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Some remarks about the Schrödinger operator with a particular singular complex potential. (Quelques remarques sur l’opérateur de Schrödinger avec un potentiel complexe singulier particulier.) (French) Zbl 1040.35017
In this paper Schrödinger operators \(S=-\Delta+V\) in \({\mathbb R}^n\) are considered for which the sum in fact cannot be defined since \(D(\Delta) \cap D(V)=\{0\}\). It is assumed that \(V\in L^1({\mathbb R}^N)\), \(V\notin L^2_{\text{loc}}({\mathbb R}^N)\) and \(\operatorname{Re} (V) > 0\) (if \(N<4\) this implies that \(S\) is defined only for \(\{0\}\)). The author studies extensions of \(S\) by using the form sum of \(-\Delta\) and \(V\). It is shown that for sufficiently large \(\lambda\) the equation \((-\Delta \oplus V)u + \lambda u = v\) with given \(v\in L^2({\mathbb R}^N)\) has a unique solution in \(H^1({\mathbb R}^N)\). For this a fixed point argument is used.

MSC:
35J10 Schrödinger operator, Schrödinger equation
47B44 Linear accretive operators, dissipative operators, etc.
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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