Dispersive estimates for Schrödinger operators: a survey. (English) Zbl 1143.35001

Schlag, W.

Bourgain, Jean (ed.) et al., Mathematical aspects of nonlinear dispersive equations. Lectures of the CMI/IAS workshop on mathematical aspects of nonlinear PDEs, Princeton, NJ, USA, 2004. Princeton, NJ: Princeton University Press (ISBN 978-0-691-12955-6/pbk; 978-0-691-12860-3/hbk). Annals of Mathematics Studies 163, 255-285 (2007).

This survey article is devoted to dispersive estimates for the Schrödinger flow
\[ e^{itH}P_c,\;H=-\Delta+ V\quad\text{on }\mathbb{R}^d,\;d\geq 1, \]
here \(P_c\) is the projection onto the continuous spectrum of \(H\). The real-valued potential \(V= V(x)\) satisfies some decay conditions at infinity of the form \(|V(x)|\leq C\langle x\rangle^{-\beta}\), \(\beta> 0\). The following estimates are of interest: dispersive estimates
\[ \sup_{t\neq 0}\,|t|^{d({1\over 2}-{1\over p})}\| e^{itH}P_cf\|_{L^q(\mathbb{R}^d)}\leq C\| f\|_{L^p(\mathbb{R}^d)} \]
for all \(f\in L^1(\mathbb{R}^d)\cap L^2(\mathbb{R}^d)\), \(p\in[1, 2]\) and \({1\over p}+{1\over q}= 1\); and Strichartz estimates \[ \| e^{itH}P_c f\|_{L^q_t(L^p_x)}\leq C\| f\|_{L^2(\mathbb{R}^d)} \] for all \({2\over q}+ {d\over p}= {d\over 2}\), \(q\in (2,\infty]\).
Content:
1. \(d\geq 3\) (perturbation argument, Duhamels formula, spectral resolution, properties of resolvent, representation of resolvent, role of resonances);
2. \(d= 1\) (weighted dispersive estimates);
3. \(d= 2\) (zero as a regular point of the spectrum of \(-\Delta+ V\));
4. time-dependent potentials (overview).
For the entire collection see [Zbl 1113.35005].

Reviewer: Michael Reissig (Freiberg)

MSC:

35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
35B45	A priori estimates in context of PDEs
35Q40	PDEs in connection with quantum mechanics
35B34	Resonance in context of PDEs

Keywords:

Schrödinger operators; dispersive estimates; Strichartz estimates; properties of potential; real-valued potential; Duhamels formula; role of resonances; time-dependent potentials