On quantum dynamics on $C^*$-algebras

I. V. Volovich ; V. Zh. Sakbaev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis, mathematical physics, and applications, Tome 301 (2018), pp. 33-47.

Résumé

We consider the problem of constructing quantum dynamics for symmetric Hamiltonian operators that have no self-adjoint extensions. For an earlier studied model, it was found that an elliptic self-adjoint regularization of a symmetric Hamiltonian operator allows one to construct quantum dynamics for vector states on certain $C^*$-subalgebras of the algebra of bounded operators in a Hilbert space. In the present study, we prove that one can extend the dynamics to arbitrary states on these $C^*$-subalgebras while preserving the continuity and convexity. We show that the obtained extension of the dynamics of the set of states on $C^*$-subalgebras is the limit of a sequence of regularized dynamics under removal of the elliptic regularization. We also analyze the properties of the limit dynamics of the set of states on the $C^*$-subalgebras.

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