Essential spectra and global hypoellipticity of pseudo-differential operators

伪微分算子的本质谱和全局亚椭圆性

基本信息

批准号：
8562-2006
负责人：
Wong, ManWah
金额：
$ 0.66万
依托单位：
York University
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2008
资助国家：
加拿大
起止时间：
2008-01-01 至 2009-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=389785
关键词：
Essential spectra global hypoellipticity pseudo

项目摘要

The first objective is to compute the spectra and essential spectra for a class of elliptic pseudo-differential operators, which contains the heat operator as a canonical example. The Fredholmness of these operators are investigated. The global regularity of these operators in terms of weighted Sobolev spaces will also be studied. A related objective is to look at the same problems for the corresponding globally elliptic problems. Another objective is to consider the twisted Laplacian, which comes up as a quantum mechanical Hamiltonian with a magnetic potential. The twisted Laplacian is a degenerate elliptic operator, which is not globally elliptic. But it is globally hypoelliptic in the sense of Schwartz distributions. More refined notions of global hypoellipticity are explored. A very challenging problem is to find a class of degenerate pseudo-differential operators that contains the twisted Laplacian as a canonical example. The results arising in this project are significant ingredients towards an understanding of degenerate partial differential equations on noncompact manifolds. The first two objectives play an important role towards constructions of new classes of pseudo-differential operators for which ellipticity implies Fredholmness and global hypoellipticity in the Schwartz sense. The twisted Laplacian, like the heat operator in the first objective and the Hermite operator in the second objective, will turn out to be a canonical example of a new class of degenerate pseudo-differential operators, which are globally hypoelliptic in the Schwartz sense.

第一个目标是计算一类椭圆伪微分算子的谱和基本谱，其中包含热算子作为典型示例。对这些操作员的 Fredholmness 进行了调查。还将研究这些算子在加权 Sobolev 空间方面的全局规律性。一个相关的目标是针对相应的全局椭圆问题来研究相同的问题。另一个目标是考虑扭曲拉普拉斯算子，它是具有磁势的量子力学哈密顿量。扭曲拉普拉斯算子是一个简并椭圆算子，它不是全局椭圆。但从施瓦茨分布的意义上来说，它是全局亚椭圆的。探索了更精确的全局低椭圆性概念。一个非常具有挑战性的问题是找到一类包含扭曲拉普拉斯作为典型示例的退化伪微分算子。该项目中产生的结果是理解非紧流形上的简并偏微分方程的重要组成部分。前两个目标对于构造新类别的伪微分算子起着重要作用，其中椭圆性意味着 Fredholmness 和 Schwartz 意义上的全局亚椭圆性。扭曲的拉普拉斯算子，就像第一个目标中的热算子和第二个目标中的埃尔米特算子一样，将成为一类新的简并伪微分算子的典型例子，这些算子在施瓦茨意义上是全局亚椭圆的。