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Strichartz estimates for Schrödinger equations with variable coefficients. (English) Zbl 1097.35002

Mém. Soc. Math. Fr., Nouv. Sér. 101-102, 208 p. (2005).
In this book the authors provide a proof of the full (local time) Strichartz estimates for the Schrödinger operator related to a nontrapping asymptotically flat perturbation of usual Laplacian in \(\mathbb{R}^n\). For \(\sigma\in(0, 1)\) define
\[ {\mathfrak B}_\sigma= \Biggl\{a\in C^\infty(\mathbb{R}^n):\forall \alpha\in \mathbb{N}^n, \exists C_\alpha: |\partial^\alpha a(x)|\leq {C_\alpha\over\langle x\rangle^{1+ |\alpha|+\sigma}}\Biggr\}, \] where \(\langle x\rangle= \sqrt{1+|x|^2}\). Let \(P\) be a second-order differential operator, \[ P= \sum^n_{j,k=1} D_j g^{jk}(x) D_k+ \sum^n_{j=1} (D_j b_j(x)+ b_j(x) D_j)+ V(x),\;D_j= -i\partial_x, \] with principal symbol \(p(x,\xi)= \sum^n_{j,k=1} G^{jk}(x) \xi_j\xi_k\) \((g^{jk}= g^{kj})\). Assume that the coefficients \(g^{jk}\), \(b_j\), \(V\) are real valued, there is \(\sigma> 0\) such that \(g^{jk}- \delta_{jk}\), and \(b_j\) are in \({\mathfrak B}_\sigma\), where \(\delta_{jk}\) is the Kronecker symbol and \(V\in L^\infty(\mathbb{R}^n)\). Then if the Hamilton flow associated to the principal symbol \(p(x,\xi)\) is nontrapping, it is proved that for \(T> 0\) and \((q, r)\) such that \(q> 2\) and \({2\over q}= {n\over 2}- {n\over r}\), there is \(C> 0\) satisfying
\[ \| e^{-itP}u_0\|_{L^q([-T, T], L^r(\mathbb{R}^n))}\leq C\| u_0\|_{L^2(\mathbb{R}^n)} \]
for all \(u_0\) in \(L^2(\mathbb{R}^n)\).
The authors prove the above theorem by constructing microlocal parametrics of the unitary operator \(e^{-itP}\) by use of FBI transform.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35J10 Schrödinger operator, Schrödinger equation
35A17 Parametrices in context of PDEs
35A22 Transform methods (e.g., integral transforms) applied to PDEs
35Q40 PDEs in connection with quantum mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)