95-272 Kiselev A., Simon B.

Rank One Perturbations with Infinitesimal Coupling (31K, AMSTeX) Jun 14, 95

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Abstract. We consider a positive self-adjoint 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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