Proof of a Spectral Property related to the singularity formation for the L2L2 critical nonlinear Schrödinger equation

We give a proof of a Spectral Property related to the description of the singularity formation for the L2L2 critical nonlinear Schrödinger equation iut+Δu+u|u|4N=0 in dimensions N=2,3,4N=2,3,4.Assuming this property, the rigorous mathematical analysis developed in a recent series of papers by Merle and Raphaël provides a complete description of the collapse dynamics for a suitable class of initial data. In particular, this implies in dimension N=2N=2 the existence of a large class of solutions blowing up with the log–log speed |u(t)|H1∼log|log(T−t)T−t where T>0T>0 is the blow up time.This Spectral Property is equivalent to the coercivity of some Schrödinger type operators. An analytic proof is given in [F. Merle, P. Raphaël, Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, Ann. of Math. 161 (1) (2005) 157–222] in dimension N=1N=1 and in this paper, we give a computer assisted proof in dimensions N=2,3,4N=2,3,4. We propose in particular a rigorous mathematical frame to reduce the check of this type of coercivity property to accessible and robust numerical results.