Title: Self-adjoint approximations of degenerate Schrodinger operator,
Abstract: The problem of construction a quantum mechanical evolution for the
Schrodinger equation with a degenerate Hamiltonian which is a symmetric
operator that does not have self-adjoint extensions is considered. Self-adjoint
regularization of the Hamiltonian does not lead to a preserving probability
limiting evolution for vectors from the Hilbert space but it is used to
construct a limiting evolution of states on a C*-algebra of compact operators
and on an abelian subalgebra of operators in the Hilbert space. The limiting
evolution of the states on the abelian algebra can be presented by the Kraus
decomposition with two terms. Both of this terms are corresponded to the
unitary and shift components of Wold's decomposition of isometric semigroup
generated by the degenerate Hamiltonian. Properties of the limiting evolution
of the states on the C*-algebras are investigated and it is shown that pure
states could evolve into mixed states.