×

zbMATH — the first resource for mathematics

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

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. S. Bakai and Yu. P. Stepanovskii, Adiabatic Invariants [in Russian], Naukova Dumka, Kiev (1981).
[2] M. Born, Lectures on Atomic Mechanics [Russian translation], ONTI, Moscow (1934).
[3] L. Schiff, Quantum Mechanics [Russian translation], IL, Moscow (1957).
[4] V. P. Maslov, Perturbation Theory and Asymptotic Methods [in Russian], Moscow State Univ. (1965). · Zbl 0653.35002
[5] R. Bott and S. S. Chern, ”Hermitian vector bundles and the equidistribution of the zeros of their holomorphic sections,” Acta Math.,114, 71-112 (1965). · Zbl 0148.31906 · doi:10.1007/BF02391818
[6] B. Simon, ”Holonomy, the quantum adiabatic theorem and Berry’s phase,” Phys. Rev. Lett.,51, 2167-2170 (1965). · doi:10.1103/PhysRevLett.51.2167
[7] M. V. Berry, ”Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London,A392, 45-57 (1984). · Zbl 1113.81306
[8] M. V. Berry, ”Classical adiabatic angles and quantal adiabatic phase,” J. Phys. A, Math. Gen.,18, 15-27 (1985). · Zbl 0569.70020 · doi:10.1088/0305-4470/18/1/012
[9] A. V. Popov, ”Transformation of modes for a piecewise analytic waveguide transition (the waveguide approximation),” IZMIRAN, Preprint No. 11 (1978).
[10] V. A. Borovikov, ”Fields in smoothly irregular waveguides and the problem of the variation of the adiabatic invariant,” Inst. Prikl. Mat., Akad. Nauk SSSR, Preprint No. 99 (1978).
[11] V. S. Buslaev, ”Quasiclassical approximation for equations with periodic coefficients,” Usp. Mat. Nauk,42, No. 6, 77-98 (1987).
[12] L. V. Berlyand and S. Yu. Dobrokhotov, ”Operator separation of variables for problems on short-wave asymptotics of differential equations with quickly oscillating coefficients,” Dokl. Akad. Nauk SSSR,297, No. 1, 80-84 (1987).
[13] L. D. Faddeev and S. L. Shatashvili, ”Algebraic and Hamiltonian methods in the theory of non-Abelian anomalies,” Teor. Mat. Fiz.,60, No. 2, 206-217 (1984).
[14] M. Jezabek and M. Praszalowics (eds.), Workshop on Skyrmions and Anomalies, World Scientific (1987).
[15] M. V. Karasev, ”Poisson algebras of symmetries and asymptotic behavior of spectral series,” Funkts. Anal. Prilozhen.,20, No. 1, 21-32 (1986).
[16] M. V. Karasev, ”Connectivities on Lagrange manifolds and some problems of the quasiclassical approximation,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Instit. V. A. Steklova,172, 41-54 (1988).
[17] M. V. Karasev, ”Lagrange rings. Multiscale asymptotics of the spectrum near a resonance,” Funkts. Anal. Prilozh.,21, No. 1, 78-79 (1987). · Zbl 0684.58034
[18] M. V. Karasev, ”To the Maslov theory of quasiclassical asymptotics. Examples of new global quantization formula applications,” ITF, Kiev, Preprint No. ITF-89-78E (1989).
[19] M. V. Karasev, ”Quantum reduction to the orbits of the symmetry algebra and the Erenfest’s problem,” ITF, Akad. Nauk Ukr. SSR, Kiev, Preprint No. ITF-87-157R (1987).
[20] V. M. Babich, V. S. Buldyrev, and I. A. Molotkov, The Space-Time Ray Method. Linear and Nonlinear Waves [in Russian], Leningrad State Univ. (1985). · Zbl 0678.35002
[21] V. P. Maslov, The Complex WKB Method in Nonlinear Equations [in Russian], Nauka, Moscow (1977). · Zbl 0449.58001
[22] V. I. Arnol’d, ”Small denominators and problems of stability of motion in classical and celestial mechanics,” Usp. Mat. Nauk,18, No. 6, 91-192 (1963).
[23] V. I. Arnol’d, V. V. Kozlov, and A. I. Neidhtadt, ”Mathematical aspects of classical and celestial mechanics,” in: Current Problems in Mathematics, Fundamental Directions [in Russian], Vol. 3, Vsesoyuz. Inst. Nauchn. Tekhn. Informatsii, Moscow (1985).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.