Strichartz Estimates for Schrödinger Equations with Variable Coefficients
This book is devoted to the proof of regularity results for the solution of the Schrödinger equation i ∂t u - A u + V(x) = 0. Coefficients of the second order differential operator A are allowed to depend on the space variable x but A is assumed to be asymptotically (as |x| ‒> +∞) a perturbation of the Laplacian. For solutions of the equation the authors show the Strichartz estimate. The main step of the proof is to express the solution for an auxiliary problem via a Fourier integral operator with complex phase, which is described in chapter 6. It relies on a careful study of phase and transport equations, presented in chapter 4 and 5. When the authors have the explicit formula for the solution, they then deduce the dispersion estimate and use a general result [M. Kell, T. Tao: End point Strichartz estimate, Amer. J. Math. 120 (1998), p. 955-980] to conclude the proof in chapter 7. The proof of the theorem is quite technically demanding but the book is well ordered and carefully written. It is very interesting and experts in the field will surely appreciate it.