\input amstex \documentstyle{amsppt} \loadbold \magnification=1200 \baselineskip=15 pt \TagsOnRight \topmatter \title Rank One Perturbations with Infinitesimal Coupling \endtitle \author A.~Kiselev and B.~Simon$^{*}$ \endauthor \leftheadtext{A.~Kiselev and B.~Simon} \affil Division of Physics, Mathematics, and Astronomy \\ California Institute of Technology \\ Pasadena, CA 91125 \endaffil \thanks $^{*}$ This material is based upon work supported by the National Science Foundation under Grant No.~DMS-9101715. The Government has certain rights in this material. \endthanks \thanks To appear in \it{J.~Funct.~Anal.} \endthanks \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. \endabstract \endtopmatter \document \bigpagebreak \flushpar {\bf \S 1. Introduction} \medpagebreak There has recently been considerable interest in the study of rank one perturbations of positive self-adjoint operators (see [11] and references therein). Let $A\geq 0$ on a Hilbert space $\Cal H$ and consider $$B=A+\alpha(\varphi, \cdot)\varphi. \tag 1.1$$ Simon-Wolff [12] first pointed out that a natural framework for this was to consider $\varphi\in\Cal H_{-1}(A)$ where $\Cal H_{s}(A)$ is the usual scale of spaces associated to $A$; that is, if $s\geq 0$, $\Cal H_{s}(A)=D(|A|^{s/2})$ with the norm $\|\cdot\|$ given by $$\|\varphi\|^{2}_{s} =\langle\varphi, (A+1)^{s}\varphi\rangle,$$ and if $s<0$, $\Cal H_{s}(A)$ is the completion of $\Cal H$ in the $\|\cdot\|_{s}$ norm. $\Cal H_{s}\subset\Cal H_{t}$ if $s>t$ and one can define $\Cal H_{\infty}(A)=\operatornamewithlimits{\cap} \limits_{s}\Cal H_{s}(A)$ and $\Cal H_{-\infty}(A)= \operatornamewithlimits{\cup}\limits_{s}\Cal H_{s}(A)$. $\Cal H^{*}_{s}=\Cal H_{-s}$ in a natural way. When $\varphi\in\Cal H_{-1}(A)$, $\psi\mapsto |(\psi, \varphi)|^{2}$ defines a quadratic form on $Q(A)=\Cal H_{+1}(A)$, which is $A$-bounded with relative bound zero. So the standard form perturbation theory ([7,10]) lets one define (1.1) for any $\alpha\in \Bbb R$. Define \align F_{\alpha}(z) &=(\varphi, (A_{\alpha}-z)^{-1}\varphi) \tag 1.2a \\ F(z) &=F_{\alpha=0}(z). \tag 1.2b \endalign One easily proves the formulae (going back to Krein and Aronszajn): $$\gather F_{\alpha}(z)=F(z)\big/ 1+\alpha F(z) \tag 1.3 \\ (A_{\alpha}-z)^{-1}\varphi = (1+\alpha F(z))^{-1}(A-z)^{-1}\varphi \tag 1.4a \\ (A_{\alpha}-z)^{-1} = (A-z)^{-1} -\alpha(1+\alpha F(z))^{-1} ((A-\bar z)^{-1}\varphi, \cdot)(A-z)^{-1}\varphi. \tag 1.4b \endgather$$ >From (1.4) one sees $\operatornamewithlimits{s-lim}\limits_{\alpha\to \infty} (A_{\alpha}-z)^{-1}$ exists. If $\varphi\notin\Cal H_{0}(A) =\Cal H$, it defines an operator $A_\infty$ on $\Cal H$. This is studied in Gesztesy-Simon [5]. Our primary goal here is two-fold: (a) To construct a family of rank one perturbations $A+\alpha (\varphi, \cdot)\varphi$ where $\varphi\notin\Cal H_{-1}(A)$ but only in $\Cal H_{-2}(A)$. Here $\alpha$ is infinitesimal. (b) Every pair of semibounded operators with $(A+i)^{-1}-(B+i)^{-1}$ rank one can be written using the $\alpha(\varphi, \cdot)\varphi$ construction with $\varphi\in\Cal H_{-1}$ and $\alpha$ finite or infinite. These two apparently paradoxical statements are not paradoxical because in (b) we did not specify if $B$ is a perturbation of $A$ or vice-versa. In fact, one can always label them so that $A\leq B$. Then we will show that $B=A+\alpha(\varphi, \cdot)\varphi$ with $\varphi\in \Cal H_{-1}(A)$ with $\alpha\in [0, \infty]$. If $\alpha <\infty$, then $A$ can be obtained from $B$ by a rank one perturbation with $\varphi\in\Cal H_{-1}(B)$. But if $\alpha=\infty$, it is necessary to use the $\Cal H_{-2}(B)$ construction to recover $A$ from $B$. At first, it is comforting that infinitesimal coupling is needed to undo infinite coupling, but that feeling is unfounded. For multiplicative perturbations, infinitesimal should undo infinite, but these perturbations are additive. In fact, $(\eta, \cdot)\eta$ with $\eta\in\Cal H_{-2}(B)/\Cal H_{-1}(B)$ is so infinite we need infinitesimal coupling to undo $\infty (\varphi, \cdot)\varphi$ with $\varphi\in\Cal H_{-1}(A)$. A theme that we will explore in this paper is that if $A,B$ have resolvents that differ by a rank one, then there exists a symmetric operator $C$ with deficiency indices $(1, 1)$ so that $A$ and $B$ are both self-adjoint extensions of $C$. To say that $B$ is $A+\alpha(\varphi, \cdot)\varphi$ with $\alpha =\infty$ and $\varphi\in\Cal H_{-1}(A)$ (equivalently that $A$ is $B+\alpha(\varphi, \cdot)\varphi$ with $\varphi\in\Cal H_{-2}(B)/\Cal H_{-1}(A)$ and $\alpha$ infinitesimal) is equivalent to saying that $B$ is the Friedrich's extension. From this point of view, our assertion (b) above is a special case of the Birman-Krein-Vishik theory of quadratic forms of positive self-adjoint extensions [3,8,13,6,2]. In \S2, we present the construction of rank one perturbations with $\varphi\in\Cal H_{-2}$. In \S3, we use resolvent ordering to prove assertion (b). In \S4, we explain the relation of infinite and infinitesimal coupling. In \S5, we consider fairly general situations $A_{n}=A+\alpha_{n}(\varphi_{n}, \cdot)\varphi_{n}$ with $\varphi_{n}$ a cutoff of $\varphi\in\Cal H_{-\infty}(A)$ and show that as $n\to\infty$, $A_n$ converges to $A$ in strong resolvent sense unless $\varphi\in\Cal H_{-1}(A)$ or $\varphi\in\Cal H_{-2}(A)$, $\alpha_{n}<0$ and $\alpha_{n}\to 0$ at a suitable rate. This provides another view of the fact that the only rank one perturbations are the $\Cal H_{-1}(A)$ and $\Cal H_{-2}(A)$ constructions. In \S6, we discuss the connection to the theory of self-adjoint extensions of deficiency indices $(1, 1)$. Finally, \S7 presents some simple examples. \bigpagebreak \flushpar {\bf \S2. The Basic $\Cal H_{\boldkey -\bold2}\boldkey( \boldkey A\boldkey)$ Construction} \medpagebreak Let $\varphi\in\Cal H_{-2}(A)$ so $(A-z)^{-1}\varphi$ makes sense for any $z\notin\text{spec}(A)$ and in particular, for $\text{Im}\,z\neq 0$. Motivated by (1.4), we try to construct a self-adjoint operator whose resolvent $R(z)$ obeys $$R(z)=(A-z)^{-1} -\sigma (z)K(z) \tag 2.1a$$ where $$K(z)=((A-\bar z)^{-1}\varphi, \cdot)(A-z)^{-1}\varphi. \tag 2.1b$$ The idea is to define $R(z)$ by (2.1) and then to pick the unknown function $\sigma (z)$ in order that $R$ obey the equation obeyed by any resolvent: $$\frac{dR}{dz} = R(z)^{2}. \tag 2.2$$ Since $\frac{dK}{dz}=(A-z)^{-1}K+K(A-z)^{-1}$ and $\frac{d}{dz}(A- z)^{-1}\equiv (A-z)^{-2}$, (2.2) is equivalent to $$\frac{d\sigma}{dz}\,K(z)=-\sigma (z)^{2}K(z)^{2}. \tag 2.3$$ But $K(z)^{2}=K(z)(\varphi, (A-z)^{-2}\varphi)$. Thus (2.2) is equivalent to $$\frac{d}{dz} \,\sigma^{-1}(z)=(\varphi, (A-z)^{-2}\varphi). \tag 2.4$$ Supposing that $A\geq 0$, we note that (2.4) shows that $\sigma^{-1}$, originally defined for $\text{Im}\,z\neq 0$, can be continued through $(-\infty, 0)$. Self-adjointness for $R$, that is, $R^{*}(z)=R(\bar z)$ requires $\sigma^{-1}$ be real there; and thus the solutions can be written $$\sigma^{-1}(z)=\beta +(\varphi, [(A-z)^{-1}-(A+1)^{-1}]\varphi) \tag 2.5$$ with $\beta$ real and equal to $\sigma^{-1}(-1)$. This motivates: \proclaim{Theorem 2.1} Fix $\beta\in\Bbb R$. Suppose $A\geq 0$ and $\varphi\in\Cal H_{-2}(A)$. For $\text{\rom{Im}}\,z\neq 0$, define $R_{\beta}(z)$ by \rom{(2.1)} with $\sigma (z)$ given by \rom{(2.5)}. Then there is a self-adjoint operator $\widetilde A_{\beta}$ with $R_{\beta} (z)=(\widetilde A_{\beta}-z)^{-1}$. \endproclaim \demo{Proof} Let $$G(z)\equiv (\varphi,[(A-z)^{-1}-(A+1)^{-1}]\varphi). \tag 2.6$$ Then for $y\in (-\infty, 0)$, $\frac{dG}{dy}=(\varphi, (A-y)^{-2} \varphi)>0$. Thus, there is at most one $y<0$, call it $y_{0}$, so $\sigma (y)^{-1}=0$. Therefore, $R_{\beta}(z)$ extends to $\Bbb C \backslash [0, \infty)\cup\{y_{0}\}$ with $R_{\beta}(y)$ self-adjoint if $y\in\Bbb R\backslash [0, \infty)\cup\{y_{0}\}$. Fix any $y_{1}<0$ with $y_{1}\neq y_{0}$ and define $A_{\beta}\equiv R_{\beta}(y_{1})^{-1}-y_{1}$. Then $R_{\beta}(z)$ and $(A_{\beta}-z)^{-1}$ obey the same differential equation (1.2) and same initial conditions at $y=y_{1}$, and so they are equal on $\text{Im}\,z\neq 0$. \qed \enddemo \remark{Remark} One can think of (2.1) in the form $$\gather (\widetilde A_{\beta}-z)^{-1}=(A-z)^{-1}-\sigma_{\beta}(z)K(z) \tag 2.1c \\ \sigma_{\beta}(z)^{-1}=\beta +(\varphi, ((A-z)^{-1}-(A+1)^{-1}\varphi) \endgather$$ as a renormalized form of (1.4), which can be written \align (A_{\alpha}-z)^{-1} &= (A-z)^{-1}-\widehat{\sigma}_{\alpha}(z)K(z) \\ \widehat{\sigma}_{\alpha}(z)^{-1} &= \alpha^{-1} +(\varphi, (A-z)^{-1}\varphi). \endalign If $\varphi\in\Cal H_{-1}(A)$, then $\widetilde A_{\beta}=A_{\alpha}$ where $\beta$ and $\alpha$ are related by $$\beta=\alpha^{-1}+(\varphi, (A+1)^{-1}\varphi). \tag 2.7$$ If $\varphi\notin\Cal H_{-1}$, in essence we need to take $\alpha^{- 1}=-\infty$ to undo the divergence of $(\varphi, (A+1)^{-1}\varphi)$, and $\alpha$ is infinitesimal and negative. The condition $\varphi\in\Cal H_{-2}(A)$ is required for the single renormalization to work. \endremark \proclaim{Theorem 2.2} If $\varphi\notin\Cal H_{-1}(A)$, then each operator $A_\beta$ defined in Theorem \rom{2.1} obeys $\widetilde A_{\beta}\leq A$ with $\widetilde A_{\beta}\neq A$. If $\varphi\in \Cal H_{-1}(A)$, there exist $\widetilde A_{\beta}$'s with $\widetilde A_{\beta}\geq A$ with $\widetilde A_{\beta}\neq A$. \endproclaim \remark{Remark} Recall ([7]) that we say $A,B$ obey $A\geq B$ if and only if there is $a\in\Bbb R$ with $A\geq a1$, $B\geq a1$; and for $z0$, that proves the $\Cal H_{-1}$ result. If $\varphi\notin\Cal H_{-1}$, then $G(-y)\to -\infty$ as $y\to\infty$. Thus, there is some $y_{0}\in (-\infty, 0)$, so $G(y)+\beta <0$ for all $y\leq y_{0}$. By (2.5) and (2.1c), $(\widetilde A_{\beta}-y)^{-1} \geq (A-y^{-1})>0$ for such $y$, so $\widetilde A_{\beta}\geq y_{0}$, $A\geq y_{0}$, and $\widetilde A_{\beta}\leq A$. \qed \enddemo \bigpagebreak \flushpar {\bf \S3. Every Rank One Perturbation Is $\Cal H_{\boldkey - \bold1}\boldkey(\boldkey A\boldkey)$-bounded} \medpagebreak In this section, we want to consider pairs of operators $A,B$ so that $(A+i)^{-1}-(B+i)^{-1}$ is rank one. We start with two results that illuminate the notion: \proclaim{Proposition 3.1} Let $A,B$ be self-adjoint operators. Then $Q(z)=(A-z)^{-1}-(B-z)^{-1}$ is rank one for one $z$ with $\text{\rom{Im}}\,z\neq 0$ if and only if it is rank one for all such $z$. \endproclaim \demo{Proof} $$(A-z)^{-1}=(1+(w-z)(A-z)^{-1})(A-w)^{-1}, \tag 3.1$$ so using the fact that $$(\varphi, (A-z)^{-1}-(B-z)^{-1}\psi) = ((A-\bar z)^{-1}\varphi, B(B-z)^{-1}\psi)-(A(A-\bar z)^{-1}\varphi, (B-z)^{-1}\psi),$$ we see that $$Q(z)=(1+(w-z)(A-z)^{-1})Q(w)(1+(w-z)(B-z)^{-1})$$ and so $\text{Rank}\,Q(z)\leq\text{Rank}\,Q(w)$. \qed \enddemo \proclaim{Proposition 3.2} Suppose that $A,B$ are self-adjoint, $A\geq 0$, and $(A+i)^{-1}-(B+i)^{-1}$ is rank one. Then $B$ is bounded from below. \endproclaim \demo{Proof} By (3.1) for $B$, $w\in (-\infty, 0)$ is in $\text{spec}(B)$ if and only if $1+(w-i)(B-i)^{-1}$ is not invertible. But \align L(w)=1+(w-i)(B-i)^{-1} &= 1+(w-i)(A-i)^{-1} + (w-i)((B-i)^{-1}-(A- i)^{-1}) \\ &= L_{1}(w)+L_{2}(w), \endalign where $L_{1}=1+(w-i)(A-i)^{-1}=(A-w)(A-i)^{-1}$ is invertible for $w\in (-\infty, 0)$ and $L_{2}=(w-i)((B-i)^{-1}-(A+i)^{-1})$ is rank one. Thus, $L(w)$ is invertible if and only if $1+L_{1}(w)^{-1}L_{2}(w)$ is invertible. By (3.1), $w\in\text{spec}(B)$ if and only if $1+L_{1} (w)^{-1}L_{2}(w)$ is not invertible. Thus, since $L_{2}$ is rank one, $w\in\text{spec}(B)$ if and only if $F(w)\equiv\text{Tr}(L_{1}(w)^{-1} L_{2}(w))=-1$. $F$ is an entire analytic function with $F(w)\neq -1$ if $\text{Im}\,w\neq 0$. We conclude $B$ has isolated point spectrum on $(-\infty, 0)$. Thus, there exist real $w_0$ with $F(w_{0})\neq -1$ and so $(B- w_{0})^{-1}-(A-w_{0})^{-1}$ is rank one. For rank one perturbations of self-adjoint operators, eigenvalues intertwine. Since $A$ has no eigenvalues in $(-\infty, 0)$, $B$ can have only one eigenvalue in $(-\infty, 0)$; that is, $B$ is bounded from below. \qed \enddemo \proclaim{Corollary 3.3} If $A\geq 0$ and $(A+i)^{-1}-(B+i)^{-1}$ is rank one, then either $A\geq B$ or $B\geq A$. \endproclaim \demo{Proof} Pick $w$ below $\text{spec}(A)\cup\text{spec}(B)$. Then $(A-w)^{-1}\geq 0$, $(B-w)^{-1}\geq 0$, and since $(A-w)^{-1}-(B-w)^{- 1}$ is rank one and self-adjoint, either $(A-w)^{-1}\geq (B-w)^{-1}$ or $(B-w)^{-1}\geq (A-w)^{-1}$. It follows that either $A\geq B$ or $B\geq A$. \qed \enddemo \proclaim{Theorem 3.4} Let $A,B$ be self-adjoint operators with $B\geq A\geq 0$. Suppose that $(A+1)^{-1}-(B+1)^{-1}$ is rank one. Then $B=A+\alpha(\varphi, \cdot)\varphi$ with $\varphi\in\Cal H_{-1}(A)$ and $\alpha\in [0, \infty]$ \rom(with $\alpha=\infty$ allowed\rom). \endproclaim \demo{Proof} Write $$(A+1)^{-1}=(B+1)^{-1}+(\eta, \cdot)\eta, \tag 3.2$$ which we can do because $(A+1)^{-1}\geq (B+1)^{-1}$. We claim that $\eta\in\Cal H_{+1}(A)$ with $(\eta, (A+1)\eta)\leq 1$; see Lemma 3.5 below. Define $\varphi=(A+1)\eta$ so (3.2) becomes $$(B+1)^{-1}=(A+1)^{-1}-((A+1)^{-1}\varphi, \cdot)(A+1)^{-1}\varphi,$$ which is just (1.4) if $$\frac{\alpha}{1+\alpha(\varphi, (A+1)^{-1}\varphi)} = 1$$ or $$\alpha = \frac{1}{1-(\eta, (A+1)\eta)} \tag 3.3$$ where $(\eta, (A+1)\eta)=1$ corresponds to $\alpha=\infty$. (1.4) at $z=-1$ implies the general relation for all $z$. \qed \enddemo \proclaim{Lemma 3.5} Let $A\geq 0$ be self-adjoint. Suppose $\eta\in\Cal H$ with $(\eta, \cdot)\eta\leq (A+1)^{-1}$. Then $\eta\in\Cal H_{+1}(A)$ with $(\eta, (A+1)\eta)\leq 1$. \endproclaim \demo{Proof} Let $E_k$ be the spectral projection $E_{[0,k]}(A)$. Let $\varphi_{k}=(A+1)E_{k}\eta$. Then, by hypothesis, $$|(\eta, \varphi_{k})|^{2}\leq (\varphi_{k}, (A+1)^{-1}\varphi_{k}). \tag 3.4$$ (3.4) is equivalent to $$(\eta, E_{k}(A+1)\eta)^{2}\leq (\eta, E_{k}(A+1)\eta)$$ or $$(\eta, E_{k}(A+1)\eta)\leq 1.$$ Taking $k\to\infty$, we see $\eta\in\Cal H_{+1}(A)$ and $(\eta, (A+1)\eta)\leq 1$. \qed \enddemo \remark{Remark} It may seem puzzling that the $\alpha$ in (3.3) obeys $1<\alpha\leq\infty$. How about $B=A+\alpha(\varphi, \cdot)\varphi$ with $\alpha <1$? The resolution is that until we normalize $\varphi$ in some way, the scale of $\alpha$ is irrelevant. If we demand $\widetilde{\varphi}$ obey $(\widetilde{\varphi}, (A+1)^{-1} \widetilde{\varphi})=1$, then we take $\widetilde{\varphi}=\varphi/ (\eta, (A+1)\eta)^{1/2}$ and $\alpha(\varphi, \cdot)\varphi=\widetilde {\alpha}(\widetilde{\varphi}, \cdot)\widetilde{\varphi}$ where now $$\widetilde{\alpha} =\frac{(\eta, (A+1)\eta)}{1-(\eta, (A+1)\eta)}.$$ As $(\eta, (A+1)\eta)$ runs from $0$ to $1$, $\widetilde{\alpha}$ runs from $0$ to infinity. \endremark As an application of Lemma 3.5, we return to the construction of \S2: \proclaim{Theorem 3.6} Suppose $A\geq 0$, $\varphi\in\Cal H_{-2}(A)$ but $\varphi\notin\Cal H_{-1}(A)$, and that $\widetilde A_{\beta}$ is the operator of Theorem \rom{2.1}. Then \roster \item"\rom{(i)}" $\Cal H_{+1}(\widetilde A_{\beta})\supset \Cal H_{+1}(A)$ \item"\rom{(ii)}" $\Cal H_{+1}(\widetilde A_{\beta})\neq \Cal H_{+1}(A)$. \endroster \endproclaim \remark{Remark} We'll see later in \S6 that $\Cal H_{+1}(A)$ has codimension 1 in $\Cal H_{+1}(\widetilde A_{\beta})$. \endremark \demo{Proof} By Theorem 2.2, $\widetilde A_{\beta}\leq A$ which implies (i). To see (ii), note that by the construction in \S2 for all sufficiently large $c>0$, $$(A_{\beta}+c)^{-1} =(A+c)^{-1}-\sigma(c)((A+c)^{-1}\varphi, \cdot) (A+c)^{-1}\varphi$$ with $\sigma(c)<0$. Thus by Lemma 3.5, $(A+c)^{-1}\varphi\in\Cal H_{+1}(\widetilde A_{\beta})$. Since $\varphi\notin\Cal H_{-1}(A)$, we have that $(A+c)^{-1}\varphi\notin\Cal H_{+1}(A)$. \qed \enddemo \bigpagebreak \flushpar {\bf \S4. Relation to Infinite Coupling} \medpagebreak Suppose $B=A+\alpha(\varphi, \cdot)\varphi$ with $\varphi\in\Cal H_{- 1}(A)$. If $\alpha<\infty$, then $\Cal H_{+1}(B)=\Cal H_{+1}(A)$ and $A=B-\alpha(\varphi, \cdot)\varphi$ so $A$ can be recovered from $B$ by the $\Cal H_{-1}$ construction. Our goal here is to show that when $\alpha=\infty$, $A$ can be recovered from $B$ by the $\Cal H_{-2}(B)$ construction of \S2, and vice-versa that the $A\to\widetilde A_{\beta}$ construction can be undone with infinite coupling. Recall ([5]) if $\varphi\in\Cal H_{-1}(A)$ but $\varphi\notin\Cal H$ and $A_{\infty}=A+\infty(\varphi, \cdot)\varphi$, then there exists a natural $\eta\in\Cal H_{-2}(A_{\infty})$ which obeys $$(A_{\infty}-z)^{-1}\eta = F(z)^{-1}(A-z)^{-1}\varphi \tag 4.1$$ with $F$ given by (1.2b). \proclaim{Proposition 4.1} Supose $A\geq 0$, $\varphi\in\Cal H_{-1} (A)$ but $\varphi\notin\Cal H$, and $\eta$ is given by \rom{(4.1)}. Then $\eta\notin\Cal H_{-1}(A_{\infty})$. \endproclaim \demo{Proof} $\eta\in\Cal H_{-1}(A_{\infty})$ if and only if $\operatornamewithlimits{\lim}\limits_{c\to\infty}\big(\eta, \frac {c}{A_{\infty}+c}\,\frac{1}{A_{\infty}+1}\,\eta\big)$ is finite. But by (4.1) $$\biggl(\eta, \frac{c}{A_{\infty}+c}\ \frac{1}{A_{\infty}+1}\, \eta\biggr)=\frac{1}{F(-1)F(-c)}\,\biggl(\varphi, \frac{c}{A+c}\ \frac{1}{A+1}\,\varphi\biggr).$$ The expectation on the right side of this equation has a non-zero limit as $c\to\infty$ since $\varphi\in\Cal H_{-1}(A)$. But $F(-c)\to 0$ as $c\to\infty$ so the limit is infinity; that is, $\eta\notin\Cal H_{-1} (A_{\infty})$. \qed \enddemo \proclaim{Theorem 4.2} Suppose $A\geq 0$ and $\varphi\in\Cal H_{-1} (A)$ but $\varphi\notin\Cal H$. Let $B\equiv A_{\infty}=A+\alpha (\varphi, \cdot)\varphi$. Then for some $\beta$ and the perturbation $\eta$, $\widetilde B_{\beta}=A$; that is, $A$ can be recovered from $B$ by the construction of \rom{\S2}. \endproclaim \demo{Proof} By (1.4b) in the limit $$(B-z)^{-1}=(A-z)^{-1}-F(z)^{-1}((A-\bar z)^{-1}\varphi,\cdot)(A-z)^{-1} \varphi.$$ By (4.1) $$(A-z)^{-1}=(B-z)^{-1}+F(z)((B-\bar z)^{-1}\eta, \cdot)(B-z)^{-1} \eta$$ which shows that $(A+1)^{-1}$ is a $(\widetilde B_{\beta}+1)^{-1}$. \qed \enddemo \remark{Remark} By \S2, the coefficient in front of $((B-\bar z)^{-1} \varphi, \cdot)(B-z)^{-1}\varphi$ should be $(\beta +G(z))^{-1}$ where $G(z)=(\eta, [(A_{\infty}-z)^{-1}-(A_{\infty}+1)^{-1}]\eta)$. The resulting relation of $\text{Im}\,F(z)^{-1}$ and $\text{Im}\,(G(z))$ is exactly what was found in [5]. \endremark \bigpagebreak \flushpar {\bf \S5. Limits} \medpagebreak We've shown in the last two sections that if $(A-z)^{-1}-(B-z)^{-1}$ is rank one (and both are bounded below), then $B$ can be recovered from $A$ via either a $\varphi\in\Cal H_{-1}(A)$ construction with $\alpha\in (-\infty, \infty]$ or else by the $\varphi\in\Cal H_{-2}(A) \backslash \Cal H_{-1}(A)$ construction with $\alpha$ infinitesimal. Thus it should be impossible to define $A+\alpha(\varphi,\cdot)\varphi$ if $\varphi\notin\Cal H_{-2}(A)$. That is what we'll prove in this section. \proclaim{Theorem 5.1} Let $A\geq 0$ and $\varphi\in\Cal H_{-\infty} (A)$. Let $\varphi_{n}=E_{[0, n]}(A)\varphi$ and $$A_{n}=A+\alpha_{n}(\varphi_{n}, \cdot)\varphi_{n}.$$ Then: \roster \item"\rom{(i)}" If $\varphi\notin\Cal H_{-2}(A)$, then for any choice of $\alpha_{n}$, $(A_{n}-z)^{-1}$ converges to $(A-z)^{-1}$ strongly as $n\to\infty$ for any $z\in\Bbb C\backslash\Bbb R$. \item"\rom{(ii)}" If $\varphi\notin\Cal H_{-1}(A)$ and $\alpha_{n}\geq 0$, then for any choice of $\alpha_{n}$ \rom(subject to $\alpha_{n}\geq 0$\rom), $(A_{n}-z)^{-1}$ converges to $(A-z)^{-1}$ strongly as $n\to\infty$ for any $z\in\Bbb C\backslash\Bbb R$. \item"\rom{(iii)}" If $\varphi\notin\Cal H_{-1}(A)$ and $\alpha_{n}\to \alpha_{\infty}\neq 0$, then for any choice of $\alpha_{n}$ \rom(subject to $\alpha_{n}\to\alpha_{\infty}$\rom), $(A_{n}-z)^{-1}$ strongly to $(A-z)^{-1}$ as $n\to\infty$ for any $z\in\Bbb C\backslash\Bbb R$. \endroster \endproclaim \remark{Remarks} 1. Thus to get a non-trivial limit, we either need $\varphi\in\Cal H_{-1}(A)$ or else $\varphi\in\Cal H_{-2}(A)$ and $\alpha_{n}$ negative and infinitesimal. 2. In cases (ii) and (iii), if $\varphi\in\Cal H_{-2}(A)$, our proof shows norm convergence. \endremark \demo{Proof} By general principles [9], weak convergence of resolvents implies strong convergence. Since the $\{(A_{n}-z)^{-1}\}$ are uniformly bounded for fixed $z\in\Bbb C\backslash\Bbb R$, it suffices to prove convergence of $(\psi_{1}, (A_{n}-z)^{-1}\psi_{2})$ for $\psi_{i}\in\Cal H_{\infty}$. By (1.4b), $$(A_{n}-z)^{-1}=(A-z)^{-1} - [\alpha^{-1}_{n}+(\varphi_{n}(A-z)^{-1} \varphi_{n})]^{-1}((A-\bar z)^{-1}\varphi_{n},\cdot)(A-z)^{-1}\varphi_{n}. \tag 5.1$$ Since $(\psi, (A-z)^{-1}\varphi_{n})$ is uniformly bounded if $\psi\in \Cal H_{+\infty}(A)$ (since $\varphi\in\Cal H_{-\infty}(A)$), strong convergence is equivalent to $$|\gamma_{n}|\equiv |\alpha^{-1}_{n}+(\varphi_{n}, (A-z)^{-1} \varphi_{n})|\to\infty.$$ Now $$\text{Im}\,\gamma_{n}=(\text{Im}\,z)\|(A-z)^{-1}\varphi_{n}\|^{2}$$ goes to infinity as $n\to\infty$ if $\varphi\notin\Cal H_{-2}$, so (i) is proven. Suppose now $\varphi\in\Cal H_{-2}$. Since $$\text{Re}\,\gamma_{n}=\alpha^{-1}_{n}+(\varphi_{n}, A[(A-\text{Re} \,z)^{2}+(\text{Im}\,z)^{2}]^{-1}\varphi_{n})-\text{Re}\,z \|(A-z)^{- 1}\varphi_{n}\|^{2},$$ we see that if $\alpha_{n}>0$ and $\varphi_{n}\notin\Cal H_{-1}(A)$, then $\text{Re}\,\gamma_{n}\to\infty$, and similarly if $\alpha^{- 1}_{n}$ has a finite limit $\text{Re}\,\gamma_{n}\to\infty$. \qed \enddemo \remark{Remark} Friedman [4] has shown that if $V_n$ are functions on $\Bbb R^{\nu}$ with $\text{supp}\,V_{n}\subset \{x\mid |x| \nu$. Thus $\delta_{0}\in\Cal H_{-1}$ only if $\nu <2$ and $\delta_{0}\in\Cal H_{-2}$ if and only if $\nu <4$. We can therefore regard Theorem 5.1 as a kind of analog of Friedman's results. \endremark \bigpagebreak \flushpar {\bf \S6. Self-Adjoint Extensions} \medpagebreak The punchline of this section is that rank one perturbations of $A\geq 0$ is really the same as the theory of self-adjoint extensions of deficiency indices $(1, 1)$ of a positive operator. From this point of view, the $\alpha =\infty$ operator found by Gesztesy-Simon [5] is exactly the Friedrich's extension. Let $A\geq 0$ and $\varphi\in\Cal H_{-2}(A)$. Whatever $A_{\alpha}= A+\alpha(\varphi, \cdot)\varphi$ is to mean $A_{\alpha}\psi$ should equal $A\psi$ if $(\varphi, \psi)=0$. Thus, define $$D_{\varphi}=\{\psi\in D(A)\mid (\varphi, \psi)=0\}.$$ Since $\varphi\in\Cal H_{-2}(A)$, $(\varphi, \psi)$ is defined for $\psi\in D(A)=\Cal H_{+2}(A)$. \proclaim{Lemma} Let $A_{0}=A\restriction D_{\varphi}$ with domain $D_\varphi$. Then $A_{0}$ has deficiency indices $(1, 1)$. \endproclaim \demo{Proof} It suffices to prove that $\text{Ran}(A_{0}+1)$ has codimension 1. But by definition, $\psi\in D_\varphi$ if and only if $(A+1)\psi$ is orthogonal to $(A+1)^{-1}\varphi$; that is, $\text{Ran}(A_{0}+1)=\{(A+1)^{-1}\varphi\}^{\perp}$ has codimension 1. \qed \enddemo \medpagebreak The rank one perturbations are thus the self-adjoint extensions of $A_0$. Deficiency one extension of semibounded operators (and generally semibounded extensions of semibounded operators) have been studied extensively [3,8,13,6,2]. The result of this theory is that these are parametrized by a single parameter $\gamma$ which runs in $(-\infty, \infty]$ with $+\infty$ allowed. They are best described in terms of quadratic forms. The operator $A^{(\infty)}$ is the Friedrich's extension and has form domain $Q(A^{(\infty)})$. There is a vector $\xi$ defined by $(A_{0}+1)^{*}\xi=0$ and for $\gamma\neq\infty$, $$Q(A^{(\gamma)})=Q(A^{(\infty)})\dotplus\{\lambda\xi\}_{\lambda\in\Bbb C},$$ where $\dotplus$ means disjoint sums and $$((\psi+\lambda\xi), A^{(\gamma)}(\psi+\lambda\xi))=(\psi, A^{(\infty)}\psi)+\lambda^{2}\gamma.$$ $\xi$ is easily seen to be $(A+1)^{-1}\varphi$. The original operator $A$ is some $A^{(\gamma_0)}$. If $A=A^{(\gamma_0)}$ with $\gamma_{0}\neq\infty$, then the $A^{(\gamma)}$ are precisely $\{A+c(\gamma-\gamma_{0})(\varphi, \cdot)\varphi\}$ for a suitable constant $c$ ($=(\varphi, (A+1)^{-1}\varphi)$). The $\gamma =\infty$ operator is exactly a Friedrich's extension. If $\gamma_{0}=\infty$, we see in this situation where the other $A^{(\gamma)}$'s are gotten by the construction in \S2. \bigpagebreak \flushpar {\bf \S7. Examples} \medpagebreak \example{Example 1} Take $A=-\Delta$ on $L^{2}(\Bbb R^{\nu})$. We want to see what $\varphi$ can be used for rank one perturbations defined at a single point $0$. Since $\varphi$ is supported at $0$, $\varphi\in\Cal H_{-\infty}(A)$ means $\varphi$ is a distribution, so its Fourier transform is a polynomial $P$ in $p$. For $\varphi\in\Cal H_{-1}(A)$, we need $$\int \frac{d^{\nu}p\, |P(p)|^{2}}{(p^{2}+1)} < \infty. \tag 7.1$$ This can only happen if $\nu =1$ and $P$ has degree $0$, that is, $\varphi=\delta(x)$. For $\varphi$ to be in $\Cal H_{-2}(A)$, we need the analog of (7.1) with $(p^{2}+1)$ replaced by $(p^{2}+1)^{2}$. This allows $P$ of degree $0$ if $\nu =2,3$ and degree $1$ if $\nu =1$. Thus, the rank one theory works exactly for $\delta(x)$ in $\nu =1,2,3$ and $\delta'(x)$ in $\nu =1$. The $\Cal H_{-2}(A)$ construction exactly corresponds to point interactions as discussed extensively (see [1] and references therein.) Of course, our construction specialized to this case is just the standard one for point interactions; so our construction in \S2 can be viewed as an abstraction of that method. One thing one can look at is undoing the point interaction in dimension 2 and 3. For concreteness, take $\nu =3$. Then $\Cal H_{+1} (\widetilde A_{\beta})$ is strictly bigger than $\Cal H_{+1}(A)$. The extra functions have a Coulomb singularity at $x=0$; that is, $\psi\in\Cal H_{+1}(\widetilde A_{\beta})$ has the form $$\psi(x)= ce^{-\mu|x|} |x|^{-1}+\widetilde{\psi}$$ with $\widetilde{\psi}\in\Cal H_{+1}(-\Delta)$. $\mu$ is a convenient parameter; $c$ is independent of $\mu$. One can think of $c$ is formally given by $\lim\limits_{|x|\to 0}\,|x|\psi(x)$. Since $\psi$ is not bounded, we can't use that definition but can use $$c(\psi) =\lim\limits_{r\to 0}\, r\frac{3}{4\pi r^{3}} \int\limits_{|x|\leq r} \psi(x)\, d^{3}x.$$ So $c$ defines a vector $\varphi\in\Cal H_{-1}(\widetilde A_{\beta})$ and the various $\widetilde A_{\beta}$'s are just $\widetilde A_{\beta_0}+\alpha(\varphi, \cdot)\varphi$ for $\alpha\in (-\infty, \infty)$. $\alpha =\infty$ recovers the original Laplacian. \endexample \example{Example 2} Let $A$ be $-\frac{d^2}{dx^2}$ on $L^{2}(0, \infty)$ with Neumann boundary condition at zero. Let $\varphi(x) =\delta(x)\in\Cal H_{-1}(A)$. Then $A+\alpha(\varphi,\cdot)\varphi$ precisely corresponds to the boundary conditions $$\sin(\theta)u'(0)+\cos(\theta)u(0)=0$$ where $\alpha =-\cot(\theta)$. $\alpha =\infty$ corresponds to Dirichlet boundary condition. The corresponding $\eta$ as discussed in [5] is just $\delta'(x)$; that is, $\delta'\in\Cal H_{-2} (A_{\infty})$. The construction in \S2 tells us how to reconstruct $A_\theta$ from $A_\infty$. \endexample \vskip 0.3in \Refs \widestnumber\key{13} \vskip 0.1in \ref\key 1 \by S.~Albeverio, F.~Gesztesy, R.~H\o egh-Krohn, and H.~Holden \book Solvable Models in Quantum Mechanics \publ Springer \yr 1988 \endref \ref\key 2 \by A.~Alonso and B.~Simon \paper The Birman-Krein-Vishik theory of self-adjoint extensions of semibounded operators \jour J.~Operator Theory \vol 4 \yr 1980 \pages 251--270 \endref \ref\key 3 \by M.S.~Birman \paper On the self-adjoint extensions of positive definite operators \lang Russian \jour Math.~Sb. \vol 38 \yr 1956 \pages 431--450 \endref \ref\key 4 \by C.N.~Friedman \paper Perturbations of the Schr\"odinger equation by potentials with small support \jour J.~Funct. Anal. \vol 10 \yr 1972 \pages 346--360 \endref \ref\key 5 \by F.~Gesztesy and B.~Simon \paper Rank one perturbations at infinite coupling \jour J.~Funct.~Anal. \vol 128 \yr 1995 \pages 245--252 \endref \ref\key 6 \by G.~Grubb \paper A characterization of the non-local boundary value problems associated with an elliptic operator \jour Ann.~Scuola Norm.~Sup.~Pisa \vol 22 \yr 1968 \pages 425--513 \endref \ref\key 7 \by T.~Kato \book Perturbation Theory for Linear Operators \bookinfo 2nd edition \publ Springer \yr 1980 \endref \ref\key 8 \by M.~Krein \paper The theory of self-adjoint extensions of semibounded Hermitian transformations and its applications, I. \jour Rec.~Math. (Math.~Sb.) \vol 20 \yr 1947 \pages 431--495 \endref \ref\key 9 \by M.~Reed and B.~Simon \book Methods of Modern Mathematical Physics, I.~Functional Analysis \publ Academic Press \yr 1972 \endref \ref\key 10 \bysame \book Methods of Modern Mathematical Physics, II.~Fourier Analysis, Self-Adjointness \publ Academic Press \yr 1975 \endref \ref\key 11 \by B.~Simon \paper Spectral analysis of rank one perturbations and applications \jour Proc.~1993 Vancouver Summer School in Mathematical Physics \toappear \endref \ref\key 12 \by B.~Simon and T.~Wolff \paper Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians \jour Commun.~Pure Appl.~Math. \vol 39 \yr 1986 \pages 75--90 \endref \ref\key 13 \by M.~Vishik \paper On general boundary conditions for elliptic differential equations \jour Trudy Moskov.~Mat. Ob\v s\v c. \lang Russian \vol 1 \yr 1952 \pages 187--246 \endref \endRefs \enddocument