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On the asymptotic behavior of solutions of Schrödinger type equations in unbounded domains. (English) Zbl 0681.35010
Analyse mathématique et applications, Contrib. Honneur Jacques-Louis Lions, 1-22 (1988).
[For the entire collection see Zbl 0651.00008.]
The author studies the asymptotic behavior of solutions $$u\in L^ 2(\Omega_ R)$$ of the Schrödinger equation $$-\Delta u+Vu=\lambda u$$ defined in the domain $$\Omega_ R=\{x\in {\mathbb{R}}^ n:\quad | x| >R\}$$ for complex valued, suitably decaying potentials V and for complex, non-positive and non-zero eigenvalues $$\lambda$$.
The author obtains a sharp bound in the sense that $0<\lim_{r\to \infty}\inf r^{(n-1)/2} \exp (kr)\| u(r,\cdot)\| \leq const,$ where $$r=| x|$$, $$k=Re(-\lambda)^{1/2}>0$$, and $$\| \cdot \|$$ is the $$L^ 2$$-norm in $$S^{n-1}.$$
The conclusions are derived from asymptotic results which hold for a general class of solutions of second order differential inequalities in a Hilbert space. These abstract results admit applications also to other situations not considered in the paper.
Reviewer: R.Weikard

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35J10 Schrödinger operator, Schrödinger equation 35P99 Spectral theory and eigenvalue problems for partial differential equations
##### Keywords:
sharp bound; differential inequalities
Zbl 0651.00008