Normal forms for semilinear superquadratic quantum oscillators

On the real line, we consider nonlinear Hamiltonian Schrödinger equations with the superquadratic oscillator −d2/dx2+x2p+η(x)+M, where p is an integer ⩾2, η is a polynomial of degree <2p such that inf(x2p+η(x))⩾0, and M is a multiplier (i.e. simultaneously diagonalized with −d2/dx2+x2p+η(x)). A previous article (Grébert et al. (2009) [11], ) contains the case p=1 in Rd. Here we deal with d=1 but we authorize any superquadratic potential. Under generic conditions on M related to the nonresonance of the linear part, such a Hamiltonian equation admits, in a neighborhood of the origin, a Birkhoff normal form at any order. Consequently we deduce long time existence for solutions of the above equation with small Cauchy data in the high Sobolev spaces. As spectral analysis (spectrum and eigenfunctions) of the linear part is not explicit, we use Helffer–Robert and Yajima–Zhangʼs results (Helffer and Robert (1982) [12], , Yajima and Zhang (2001) [21]) to understand asymptotic behavior of both spectrum and eigenfunctions.