# Absolutely continuous and pure point spectra of discrete operators with sparse potentials

@inproceedings{Molchanov2021AbsolutelyCA,
title={Absolutely continuous and pure point spectra of discrete operators with sparse potentials},
author={Stanislav Molchanov and Oleg Safronov and Boris Vainberg},
year={2021}
}
• Published 17 October 2021
• Physics, Mathematics
We consider the discrete Schr\”odinger operator $H=-\Delta+V$ with a sparse potential $V$ and find conditions guaranteeing either existence of wave operators for the pair $H$ and $H_0=-\Delta$, or presence of dense purely point spectrum of the operator $H$ on some interval $[\lambda_0,0]$ with $\lambda_0<0$.

## References

SHOWING 1-10 OF 18 REFERENCES
ABSOLUTELY CONTINUOUS SPECTRUM OF A ONE-PARAMETER FAMILY OF SCHRÖDINGER OPERATORS
Under certain conditions on the potential V , it is shown that the absolutely continuous spectrum of the Schrödinger operator −Δ + αV is essentially supported on [0,+∞) for almost every α ∈ R. §1.
One-dimensional Schrödinger operators with random decaying potentials
• Mathematics
• 1988
AbstractWe investigate the spectrum of the following random Schrödinger operators: $$H(\omega ) = - \frac{{d^2 }}{{dt^2 }} + a(t)F(X_t (\omega )),$$ whereF(Xt(ω)) is a Markovian potential studied by
Absolutely Continuous Spectrum of Schrödinger Operators with Slowly Decaying and Oscillating Potentials
• Mathematics
• 2005
The aim of this paper is to extend a class of potentials for which the absolutely continuous spectrum of the corresponding multidimensional Schrödinger operator is essentially supported by [0,∞). Our
Singular continuous spectrum under rank one perturbations and localization for random hamiltonians
• Mathematics
• 1986
We consider a selfadjoint operator, A, and a selfadjoint rank-one projection, P, onto a vector, φ, which is cyclic for A. In terms of the spectral measure dμAφ, we give necessary and sufficient
Radiation conditions for the difference schrödinger operators
• Mathematics
• 2001
The problem of determining a unique solution of the Schrödinger equation on the lattice is considered, where Δ is the difference Laplacian and both f and q have finite supports. It is shown that
Geometric aspects of functional analysis : Israel Seminar 2001-2002
• Mathematics
• 2003
Preface.- F. Barthe, M. Csornyei and A. Naor: A Note on Simultaneous Polar and Cartesian Decomposition.- A. Barvinok: Approximating a Norm by a Polynomial.- S.G. Bobkov: Concentration of
Anderson model with decaying randomness: mobility edge
• Mathematics
• 2000
Abstract. In this paper we consider the Anderson model with decaying randomness $a_nq_{\omega}(n)$, $a_n > 0, n \in {\mathbb Z}^{\nu}$ and $q_{\omega}(n)$, i.i.d. random variables with an
Multiscale averaging for ordinary differential equation. Applications to the spectral theory of onedimensional Schrödinger operators with sparse potentials Ser