Growth of Sobolev norms in linear Schrödinger equations with quasi-periodic potential. (English) Zbl 0938.35150

Bourgain, Jean

Commun. Math. Phys. 204, No. 1, 207-247 (1999).

Denote by \(S(t)\) the unitary evolution operator on \(L^2\) associated to a Schrödinger operator with time-dependent electric potential, periodic in the space variables \(x\) and quasi periodic in the time variable \(t\). Then the author studies the behaviour of \(S(t)\) on the Sobolev space \(H^s\) for \(s>0\) and \(t\) large. In particular, he obtains an upper bound on its norm on \(H^s\) of the form \({\mathcal O}(\log(2+|t|)^{C_s}\) in the case where the space-dimension is one and in the case where the space-dimension is 2 and the electric potential is small enough.

Reviewer: André Martinez (Bologna)

MSC:

35Q40	PDEs in connection with quantum mechanics
35J10	Schrödinger operator, Schrödinger equation
47D06	One-parameter semigroups and linear evolution equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

time evolution; Schrödinger operator; periodic potential