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Local properties of solutions of Schrödinger equations. (English) Zbl 0783.35054
This paper is concerned with the local behaviour of distributional solutions \(u\) to \[ -\Delta u+Vu=0 \tag{1} \] in some open set \(\Omega \subset \mathbb{R}^ n\). For the potential, \(V \in K^{n,\delta}(\Omega)\) is assumed with some \(\delta>0\). The class \(V \in K^{n,\delta}(\Omega)\) consists of those potentials \(V\) for which \[ \lim_{\varepsilon \downarrow 0}\sup_{x\in\Omega} \int_{\{y \in \Omega:| x-y |<\varepsilon\}}V(y) | x-y|^{-n-\delta+2}dy=0 \] holds. The authors show that near some point in \(\Omega\), say 0, any solution \(u \not\equiv 0\) of (1) may be represented as \[ u=P_ M+\Phi \tag{2} \] where \(P_ M\) is a nonzero homogeneous harmonic polynomial of some degree \(M \in \mathbb{N}\), and \(\Phi(x)=O\bigl( | x |^{M+\min\{1;\delta'\}}\bigr)\) as \(x\to 0\) for any \(\delta'<\delta\). Conversely for given \(P_ M\), equation (1) has a solution of the form (2) in some sufficiently small ball. In the same spirit, a representation result is proved for certain inhomogeneous equations.
Reviewer: K.J.Witsch (Essen)

MSC:
35Q40 PDEs in connection with quantum mechanics
35B40 Asymptotic behavior of solutions to PDEs
35J10 Schrödinger operator, Schrödinger equation
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