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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