Spectra and semigroup smoothing for non-elliptic quadratic operators
type: Item
instance of: scholarly article
title: Spectra and semigroup smoothing for non-elliptic quadratic operators
zbMATH Open document ID: 1171.47038
author: Michael Hitrik
author: K. Pravda-Starov
published in: Mathematische Annalen
publication date: 2009-07-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/0712.0819
review text: In this paper, the authors study spectral and semigroup properties for pseudodifferential operators defined by the Weyl quantization formula \[ q(x,\xi)^{w}u(x)=\frac{1}{(2 \pi)^{n}}\int_{\mathbb{R}^{2n}}e^{i(x-y)\cdot\xi} q \biggl(\frac{x + y}{2},\xi\biggl)u(y) \,dy \,d\xi \] for some symbols \(q(x,\xi)\), \((x,\xi)\in\mathbb{R}^{n}\times\mathbb{R}^{n}\), which are complex-valued quadratic forms. First, it is pointed out the existence of a linear subspace \(S\) in \(\mathbb{R}_{x}^{n}\times\mathbb{R}_{\xi}^{n}\), called a singular space, which is defined in terms of the Hamilton map of the Weyl symbol \(q\), such that when \(S\) has a symplectic structure, the associated heat equation \[ \frac{\partial u}{\partial t}-q(x,\xi)^w u=0 \] with \(u(t,\cdot)/_{t=0}=u_{0}\in L^{2}(\mathbb{R}^{n})\) is smoothing in every direction of the orthogonal complement \(S^{\sigma \bot}\) of \(S\) with respect to the canonical symplectic form \(\sigma\) in \(\mathbb{R}^{2 n}\). The second result concerns the structure of the spectra for the considered pseudodifferential operators. It is proved that, if the real part of \(q\) is a non-positive quadratic form and if \(q\) is elliptic on \(S\), then the spectrum of the quadratic pseudodifferential operator \(q (x,\xi)^{w}\) is only composed of eigenvalues of finite multiplicity, and \(\sigma (q (x,\xi)^{w})\) coincides with the set of all points represented in the form \(\sum_{\lambda} (r_{\lambda} + 2 k_{\lambda})(-i \lambda)\), where the sum is taken over all \(\lambda \in \sigma (F)\) such that \(-i \lambda \in \mathbb{C}_{-} \cup (\sum (q \mid s)\backslash \{ 0 \})\), \(F\) is the Hamilton map associated to the quadratic form \(q\), \(r_{\lambda}\) denotes the dimension of the space of generalized eigenvalues of \(F\) in \(\mathbb{C}^{2 n}\) belonging to the eigenvalue \(\lambda \in \mathbb{C}\), \(\sum(q\mid S)=\overline{q (S)}\), and \(\mathbb{C}_{-}=\{z\in \mathbb{C}:\text{Re} z < 0 \}\). The large time behavior of contraction s…
reviewed by: Petru A. Cojuhari
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 35P05
zbMATH DE Number: 5586456
MaRDI profile type: Publication
OpenAlex ID: W2130886383
arXiv ID: 0712.0819
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DOI: 10.1007/S00208-008-0328-Y
description: scientific article