Essential spectra and global hypoellipticity of pseudo-differential operators

第一个目标是计算一类椭圆型伪微分算子的谱和本质谱，其中包含热算子作为典型例子。研究了这些算子的Fredholmness性。这些算子在加权Sobolev空间中的全局正则性也将被研究。一个相关的目标是看相同的问题，相应的全球椭圆问题。另一个目标是考虑扭曲的拉普拉斯算子，它是一个具有磁势的量子力学哈密顿算子。扭拉普拉斯算子是一个退化的椭圆算子，它不是整体椭圆的。但它在Schwartz分布意义下是全局亚椭圆的。更精细的概念，全球hypoellipticity进行了探讨。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，其中包含扭曲的拉普拉斯算子作为一个典型的例子。本计画的结果对于了解非紧流形上退化的偏微分方程具有重要的意义。前两个目标发挥了重要作用，对建设新的类的伪微分算子的椭圆性意味着Fredholmness和全球hypoellipticity在施瓦茨意义上。扭曲拉普拉斯算子，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，将成为一类新的退化伪微分算子的典型例子，它们在施瓦茨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.