Stochastic properties of degenerated quantum systems. (English) Zbl 1191.35221
Summary: We study the Schrödinger equation with degenerated symmetric but not self-adjoint Hamiltonian. The above properties of the quantum Hamiltonian arise in the description of the asymptotic behavior of the regularizing self-adjoint Hamiltonians sequence. A quantum dynamical semigroup corresponding to degenerated Hamiltonian is defined by means of the passage to the limit for the sequence of the regularizing dynamical semigroups. These semigroups are generated by the regularizing self-adjoint Hamiltonians. The necessary and sufficient conditions are obtained for the convergence of the regularizing semigroups sequence. The description of the divergent sequence of semigroups requires the extension of the stochastic process concept. We extend the stochastic process concept onto the family of measurable functions defined on the space endowed with finite additive measure. The above extension makes it possible to describe the structure of the partial limits set of the regularizing semigroups sequence.
MSC:
| 81S25 | Quantum stochastic calculus |
| 47N50 | Applications of operator theory in the physical sciences |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 35Q40 | PDEs in connection with quantum mechanics |
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