Invariant tori, effective stability, and quasimodes with exponentially small error terms. II: Quantum Birkhoff normal forms. (English) Zbl 1002.37028

Popov, G.

Ann. Henri Poincaré 1, No. 2, 249-279 (2000).

In the first part of this paper [cf. ibid. 1, No. 2, 223-248 (2000; Zbl 0970.37050)], the effective stability of quasi periodic orbits in nearly integrable classical Hamiltonian systems has been shown for a set \(\Lambda\) of invariant tori whose frequencies obey a certain Diophantine condition. Using this result, the authors obtain quasimodes for a Schrödinger type operator \(P_h\) with exponentially small error terms in the semiclassical limit by constructing a quantum Birkhoff normal form for this operator around \(\Lambda\). Applying this result, sharp lower bounds for the number of resonances of the operator are found. The theory is also applied to the case of discrete spectrum of \(P_h\).

Reviewer: Simos Ichtiaroglou (Thessaloniki)

MSC:

37J40	Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
81Q15	Perturbation theories for operators and differential equations in quantum theory
70H33	Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
70H08	Nearly integrable Hamiltonian systems, KAM theory

Keywords:

Hamiltonian systems; Schrödinger type operators; quantum Birkhoff normal forms; quasimodes; discrete spectrum

Citations:

Zbl 0970.37050