Decay Bounds for Functions of Hermitian Matrices with Banded or Kronecker Structure

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

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Eigenbranes in Jackiw-Teitelboim gravity

Journal of High Energy Physics ◽

10.1007/jhep02(2021)168 ◽

2021 ◽

Vol 2021 (2) ◽

Author(s):

Andreas Blommaert ◽

Thomas G. Mertens ◽

Henri Verschelde

Keyword(s):

Finite Volume ◽

Chaotic System ◽

Path Integral ◽

Hermitian Matrices ◽

The Matrix ◽

Integral Contour

Abstract It was proven recently that JT gravity can be defined as an ensemble of L × L Hermitian matrices. We point out that the eigenvalues of the matrix correspond in JT gravity to FZZT-type boundaries on which spacetimes can end. We then investigate an ensemble of matrices with 1 ≪ N ≪ L eigenvalues held fixed. This corresponds to a version of JT gravity which includes N FZZT type boundaries in the path integral contour and which is found to emulate a discrete quantum chaotic system. In particular this version of JT gravity can capture the behavior of finite-volume holographic correlators at late times, including erratic oscillations.

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Direct Scaling Analysis of Fermionic Multiparticle Correlated Anderson Models with Infinite-Range Interaction

Advances in Mathematical Physics ◽

10.1155/2016/2129682 ◽

2016 ◽

Vol 2016 ◽

pp. 1-17 ◽

Cited By ~ 3

Author(s):

Victor Chulaevsky

Keyword(s):

Dynamical Localization ◽

Scaling Analysis ◽

Range Interaction ◽

Direct Scaling ◽

Strongly Mixing ◽

Optimal Decay ◽

Decay Bounds ◽

Multiparticle Systems ◽

Infinite Range ◽

Anderson Models

We adapt the method of direct scaling analysis developed earlier for single-particle Anderson models, to the fermionic multiparticle models with finite or infinite interaction on graphs. Combined with a recent eigenvalue concentration bound for multiparticle systems, the new method leads to a simpler proof of the multiparticle dynamical localization with optimal decay bounds in a natural distance in the multiparticle configuration space, for a large class of strongly mixing random external potentials. Earlier results required the random potential to be IID.

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Bounds for the Extreme Eigenvalues of Block 2 × 2 Hermitian Matrices

Journal of Mathematical Sciences ◽

10.1007/s10958-005-0312-y ◽

2005 ◽

Vol 129 (2) ◽

pp. 3772-3786

Author(s):

L. Yu. Kolotilina

Keyword(s):

Hermitian Matrices ◽

Extreme Eigenvalues

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A necessary and sufficient condition for simultaneous diagonalization of two hermitian matrices and its application

Glasgow Mathematical Journal ◽

10.1017/s0017089500000859 ◽

1970 ◽

Vol 11 (1) ◽

pp. 81-83 ◽

Cited By ~ 12

Author(s):

Yik-Hoi Au-Yeung

Keyword(s):

Complex Numbers ◽

Single Element ◽

Singular Matrix ◽

Necessary And Sufficient Condition ◽

Simultaneous Diagonalization ◽

Hermitian Matrices ◽

Skew Field ◽

Conjugate Transpose ◽

Necessary And Sufficient ◽

Real Quaternions

We denote by F the field R of real numbers, the field C of complex numbers, or the skew field H of real quaternions, and by Fn an n dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be hermitian if A = A*, and we say that two n × n hermitian matrices A and B with elements in F can be diagonalized simultaneously if there exists a non singular matrix U with elements in F such that UAU* and UBU* are diagonal matrices. We shall regard a vector u ∈ Fn as a l × n matrix and identify a 1 × 1 matrix with its single element, and we shall denote by diag {A1, …, Am} a diagonal block matrix with the square matrices A1, …, Am lying on its diagonal.

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