Invariant tori, effective stability, and quasimodes with exponentially small error terms. II: Quantum Birkhoff normal forms
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 \(P_h\).
scientific article; zbMATH DE number 1540533