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

2021 ◽  
Author(s):  
S. Molchanov ◽  
O. Safronov ◽  
B. Vainberg
Keyword(s):  
Point Spectrum ◽  
Pure Point ◽  
Absolutely Continuous ◽  
Wave Operators ◽  
Sparse Potentials ◽  
Discrete Operators ◽  
Sparse Potential

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 $\lambda_0<0$.

scholarly journals Pure Point Spectrum of the Laplacians on Fractal Graphs

1995 ◽  
Vol 129 (2) ◽  
pp. 390-405 ◽  
Author(s):  
L. Malozemov ◽  
A. Teplyaev
Keyword(s):  
Point Spectrum ◽  
Pure Point ◽  
Pure Point Spectrum

scholarly journals Global multiplicity bounds and spectral statistics for random operators

2020 ◽  
Vol 32 (09) ◽  
pp. 2050025
Author(s):  
Anish Mallick ◽  
Krishna Maddaly
Keyword(s):  
Lower Bound ◽  
Separable Hilbert Space ◽  
Point Spectrum ◽  
Pure Point ◽  
Pure Point Spectrum ◽  
Positive Lebesgue Measure ◽  
Spectral Multiplicity ◽  
Local Statistics ◽  
Higher Rank ◽  
Spectral Statistics

In this paper, we consider Anderson type operators on a separable Hilbert space where the random perturbations are finite rank and the random variables have full support on [Formula: see text]. We show that spectral multiplicity has a uniform lower bound whenever the lower bound is given on a set of positive Lebesgue measure on the point spectrum away from the continuous one. We also show a deep connection between the multiplicity of pure point spectrum and local spectral statistics, in particular, we show that spectral multiplicity higher than one always gives non-Poisson local statistics in the framework of Minami theory. In particular, for higher rank Anderson models with pure point spectrum, with the randomness having support equal to [Formula: see text], there is a uniform lower bound on spectral multiplicity and in case this is larger than one, the local statistics is not Poisson.

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