A uniform quantum version of the Cherry theorem

Consider in $L^2(\R^2)$ the operator family $H(\epsilon):=P_0(\hbar,\omega)+\epsilon F_0$. $P_0$ is the quantum harmonic oscillator with diophantine frequency vector \om, $F_0$ a bounded pseudodifferential operator with symbol decreasing to zero at infinity in phase space, and \ep\in\C. Then there exist \ep^\ast >0 independent of $\hbar$ and an open set \Omega\subset\C^2\setminus\R^2 such that if |\ep|<\ep^\ast and \om\in\Om the quantum normal form near $P_0$ converges uniformly with respect to $\hbar$. This yields an exact quantization formula for the eigenvalues, and for $\hbar=0$ the classical Cherry theorem on convergence of Birkhoff's normal form for complex frequencies is recovered.Comment: 17 page

On the energy exchange between resonant modes in nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equation $$i\psi_t= -\psi_{xx}\pm 2\cos 2x \ |\psi|^2\psi,\quad x\in S^1,\ t\in \R$$ and we prove that the solution of this equation, with small initial datum \psi_0=\e (\cos x+\sin x), will periodically exchange energy between the Fourier modes $e^{ix}$ and $e^{-ix}$. This beating effect is described up to time of order \e^{-9/4} while the frequency is of order \e^2. We also discuss some generalizations

Asymptotic Density of Eigenvalue Clusters for the Perturbed Landau Hamiltonian

We consider the Landau Hamiltonian (i.e. the 2D Schroedinger operator with constant magnetic field) perturbed by an electric potential V which decays sufficiently fast at infinity. The spectrum of the perturbed Hamiltonian consists of clusters of eigenvalues which accumulate to the Landau levels. Applying a suitable version of the anti-Wick quantization, we investigate the asymptotic distribution of the eigenvalues within a given cluster as the number of the cluster tends to infinity. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the Radon transform of the perturbation potential V.Comment: 30 pages. The explicit dependence on B and V in Theorem 1.6 (i) - (ii) indicated. Typos corrected. To appear in Communications in Mathematical Physic