Essential spectra and global hypoellipticity of pseudo-differential operators

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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.