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| A relation between the theory of Schur functions and a notion for recurrence in discrete time quantum systems has been recently discovered. This is the origin of a rich interplay between spectral theory, complex analysis, orthogonal polynomials theory and the issue of quantum recurrence. The above connection not only provides new analytical techniques for quantum mechanical problems, but also reveals an unexpected geometrical and a topological meaning of some recurrence properties of quantum systems. We will review some of these results and their surprising physical consequences.
The results that will be reviewed are the fruit of joint works with:
Jean Bourgain (IAS Princeton)
Alberto Gr\"unbaum (UC Berkeley)
Albert Werner (Freie Universit\"at Berlin)
Reinhard Werner (Leibniz Universit\"at Hannover)
Jon Wilkening (UC Berkeley)
REFERENCES
Recurrence for discrete time unitary evolutions,
F.A. Gr\"unbaum, L. Vel\'azquez, A.H. Werner, R.F. Werner,
Commun. Math. Phys. 320 (2013) 543-569.
Quantum recurrence of a subspace and operator-valued Schur functions,
J. Bourgain, F.A. Gr\"unbaum, L. Vel\'azquez, J. Wilkening,
Commun. Math. Phys. (in press), arXiv:1302.7286 [quant-ph]. |
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