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On commutativity of unbounded operators in Hilbert space
Theses and Dissertations
  • Feng Tian, University of Iowa
Document Type
Dissertation
Date of Degree
2011
Degree Name
PhD (Doctor of Philosophy)
Degree In
Mathematics
First Advisor
Palle 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.

Keywords
  • index theory,
  • point interaction,
  • Schrödinger equations,
  • self-adjoint extension,
  • Sturm-Liouville
Pages
vi, 117 pages
Bibliography
Includes bibliographical references (pages 113-117).
Copyright
Copyright 2011 Feng Tian
Disciplines
Citation Information
Feng Tian. "On commutativity of unbounded operators in Hilbert space" (2011)
Available at: http://works.bepress.com/feng_tian/2/