Date of Award

Spring 1-1-2016

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Alexander Gorokhovsky

Second Advisor

Carla Farsi

Third Advisor

Marty Walter

Fourth Advisor

Pedro Berrizbeitia

Fifth Advisor

Sebastian Casalaina-Martin

Abstract

The main problem we consider in this thesis is the essential self-adjointness of the symplectic Dirac operators D and ~D constructed by Katharina Habermann in the mid 1990s. Her constructions run parallel to those of the well-known Riemannian Dirac operators, and show that in the symplectic setting many of the same properties hold. For example, the symplectic Dirac operators are also unbounded and symmetric, as in the Riemannian case, with one important difference: the bundle of symplectic spinors is now infinite-dimensional, and in fact a Hilbert bundle. This infinite dimensionality makes the classical proofs of essential self-adjointness fail at a crucial step, namely in local coordinates the coefficients are now seen to be unbounded operators on L2(Rn). A new approach is needed, and that is the content of these notes. We use the decomposition of the spinor bundle into countably many finite-dimensional subbundles, the eigenbundles of the harmonic oscillator, along with the simple behavior of D and ~D with respect to this decomposition, to construct an inductive argument for their essential self-adjointness. This requires the use of ancillary operators, constructed out of the symplectic Dirac operators, whose behavior with respect to the decomposition is transparent. By an analysis of their kernels we manage to deduce the main result one eigensection at a time.

Included in

Mathematics Commons

Share

COinS