# 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} }

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

- 2013

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

- Adv. Math. Appl
- 1999

Modified Prüffer and EFGP transforms and the spectral analysis of onedimensional Schrödinger operators

- Comm. Math. Phys
- 1998

Absolutely continuous spectrum of multidimensional Schrödinger operator

- Mathematics, Physics
- 2004

We prove that a three-dimensional Schrodinger operator with slowly decaying potential has an absolutely continuous (a.c.) spectrum that fills R+. Asymptotics of Green's functions is obtained as well.