Essential spectra and global hypoellipticity of pseudo-differential operators

第一个目标是计算一类以热算子为典型例子的椭圆型拟微分算子的谱和本质谱。研究了这些算子的Fredholm性。我们还将研究这些算子在加权Sobolev空间中的全局正则性。一个相关的目标是为相应的全局椭圆型问题寻找相同的问题。另一个目标是考虑扭曲的拉普拉斯量，它是一个具有磁势的量子力学哈密顿量。扭曲拉普拉斯算子是一个退化的椭圆算子，它不是全局椭圆的。但在Schwartz分布意义下，它是整体亚椭圆的。探索了整体亚椭圆性的更精细的概念。一个非常具有挑战性的问题是找到一类退化的伪微分算子，它包含扭曲的拉普拉斯算子作为典型例子。在这个项目中产生的结果对于理解非紧流形上的退化偏微分方程组是重要的组成部分。前两个目标对于构造新的伪微分算子起着重要的作用，对于这些算子，椭圆性蕴含着Schwartz意义下的Fredholm性和整体亚椭圆性。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Hermite算子一样，将被证明是一类新的退化伪微分算子的典型例子，这些算子在Schwartz意义下是全局亚椭圆的。

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.