## Time decay of solutions to the Schrödinger equation in exterior domains. I, II.(English)Zbl 0684.35019

The time evolution given by the free particle Schrödinger operator on the complement, D, of a star-shaped bounded domain in $${\mathbb{R}}^ n$$ is studied. Dirichlet boundary conditions are considered. Upper bounds for the decay rates of the $$L^ p(D)$$-norm of the evolved state, u(t), are obtained. The bounds obtained in the first part are improved in the second one. For orientation let us reproduce one particular result: for $$n\geq 5$$, $$2\leq p\leq 2n/(n-4)$$ $\| u(t)\|_ p\leq const. I^{1/2}(1+t)^{-n(1-2/p)/2},$ where I is a functional of the initial value, u(0).
Reviewer: G.Nenciu

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35Q99 Partial differential equations of mathematical physics and other areas of application 35K10 Second-order parabolic equations

### Keywords:

exterior domains; Schrödinger operator; decay rates
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### References:

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