scholarly journals Multiplicities and Plancherel formula for the space of nondegenerate Hermitian matrices

2021 ◽  
Author(s):  
Raphaël Beuzart-Plessis
Keyword(s):  
Plancherel Formula ◽  
Hermitian Matrices

scholarly journals SPHERICAL FUNCTIONS ON THE SPACE OF p-ADIC UNITARY HERMITIAN MATRICES

2014 ◽  
Vol 10 (02) ◽  
pp. 513-558
Author(s):  
YUMIKO HIRONAKA ◽  
YASUSHI KOMORI
Keyword(s):  
Spherical Functions ◽  
Plancherel Formula ◽  
Symmetric Polynomials ◽  
Hermitian Matrices ◽  
Cartan Decomposition ◽  
Explicit Formulas ◽  
Spherical Fourier Transform ◽  
Rank 2 ◽  
Type C ◽  
Residual Characteristic

We investigate the space X of unitary hermitian matrices over 𝔭-adic fields through spherical functions. First we consider Cartan decomposition of X, and give precise representatives for fields with odd residual characteristic, i.e. 2 ∉ 𝔭. From Sec. 2.2 till the end of Sec. 4, we assume odd residual characteristic, and give explicit formulas of typical spherical functions on X, where Hall–Littlewood symmetric polynomials of type Cn appear as a main term, parametrization of all the spherical functions. By spherical Fourier transform, we show that the Schwartz space [Formula: see text] is a free Hecke algebra [Formula: see text]-module of rank 2n, where 2n is the size of matrices in X, and give the explicit Plancherel formula on [Formula: see text].

scholarly journals Joseph Fourier 250thBirthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 250
Author(s):  
Frédéric Barbaresco ◽  
Jean-Pierre Gazeau
Keyword(s):  
Fourier Analysis ◽  
Heat Equation ◽  
Harmonic Analysis ◽  
Lie Groups ◽  
Coherent States ◽  
Geometric Theory ◽  
Plancherel Formula ◽  
Group Representations ◽  
Modern Development ◽  
Xxist Century

For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern development of Fourier analysis during XXth century has explored the generalization of Fourier and Fourier-Plancherel formula for non-commutative harmonic analysis, applied to locally compact non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups (by associating coherent states to group representations that are square integrable over a homogeneous space). The name of Joseph Fourier is also inseparable from the study of mathematics of heat. Modern research on Heat equation explores geometric extension of classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. The heat equation for a general volume form that not necessarily coincides with the Riemannian one is useful in sub-Riemannian geometry, where a canonical volume only exists in certain cases. A new geometric theory of heat is emerging by applying geometric mechanics tools extended for statistical mechanics, for example, the Lie groups thermodynamics.

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

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, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

scholarly journals Eigenbranes in Jackiw-Teitelboim gravity

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.

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

1970 ◽  
Vol 11 (1) ◽  
pp. 81-83 ◽  
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.

scholarly journals Graphs and Hermitian matrices: eigenvalue interlacing

2004 ◽  
Vol 289 (1-3) ◽  
pp. 119-127 ◽  
Author(s):  
Béla Bollobás ◽  
Vladimir Nikiforov
Keyword(s):  
Hermitian Matrices
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