# Self-adjoint approximations of degenerate Schrodinger operator

## TL;DR

This paper develops a method to construct a limiting quantum evolution for degenerate Hamiltonians lacking self-adjoint extensions, using regularization and algebraic approaches, revealing that pure states can evolve into mixed states.

## Contribution

It introduces a novel approach to define quantum evolution for degenerate Hamiltonians without self-adjoint extensions via C*-algebra and Kraus decomposition.

## Key findings

- Limiting evolution can be represented by Kraus decomposition with two terms.
- Pure states can evolve into mixed states under the constructed evolution.
- Properties of the evolution on C*-algebras are characterized.

## 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.