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A Strichartz inequality for the Schrödinger equation on nontrapping asymptotically conic manifolds. (English) Zbl 1068.35119
The aim of the article is to prove for a solution \(u\) of the Schrödinger equation \(iu_t+\frac12\Delta_Mu=0\) the \(L^4\) space-time Strichartz inequality \(\int_0^1\int_M|u(t,z)|^4\,dg(z)\,dt\leq C\|u(0)\|^4_{H^{1/4}(M)}\) on any smooth three-dimensional asymptotically conic Riemannian manifold \(M\) obeying a nontrapping condition. The proof is based on the interaction Morawetz inequality as a positive commutator inequality for the tensor product \(U(t,z',z'')\equiv u(t,z')u(t,z'')\) of the solution with itself and smoothing estimates for Schrödinger solutions. It is shown that the exponent in \(H^{1/4}(M)\) is sharp.

MSC:
35Q40 PDEs in connection with quantum mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)
58J47 Propagation of singularities; initial value problems on manifolds
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