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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
Citations:
Zbl 0651.00008