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New global asymptotics and anomalies for the problem of quantization of the adiabatic invariant. (English. Russian original) Zbl 0715.58036
Funct. Anal. Appl. 24, No. 2, 104-114 (1990); translation from Funkts. Anal. Prilozh. 24, No. 2, 24-36 (1990).
It is well known that ‘transposing’ a classical Hamiltonian into a quantized one is still more an art than a science, because classical evolution equations are as a rule nonlinear, whereas quantized ones are always linear (probability amplitudes have to add by definition).
The author tries to transfer the notion of ‘chaos’ into quantum dynamics by reexamining some known technical difficulties of the theory of adiabatic approximations. He claims that in certain cases the counterpart to a nonlinear ‘anomaly’ implies an increase of dimensionality in the quantum asymptotics, i.e. implies the appearance of additional eigenvalues.
No physical justification of these extra eigenvalues is proposed.
Reviewer: I.Gumowski

58J37 Perturbations of PDEs on manifolds; asymptotics
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
81Q50 Quantum chaos
81P99 Foundations, quantum information and its processing, quantum axioms, and philosophy
Full Text: DOI
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