Dispersive estimates for the Schrodinger equation with finite rank perturbations

In this paper, we investigate dispersive estimates for the time evolution of HamiltoniansH = -& UDelta; + ⠂N j =1 (& BULL;,& phi;j)& phi;j in Rd, d> 1,where each & phi;j satisfies certain smoothness and decay condi-tions. We show that, under a spectral assumption, there exists a constant C = C(N, d, & phi;1, . . . , & phi;N) > 0 such thatIle-itH IIL1-L & INFIN; < Ct- 2d, for t > 0.As far as we are aware, this seems to provide the first study of L1 - L & INFIN; estimates for finite rank perturbations of the Lapla-cian in any dimension. We first deal with rank one perturbations (N = 1). Then we turn to the general case. By using an Aronszajn-Krein type formula for finite rank perturbations, we reduce the problem to the rank one case and solve it in a unified manner. More-over, we show that in some specific situations, the constant C(N, d, & phi;1, . . . , & phi;N) grows polynomially in N. Finally, as an application, we are able to extend the results to N = & INFIN; and deal with some trace class perturbations.& COPY; 2023 Elsevier Inc. All rights reserved.