A P ] 1 6 O Ct 2 00 6 Perturbation Expansion and Nth OrderFermi Golden RuleoftheNonlinearSchr Odinger Equations

It is wellknown that the eigenvalues em bedded in a continuous spectrum of a self-adjoint operator, L, are unstable under generic perturbations. Intuitively,one can think oftim e periodic solutions (‘bound states’) ofthe equation,i_ u = Lu,corresponding to such eigenvalues,leaking their energy to the continuousspectrum solutions(‘scattering states’)which in turn isradiated to in nity.Thecouplingbetween tim e-periodicand continuousspectrum solutions wascom puted in the second orderofperturbation theory by P.A.M .Dirac in 1927 (see [Di],resulting in an elegantexpression.Ifthisexpression isnonzero, then the decay m entioned above takes place. E.Ferm icalled this condition \G olden Rule No. 2". In physics literature it is known as the Ferm iG olden Rule(FG R).TheFG R,com puted rstforatom icand m olecularstates,appears in m any otherareasofphysicse.g.in non-equilibrium statisticalm echanics.O f course,due to energy conservation the isolated eigenvalues are stable under reasonabletim e independentperturbations. TheFerm iG olden Rulewasintroduced tononlinearHam iltonian PDE in [S] whereitwasused toprovethattim e-periodicsolutionsforlinear(and nonlinear) waveequationsareunstableundergenericnonlinearperturbations.Itwasused