New global asymptotics and anomalies for the problem of quantization of the adiabatic invariant (Q752576)

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.