On Distributions of Self-Adjoint Extensions of Symmetric Operators

In quantum probability a self-adjoint operator on a Hilbert space determines a real random variable and one can define a probability distribution with respect to a given state. In this paper we consider self-adjoint extensions of certain symmetric operators, such as momentum and Hamiltonian operators, with various boundary conditions, explicitly compute their probability distributions in some state and study dependence of these probability distributions on boundary conditions.