Stability and instability in quantum mechanics. (English) Zbl 0584.35024

Bellissard, Jean

Trends and developments in the eighties, Bielefeld Encounters Math. Phys. 4 and 5, 1-106 (1985).

[For the entire collection see Zbl 0577.00003.]
A survey (with complete proofs) devoted to both instability ”in the small” and ”in the large”. Instability ”in the small is alike to the small divisors problem in classical mechanics; it can be treated by a KAM-like technique with likewise results: the asymptotic behaviour depends on the rationality or irrationality of frequency ratios. Instability ”in the large” is instability that occurs only for sufficiently big perturbations; main example is when a beam of highly excited hydrogen atoms crosses a microwave chamber; for some critical value of field amplitude ionization rate suddenly increases. Usual perturbation theory demands hundreds of terms for description of this phenomenon and is therefore useless. Barrier penetration theory cannot be directly applied, because the states are highly excited. Classical approximation explains the averaged behaviour, but cannot explain resonances. Several strict results are proved in the survey using KAM technique and renormalization group, but in general the problem ”does there exist quantum chaos” remains open.
The problem of stability ”in the large” is analogic to Schrödinger equations for incommensurate crystals (i.e. with pseudorandom or random distribution of points). Classically such systems are stochastic, with scale invariant fractal spectrum. For some concrete system the situation is proved to be the same in the quantum case, and numerical experiments lead to a hypothesis that it is a generic type of behaviour of complicated quantum systems.

Reviewer: V.Ya.Kreǐnovich

MSC:

35J10	Schrödinger operator, Schrödinger equation
81Q15	Perturbation theories for operators and differential equations in quantum theory
35A35	Theoretical approximation in context of PDEs
81Q99	General mathematical topics and methods in quantum theory
35B35	Stability in context of PDEs

Keywords:

small divisors problem; asymptotic behaviour; KAM technique; Schrödinger equations

Citations:

Zbl 0577.00003