Normal Forms for Semilinear Quantum Harmonic Oscillators

Benoit Grebert, Rafik Imekraz, Eric Paturel

We consider the semilinear harmonic oscillator $$iψ_t=(-Δ+\va{x}^{2} +M)ψ+\partial_2 g(ψ,\bar ψ), \quad x\in \R^d, t\in \R$$ where $M$ is a Hermite multiplier and $g$ a smooth func…