 95272 Kiselev A., Simon B.
 Rank One Perturbations with Infinitesimal Coupling
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Jun 14, 95

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Abstract. We consider a positive selfadjoint operator $A$ and formal
rank one pertubrations
$$
B=A+\alpha(\varphi, \cdot)\varphi
$$
where $\varphi\in\Cal H_{2}(A)$ but $\varphi\notin\Cal H_{1}(A)$,
with $\Cal H_{s}(A)$ the usual scale of spaces. We show that $B$ can be
defined for such $\varphi$ and what are essentially negative
infinitesimal values of $\alpha$. In a sense we'll make precise, every
rank one perturbation is one of three forms: (i) $\varphi\in\Cal H_{1}
(A)$, $\alpha\in\Bbb R$; (ii) $\varphi\in\Cal H_{1}$, $\alpha =\infty$;
or (iii) the new type we consider here.
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