Asymptotics of the eigenvalues of operators with a Poisson algebra of symmetries in the highest symbol
type: Item
instance of: scholarly article
title: Asymptotics of the eigenvalues of operators with a Poisson algebra of symmetries in the highest symbol
zbMATH Open document ID: 0565.58038
DOI: 10.1007/BF01077830
publication date: 1984-01-01T00:00:00Z
review text: Let on \({\mathfrak X}\), \({\mathcal R}^ n_ q\oplus {\mathcal R}^ n_ p\) (n\(\geq 2)\) the action of the group \(R_+\) be given such that dp\(\wedge dq\), \(H_ i\) are homogeneous of the degrees 1, L-i respectively, \(H_ i\) real valued. Let \({\mathcal H}=H(q,-i\partial_ q)\) have the symbol \(H=H_ 0+H_ 1+H_ 2\) and be essentially self-adjoint in \(L^ 2\) and on the range of \(H_ 0|_{{\mathfrak X}}\) the spectrum of H be discrete. Let \(H_ 0\) be an integrable symbol and satisfy a certain non-resonance condition. Then H has an eigenvalue series of degree m provided \(H_ 1\), \(H_ 2\) satisfy certain conditions; here \(m=1,2,4\) depends on these conditions. In particular, \({\mathcal H}=-\Delta +V(q)\) on \({\mathcal S}^ n\) with an odd potential V(q) has a fourth order eigenvalue series \(E(k)=k^ 2+k(n-1)+c_ 1k^{-2}+c_ 2k^{-3}.\)
Mathematics Subject Classification ID: 58J40
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 35P20
zbMATH DE Number: 3901703
zbMATH Keywords: h-pseudo-differential operator
zbMATH Keywords: Poisson algebra of symmetries
zbMATH Keywords: non- resonance condition
zbMATH Keywords: Maslov index
zbMATH Keywords: eigenvalue series
author: M. V. Karasev
MaRDI profile type: Publication
cites work: Asymptotics of eigenvalue clusters for the Laplacian plus a potential
cites work: Sur le spectre des opérateurs elliptiques à bicaracteristiques toutes périodiques
cites work: Q3895943
published in: Functional Analysis and Its Applications
Recommended article: Poisson symmetry algebras and the asymptotics of spectral series
Recommended article: Q4292934
Recommended article: Q3217120
Recommended article: On the asymptotic distribution of the spectrum of an operator over irreducible representations of its symmetry group
Recommended article: Asymptotics of eigenvalues of large symmetric Toeplitz matrices with smooth simple-loop symbols
Recommended article: Q5749716
Recommended article: Supersymmetry in quantum mechanics: an extended view
Recommended article: Asymptotic symmetry groups and operator algebras
Recommended article: On the eigenvalue asymptotics of certain Schrödinger operators
Recommended article: Q5448149
description: scientific article; zbMATH DE number 3901703