Date of Degree

2011

Document Type

PhD diss.

Degree Name

PhD (Doctor of Philosophy)

Department

Mathematics

First Advisor

Palle E. Jorgensen

Abstract

We study several unbounded operators with view to extending von Neumann's theory of deficiency indices for single Hermitian operators with dense domain in Hilbert space. If the operators are non-commuting, the problems are difficult, but special cases may be understood with the use representation theory. We will further study the partial derivative operators in the coordinate directions on the L2 space on various covering surfaces of the punctured plane. The operators are defined on the common dense domain of C functions with compact support, and they separately are essentially selfadjoint, but the unique selfadjoint extensions will be non-commuting. This problem is of a geometric flavor, and we study an index formulation for its solution.

The applications include the study of vector fields, the theory of Dirichlet problems for second order partial differential operators (PDOs), Sturm-Liouville problems, H.Weyl's limit-point/limit-circle theory, Schrödinger equations, and more.

Pages

vi, 117

Bibliography

113-117

Copyright

Copyright 2011 Feng Tian

Included in

Mathematics Commons

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