Canonical forms for perturbations of the harmonic oscillator (Q2339312)

Moser averaging techniques are used to obtain classical and quantum Birkhoff canonical forms for the 2D perturbed harmonic oscillator. Emphasis is laid on the perturbed semiclassical quantum Hamiltonians \(H=H_{0}+h^{2} H_{2}+h^{3}H_{3}+ \dots\), where \(H_{0} = -(h^{2}/2) (\partial^{2} / \partial x_{1}^{2} + \partial^{2} / \partial x_{2}^{2}) + (1/2)(x_{1}^{2}+x_{2}^{2})\) and \(H_{i}, i\geq2\) are semiclassical pseudodifferential operators of order zero. One also obtains the asymptotic expansion of the spectral measure of \(H\) in powers of \(h\) with terms expressible in terms of the quantum Birkhoff canonical form. In the final section, other settings to which the developed techniques can be applied are shortly described.