Short time behavior of Hermite functions on compact Lie groups
type: Item
instance of: scholarly article
title: Short time behavior of Hermite functions on compact Lie groups
zbMATH Open document ID: 0928.22010
author: Jeffrey J. Mitchell
published in: Journal of Functional Analysis
publication date: 2000-01-03T00:00:00Z
review text: Let \(p_{t}(x)\) be the (Gaussian) heat kernel on \(QTR{\mathbb R}^n\) at time \(t\). The classical Hermite polynomials at time \(t\) may be defined by a Rodrigues formula, given by \(H_{\alpha }(-x,t)p_{t}(x)=\alpha p_{t}(x)\), where \(\alpha \) is a constant coefficient differential operator on \(QTR{\mathbb R}^n\). Recent work of Gross (1993) and Hijab (1994) has led to the study of a new class of functions on a general compact Lie group \(G\). In analogy with the \(QTR{\mathbb R}^n\) case, these ``Hermite functions'' on \(G\) are obtained by the same formula, wherein \(p_{t}(x)\) is now the heat kernel on the group, \(-x\) is replaced by \(x^{-1}\), and \(\alpha \) is a right invariant differential operator. Let \(QTR\mathfrak {g}\) be the Lie algebra of \(G\). Composing a Hermite function on \(G\) with the exponential map produces a family of functions on \(QTR\mathfrak {g}\). We prove that these functions, scaled appropriately in \(t\), approach the classical Hermite polynomials at time \(1\) as \(t\) tends to \(0\), both uniformly on compact subsets of \(QTR\mathfrak {g}\) and in \(L^{p}(QTR\mathfrak {g},\mu)\), where \(1\leq p<\infty \), and \(\mu \) is a Gaussian measure on \(QTR\mathfrak {g}\). Similar theorems are established when \(G\) is replaced by \(G/K\), where \(K\) is some closed, connected subgroup of \(G\).
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 33C50
zbMATH DE Number: 1309115
zbMATH Keywords: compact Lie groups
zbMATH Keywords: Hermite functions
zbMATH Keywords: heat kernel
zbMATH Keywords: constant coefficient differential operator
reviewed by: François Rouvière
MaRDI profile type: Publication
full work available at URL: jfan.1999.3404
OpenAlex ID: W1967159055
cites work: Le spectre d'une variété riemannienne. (The spectrum of a Riemannian manifold)
cites work: Q5341449
cites work: Q3994903
cites work: Q3344779
cites work: Q4003026
cites work: On the Kakutani-Itô-Segal-Gross and Segal-Bargmann-Hall isomorphisms
cites work: Uniqueness of ground states for Schrödinger operators over loop groups
cites work: Q4322878
cites work: The Segal-Bargmann ``coherent state'' transform for compact Lie groups
cites work: Q5734920
cites work: Hermite functions on compact Lie groups. I
cites work: Hermite functions on compact Lie groups. II
cites work: Multiple Wiener integral
cites work: Determination of the Spectrum of the Flow of Brownian Motion
cites work: Q5731354
cites work: On the parabolic kernel of the Schrödinger operator
cites work: Short time behavior of the heat kernel and its logarithmic deviatives
cites work: Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds
cites work: The Laplacian on a Riemannian manifold. An introduction to analysis on manifolds
cites work: Tensor Algebras Over Hilbert Spaces. I
cites work: Some estimates of the transition density of a nondegenerate diffusion Markov process
cites work: Short time behavior of logarithmic derivatives of the heat kernel
cites work: Upper bounds on derivatives of the logarithm of the heat kernel
cites work: Q4527707
DOI: 10.1006/JFAN.1999.3404
description: scientific article; zbMATH DE number 1309115