ON COMMUTATIVITY OF UNBOUNDED OPERATORS IN HILBERT SPACE

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 of representation theory. We will fur-ther study the partial derivative operators in the coordinate directions on the L2 space on various covering surfaces of the punctured plane. The operators are de-fined 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 prob-lems for second order partial differential operators (PDOs), Sturm-Liouville problems, H. Weyl’s limit-point/limit-circle theory, Schrödinger equations, and more. Abstract Approved