Decay estimates for Schrödinger operators. (English) Zbl 0743.35008

Journé, J.-L.; Soffer, A.; Sogge, C. D.

Commun. Pure Appl. Math. 44, No. 5, 573-604 (1991).

Using the spectral theorem the unitary operator \(e^{itH_ 0}\) is well defined for \(t\in\mathbb{R}\), if \(H_ 0=-(\partial/\partial x_ 1)^ 2- \cdots-(\partial/\partial x_ n)^ 2\) is minus the Laplacian. Given initial data \(f(x)\) the function \(u(\cdot,t)=e^{itH_ 0}f\) solves the time dependent Schrödinger equation \(i\partial u/\partial t+H_ 0u=0\), \(u|_{t=0}=f\). Since the kernel of \(e^{itH_ 0}\) is \((4\pi it)^{n/2}e^{| x-y|^ 2/4it}\), one sees that the solution is dispersive in the sense that \[ \| u(\cdot,t)\|_{L^{p'}(\mathbb{R}^ n)}\leq Ct^{-n(1/p-1/2)}\| f\|_{L^ p(\mathbb{R}^ n)}, t>0,\quad\text{ if } 1\leq p\leq 2, \text{ and } 1/p+1/p'=1. \] This paper replaces the free operator \(H_ 0\) by more general Hamiltonians \(H=- \Delta+V(x)\) with suitable singularity assumptions on \(V(x)\). Let \(P_ c\) denote projection onto the continuous part of the spectrum of \(H\). The norm result is: Let \(n\geq 3\). Then if 0 is neither an eigenvalue nor a resonance for \(H\), \(\| e^{itH}P_ c\psi\|_{L^{p'}(\mathbb{R}^ n)}\leq ct^{-n(1/p-1/2)}\|\psi\|_{L^ P(\mathbb{R}^ n)}\), \(t>0\).

Reviewer: I.N.Katz (St.Louis)

MSC:

35B40	Asymptotic behavior of solutions to PDEs
35G10	Initial value problems for linear higher-order PDEs
35K25	Higher-order parabolic equations
35J10	Schrödinger operator, Schrödinger equation
35Q40	PDEs in connection with quantum mechanics

Keywords:

Schrödinger equation; dispersive

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