  We prove sharp analytic regularity and decay at infinity of solutions of
variable coefficients nonlinear harmonic oscillators. Namely, we show
holomorphic extension to a sector in the complex domain, with a corresponding
Gaussian decay, according to the basic properties of the Hermite functions in
R^d. Our results apply, in particular, to nonlinear eigenvalue problems for the
harmonic oscillator associated to a real-analytic scattering, or asymptotically
conic, metric in R^d, as well as to certain perturbations of the classical
harmonic oscillator.
