Document Type


Date of Degree

Spring 2011

Degree Name

PhD (Doctor of Philosophy)

Degree In


First Advisor

Palle Jorgensen

First Committee Member

Tong Li

Second Committee Member

Surjit Khurana

Third Committee Member

Bor-Luh Lin

Fourth Committee Member

Muthukrishnan Krishnamurthy

Fifth Committee Member

Juan Gatica

Sixth Committee Member

Gerhard Strohmer

Seventh Committee Member

Wayne Polyzou


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.


index theory, point interaction, Schrödinger equations, self-adjoint extension, Sturm-Liouville


vi, 117 pages


Includes bibliographical references (pages 113-117).


Copyright 2011 Feng Tian

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Mathematics Commons