# Spectral Theory

## New submissions

[ total of 7 entries: 1-7 ]
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### New submissions for Tue, 20 Mar 18

[1]
Title: Global multiplicity bounds and Spectral Statistics Random Operators
Subjects: Spectral Theory (math.SP)

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 $\mathbb{R}$. 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 higher rank Anderson models with pure-point spectrum, with the randomness having support equal to $\mathbb{R}$, there is a uniform lower bound on spectral multiplicity and in case this is larger than one the local statistics is not Poisson.

[2]
Title: Quasi boundary triples, self-adjoint extensions, and Robin Laplacians on the half-space
Subjects: Spectral Theory (math.SP)

In this note self-adjoint extensions of symmetric operators are investigated by using the abstract technique of quasi boundary triples and their Weyl functions. The main result is an extension of Theorem 2.6 in [5] which provides sufficient conditions on the parameter in the boundary space to induce self-adjoint realizations. As an example self-adjoint Robin Laplacians on the half-space with boundary conditions involving an unbounded coefficient are considered.

### Cross-lists for Tue, 20 Mar 18

[3]  arXiv:1803.06717 (cross-list from math.AP) [pdf, other]
Title: High frequency limits for invariant Ruelle densities
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Dynamical Systems (math.DS); Spectral Theory (math.SP)

We establish an equidistribution result for Ruelle resonant states on compact locally symmetric spaces of rank one. More precisely, we prove that among the first band Ruelle resonances there is a density one subsequence such that the respective products of resonant and co-resonant states converge weakly to the Liouville measure. We prove this result by establishing an explicit quantum-classical correspondence between eigenspaces of the scalar Laplacian and the resonant states of the first band of Ruelle resonances which also leads to a new description of Patterson-Sullivan distributions.

### Replacements for Tue, 20 Mar 18

[4]  arXiv:1709.01774 (replaced) [pdf, ps, other]
Title: Multiplicity theorem of singular Spectrum for general Anderson type Hamiltonian
Subjects: Spectral Theory (math.SP)
[5]  arXiv:1711.05850 (replaced) [pdf, other]
Title: Local eigenvalue statistics of one-dimensional random non-selfadjoint pseudo-differential operators