽  
Generalized Eigenvectors ◽  
Neumann Type ◽  
Periodic Schrödinger Operator

The Schrödinger operator on the half-line with periodic background potential perturbed by a certain potential of Wigner—von Neumann type is considered. The asymptotics of generalized eigenvectors for λ ϵ ℂ+ and on the absolutely continuous spectrum is established. The Weyl—Titchmarsh-type formula for this operator is proven.

First KdV Integrals¶and Absolutely Continuous Spectrum¶for 1-D Schrödinger Operator

2001 ◽  
Vol 216 (1) ◽  
pp. 195-213 ◽  
Author(s):  
S. Molchanov ◽  
M. Novitskii ◽  
B. Vainberg
Keyword(s):  
Continuous Spectrum ◽  
Schrödinger Operator ◽  
Schrodinger Operator ◽  
Absolutely Continuous Spectrum ◽  
Absolutely Continuous
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