\input amstex \documentstyle{amsppt} \magnification=1200 \baselineskip=15 pt \TagsOnRight \topmatter \title Rank One Perturbations at Infinite Coupling \endtitle \author F. Gesztesy$^1$ and B. Simon$^2$ \endauthor \leftheadtext{F.~Gesztesy and B.~Simon} \thanks $^1$Department of Mathematics, University of Missouri, Columbia, MO 65211. E-mail: mathfg\@mizzou1.\linebreak missouri.edu \endthanks \thanks $^2$Division of Physics, Mathematics and Astronomy, California Institute of Technology, 253-37,\linebreak Pasadena, CA 91125. 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 \endtopmatter \bigpagebreak \document \block \tenrm{\smc{Abstract.}} We discuss rank one perturbations $A_{\alpha}=A+\alpha (\varphi, \cdot)\varphi, \alpha\in\Bbb R, A\geq 0$ self-adjoint. Let $d\mu_{\alpha}(x)$ be the spectral measure defined by $(\varphi, (A_{\alpha}-z)^{-1}\varphi)=\int\, d\mu_{\alpha}(x)/(x-z)$. We prove there is a measure $d\rho_{\infty}$ which is the weak limit of $(1+\alpha^{2})\,d\mu_{\alpha}(x)$ as $\alpha\to\infty$. If $\varphi$ is cyclic for $A$, then $A_\infty$, the strong resolvent limit of $A_\alpha$, is unitarily equivalent to multiplication by $x$ on $L^{2}(\Bbb R, d\rho_{\infty})$. This generalizes results known for boundary condition dependence of Sturm-Liouville operators on half-lines to the abstract rank one case. \endblock \vskip 0.4in \flushpar {\bf \S 1. Introduction} This paper is a contribution to the theory of rank one perturbations which in its natural format involves a self-adjoint operator, $A\geq 0$ in a complex separable Hilbert space $\Cal H$, and a vector, $\varphi\in\Cal H_{-1}(A)$, with $\Cal H_{s}(A)$ the scale of spaces associated to $A$. Then $q_{\varphi}(\psi, \eta)=(\psi, \varphi) (\varphi, \eta)$ defines a quadratic form on $\Cal H_{+1}(A)$ with $q_{\varphi}$ a form-bounded perturbation of $A$ with relative bound zero. Accordingly, $A_{\alpha}\equiv A+\alpha(\varphi, \cdot)\varphi, \alpha\in\Bbb R$ defines a self-adjoint operator with $\Cal H_{s} (A_{\alpha})=\Cal H_{s}(A)$ for $|s|\leq 1$. We will suppose that $\varphi$ is cyclic for $A$ in which case it is easy to see that $\varphi$ is also cyclic for each $A_\alpha$. If $d\mu_\alpha$ is the spectral measure for $\varphi$ associated to $A_\alpha$, then $A_\alpha$ is unitarily equivalent to multiplication by $x$ on $L^{2}(\Bbb R, d\mu_{\alpha})$. Define $$F_{\alpha}(z)=\int\limits_{\Bbb R}\frac{d\mu_{\alpha}(x)}{x-z}$$ where $\varphi\in\Cal H_{-1}(A_{\alpha})$ implies that $$\int\limits_{\Bbb R}\frac{d\mu_{\alpha}(x)}{|x|+1}<\infty$$ so that the integral defining $F$ converges. One has the basic formula (with $F(z)\equiv F_{\alpha=0}(z)$) $$F_{\alpha}(z)=\frac{F(z)}{1+\alpha F(z)}. \tag 1$$ We are interested here in the case $\alpha=\infty$. By the monotone convergence theorem for forms ([3,6]), we have that $\operatornamewithlimits{\text{s-lim}}\limits_{\alpha\to\infty} (A_{\alpha}-z)^{-1}$ exists (the existence also follows from the explicit formula for $(A_{\alpha}-z)^{-1}$, eq.~(6) below) and can be described as follows. Let $$\Cal H_{+1}(A_{\infty})=\{\psi\in\Cal H_{+1}\mid (\varphi, \psi)=0\}$$ and $\Cal H(A_{\infty})=\overline{\Cal H_{+1}(A_{\infty})}$. This is all of $\Cal H$ if $\varphi\notin\Cal H$ and a codimension one subspace if $\varphi\in\Cal H$. Let $A_\infty$ be the self-adjoint operator on $\Cal H(A_{\infty})$ defined by the closed quadratic form $\psi, \eta\mapsto (\psi, A\eta)$ on $\Cal H_{+1}(A_{\infty})$. If $\Cal H(A_{\infty})\neq\Cal H$, extend $(A_{\infty}-z)^{-1}$ to all of $\Cal H$ by setting it zero on $\Cal H(A_{\infty})^{\perp}$. Then $\text{s-lim}(A_{\alpha}-z)^{-1}=(A_{\infty}-z)^{-1}$. By (1), $d\mu_{\alpha}(x)\to 0$ weakly as $\alpha\to\infty$, so we do not have any obvious spectral measure of $A_\infty$. Our main goal here is to prove that $(1+\alpha^{2})\,d\mu_{\alpha}$ does have a weak limit as $\alpha\to\infty$ which is the spectral measure for a vector $\eta\in\Cal H_{-2}(A_{\infty})$. Explicitly, define $$d\rho_{\alpha}(x)=(1+\alpha^{2})\,d\mu_{\alpha}(x). \tag 2$$ Then we will prove that \proclaim{Theorem 1} There exists a vector, $\eta\in\Cal H_{- 2}(A_{\infty})$, cyclic for $A_\infty$ so that if $d\rho_{\infty}(x)$ is the spectral measure for $\eta$ with respect to $A_\infty$, then $$\int\limits_{\Bbb R} f(x)\,d\rho_{\alpha}(x)\to \int\limits_{\Bbb R} f(x)\, d\rho_{\infty}(x) \tag 3$$ for all continuous functions, $f$, of compact support. \endproclaim Note that since $\eta\in\Cal H_{-2}(A_{\infty})$, $$\int\limits_{\Bbb R}\frac{d\rho_{\infty}(x)}{(|x|+1)^{2}}<\infty. \tag 4$$ It may be that (4) fails if $(|x|+1)^{-2}$ is replaced by $(|x|+1)^{-1}$. We will see explicit examples in \S 5 where the integral diverges for $(|x|+1)^{-2+\epsilon}$. The proof will show that (3) holds if $f(x)=(|x|+1)^{-\alpha}$ with $\alpha > 2$. There will be examples when it fails if $\alpha = 2$. Another major result we'll prove is that $$d\rho_{\infty}(x)=\lim_{\epsilon\downarrow 0} \pi^{-1} \bigl[\text{Im} ((-F(x+i\epsilon))^{-1})\,dx\bigr].$$ The abstract theory appears in \S 2. We discuss boundary condition dependence of Schr\"o-dinger operators on the half-line in \S 3. In that case, $d\rho_{\alpha}$ is the Weyl spectral measure and $d\rho_{\infty}$ is the Dirichlet spectral measure. In \S4, we consider the case when $A$ is bounded. In \S 5, we discuss a further example. \bigpagebreak \flushpar {\bf \S 2. The Main Results} We begin by recalling some of the standard formulae for rank one perturbations [7]: $$\gather F_{\alpha}(z)=F(z)/[1+\alpha F(z)], \\ (A_{\alpha}-z)^{-1}\varphi=(1+\alpha F(z))^{-1} (A-z)^{-1} \varphi, \tag 5 \\ (A_{\alpha}-z)^{-1}=(A-z)^{-1} - \frac{\alpha}{1+\alpha F(z)}\, ((A-\bar z)^{-1}\varphi, \cdot) (A-z)^{-1}\varphi, \tag 6 \\ \text{Tr}[(A-z)^{-1}-(A_{\alpha}-z)^{-1}]=\int\limits^{\infty}_{E_{\alpha}} (\lambda-z)^{-2}\xi_{\alpha}(\lambda)\, d\lambda, \quad E_{\alpha}=\min(0,\inf\,\text{spec}(A_{\alpha})), \endgather$$ where $\xi_\alpha$ is the Krein spectral shift [4] given by $$\xi_{\alpha}(x)=\frac{1}{\pi} \text{ Arg}(1+\alpha F(x+i0)). \tag 7$$ For $\alpha>0$ we have $\text{Arg}(\cdot)\in [0, \pi]$ and hence $0\leq\xi_{\alpha}\leq 1$ in this case. If $\|\varphi\|=\infty$, let $P=0$, and if $\|\varphi\|<\infty$, let $P$ be the projection onto $\{c\varphi\mid c\in\Bbb C\}$. Thus, $\Cal H(A_{\infty})=\text{Ran}(1-P)$. \proclaim{Proposition 2} There exists $\eta\in\Cal H_{-2}(A_{\infty})$ so that for all $z\in\Bbb C$: $$(A_{\infty}-z)^{-1}\eta = \lim_{\alpha\to\infty} \alpha (1-P) (A_{\alpha}-z)^{-1}\varphi. \tag 8$$ If $\varphi$ is cyclic for $A$, then $\eta$ is cyclic for $A_\infty$. \endproclaim \demo{Proof} By (5), the limit on the right side of (8) exists, call it $\psi(z)$, and is given by $$\psi(z)=F(z)^{-1} (1-P)(A-z)^{-1}\varphi. \tag 9$$ We have that $$(A_{\alpha}-z)^{-1}\varphi - (A_{\alpha}-w)^{-1}\varphi = (z-w) (A_{\alpha}-z)^{-1} (A_{\alpha} - w)^{-1} \varphi. \tag 10$$ Multiply by $\alpha$, take $\alpha\to\infty$, and note that if $\|\varphi\|<\infty$, then $P(A_{\infty}-w)^{-1}\varphi=0$. We conclude that $$\psi(z)-\psi(w)=(z-w)(A_{\infty}-z)^{-1}\psi(w)$$ or $$\psi(z)=[1+(z-w)(A_{\infty}-z)^{-1}]\psi(w). \tag 11$$ Note that $\psi(z)\in\Cal H(A_{\infty})$ (because of the $1-P$) so we can define $\eta(z)\equiv (A_{\infty}-z)\psi(z)$ in $\Cal H_{-2}(A_{\infty})$. (11) precisely says that $\eta(z)=\eta(w)$, that is, it is independent of $z$; call it $\eta$. Cyclicity follows from (9) since if $\{(A-z)^{-1}\varphi\}$ is total in $\Cal H$, then clearly $\{(1-P)(A-z)^{-1}\varphi\}$ is total in $(1-P)\Cal H=\Cal H(A_{\infty})$. \qed \enddemo \remark{Remark} In \S 4, we'll prove that when $A$ is bounded, then $\eta=-(1-P)A\varphi$. \endremark \proclaim{Theorem 3} Let $d\rho_\infty$ be the spectral measure for $\eta$. Then $$\int\limits_{\Bbb R}\frac{d\rho_{\infty}(x)}{(x-z)^{2}}= F(z)^{-2}\, \frac{dF}{dz} - \frac{1}{\|\varphi\|^{2}}. \tag 12$$ \endproclaim \demo{Proof} For simplicity, suppose $z$ is real and negative. By definition of $\eta$: \align \int\limits_{\Bbb R}\frac{d\rho_{\infty}(x)}{(x-z)^{2}} &\equiv (\eta, (A_{\infty}-z)^{-2}\eta) \\ &= \bigl(\varphi, (A-z)^{-1}(1-P)(A-z)^{-1}\varphi\bigr)\big/F(z)^{2} \\ &= \biggl[(\varphi,(A-z)^{-2}\varphi)-\frac{1}{\|\varphi\|^{2}}\, \langle\varphi, (A-z)^{-1}\varphi\rangle^{2}\biggr]\big/ F(z)^{2} \endalign since $P=\|\varphi\|^{-2}(\varphi, \cdot)\varphi$. But this is precisely the right side of (12). (12) for general $z$ follows by analyticity. \qed \enddemo Recall that $d\rho_{\alpha}$ is defined by (2). Then \proclaim{Theorem 4} \roster\runinitem"\rom{(i)}" $$\lim\limits_{\alpha\to\infty}\int\limits_{\Bbb R}\frac{d\rho_{\alpha}(x)} {(x-z)^{2}}=\frac{1}{\|\varphi\|^{2}}+\int\limits_{\Bbb R}\frac{d\rho_{\infty} (x)}{(x-z)^{2}}.$$ \item"\rom{(ii)}" $$\lim\limits_{\alpha\to\infty} \int\limits_{\Bbb R} \frac{d\rho_{\alpha}(x)}{(x-z)^{3}}=\int\limits_{\Bbb R} \frac{d\rho_{\infty}(x)}{(x-z)^{3}}.$$ \item"\rom{(iii)}" For any continuous $f$ of compact support $$\lim_{\alpha\to\infty} \int\limits_{\Bbb R} f(x)\,d\rho_{\alpha}(x)= \int\limits_{\Bbb R} f(x)\, d\rho_{\infty}(x).$$ \endroster \endproclaim \demo{Proof} (ii) implies (iii) by a Stone-Weierstrass type argument. (i) implies (ii) by using the fact that both sides are analytic in $z$ on $\Bbb C\backslash\Bbb R$ so their derivatives in $z$ converge. To prove (i), use Theorem 3 and the calculation \align \int\limits_{\Bbb R}\frac{d\rho_{\alpha}(x)}{(x-z)^{2}} &= (1+\alpha^{2}) (\varphi, (A_{\alpha}-z)^{-2}\varphi) \\ &= \frac{(1+\alpha^{2})}{(1+\alpha F)^{2}}\, (\varphi, (A-z)^{- 2}\varphi) \tag 13 \\ &= \frac{1+\alpha^{2}}{(1+\alpha F)^{2}} \, \frac {dF}{dz}. \endalign (13) follows from (5). \qed \enddemo \proclaim{Theorem 5} $$d\rho_{\infty}(x)=\pi^{-1} \lim_{\epsilon\downarrow 0} \text{\rom{ Im}} \bigg[-\frac{1}{F(x+i\epsilon)}\biggr] dx.$$ \endproclaim \demo{Proof} We start with (12) and integrate, noting that $F'/F^{2}=\frac{d}{dz} (-1/F)$ to get $$\int\limits_{\Bbb R} d\rho_{\infty}(x) \biggl(\frac{1}{x-z} -\frac{1}{x+1}\biggr) =-\frac{1}{F(z)} + \frac{1}{F(-1)} - (z+1)\, \frac{1}{\|\varphi\|^2}.$$ The theorem then follows by the standard relations between a measure and the boundary values of its Borel transform. \qed \enddemo \bigpagebreak \flushpar {\bf \S 3. Variation of Boundary Condition} As an example of the general theory, we consider the case of boundary conditions variation for Schr\"odinger operators on $L^{2}(0, \infty)$. The formulae that result are well-known (see, e.g., [1,2,5,8]). The point is that they fit into a more general framework. Let $V$ be continuous and bounded below on $[0, \infty)$. Let $H_\theta$ be the operator on $L^{2}([0, \infty), dx)$ formally given by $-\frac{d^2} {dx^2}+V(x)$ with $u(0)\cos\theta+u'(\theta)\sin\theta=0$ boundary conditions. One defines the Weyl $m$-function, $m_{\theta}(z)$, and Weyl spectral measure, $d\rho_{\theta}(x)$, so that for $\theta\neq 0$: $$m_{\theta}(z)=\cot(\theta)+\int\limits_{\Bbb R} \frac{d\rho_{\theta}(x)}{x-z} \tag 14$$ and $d\rho_{\theta}\to d\rho_{\theta=0}$ as $\theta\downarrow 0$. Moreover, $$m_{\theta=0}(z)=-1\big/m_{\theta=\pi/2}(z). \tag 15$$ For $\theta\neq 0$, the Green's function, $G_{\theta}(0, 0; z)$ is related to $m_{\theta}(z)$ by $$G_{\theta}(0, 0; z)=\sin^{2}(\theta) [-\cot\theta +m_{\theta}(z)]. \tag 16$$ This fits into the general framework by taking $A=H_{\theta=\pi/2}$ and $\varphi=\delta_{0}$, the delta function at 0. Then for $\theta\neq 0$, $$H_{\theta}=A-\cot(\theta)(\varphi, \cdot)\varphi$$ and $F_{-\cot(\theta)}(z)=G_{\theta}(0, 0; z)$. By (14) and (16), $d\rho_{\theta}$ is just $(1+\alpha^{2})\,d\mu_{\alpha}$ where $\alpha=-\cot\theta$ and $\lim\limits_{\theta\to 0} d\rho_{\theta}=d\rho_{0}$ is just what we found in the last section. (15) is just Theorem 5. We want to identify the vector $\eta$. Let $\psi_{+}(x, z)$ be the solution of $(-\frac{d^2}{dx^2}+V(x)-z)\psi=0$ which is $L^2$ at infinity normalized any way that is convenient. Then from the Wronskian formula for $G_{\theta=\pi/2}$, we get $$((A-z)^{-1}\varphi)(x)=\frac{\psi_{+}(x, z)}{\psi'_{+}(0, z)}$$ and $$F(z)=-\psi_{+}(0, z)\big/ \psi'_{+}(0, z).$$ It follows that $$F(z)^{-1}(A-z)^{-1}\varphi = \psi_{+}(x, z)\big/\psi_{+}(0, z)$$ which, by the Wronskian formula for $G_{\theta=0}$, is just $$(A_{\infty}-z)^{-1}\delta'(x),$$ that is, $\eta$ is $\delta'$ (note that $P=0$ in this case) and $d\rho_{\infty}$ is the spectral function for the vector $\delta'$. We note that it is well known that $\int\limits^{\mu}_{0}d\rho_{\infty}(x) \sim C\mu^{3/2}$ as $\mu\to\infty$ so that $\int\limits^{\infty}_{0} \frac{d\rho_{\infty}(\lambda)}{(1+|\lambda|)^{k}}<\infty$ if and only if $k>\frac32$. In particular, $\eta\notin\Cal H_{-1}(A_{\infty})$. \bigpagebreak \flushpar {\bf \S4. Bounded Operators} One gets insight into the general theory by considering the case where $A$ is bounded. Since $\|\varphi\|$ is then finite, we'll suppose $\|\varphi\|=1$. We'll also get a better understanding of the $\frac{1}{\|\varphi\|^2}$ term in Theorem 4(i). We first note: \proclaim{Theorem 6} If $A$ is bounded and $\|\varphi\|=1$, then $\eta=-(1-P)A\varphi$. \endproclaim \demo{Proof} If $A$ is bounded, then $A_\infty$ is just $(1-P)A(1- P)$. Thus \align \eta &= F(z)^{-1}(1-P)(A-z)(1-P)(A-z)^{-1}\varphi \\ &= F(z)^{-1}(1-P)(A-z)(A-z)^{-1}\varphi - F(z)^{-1} [(1-P)(A- z)\varphi] F(z). \endalign The first term is zero since $(1-P)\varphi=0$. The second is $-(1- P)A\varphi$ since $(1-P)z\varphi=0$. \qed \enddemo Since $\|\varphi\|=1$, $(\varphi, \cdot)\varphi$ is just a projection, $P$. Instead of $A+\alpha P$, look at $$P+\alpha^{-1}A=B_{\alpha}.$$ $P$ has an isolated simple eigenvalue at 1 with eigenvector $\varphi$. Thus by regular perturbation theory [3], $B_\alpha$ has the eigenvalue at $1+(\varphi, A\varphi)\alpha^{-1}+O(\alpha^{-2})$ with eigenvector $$\psi_{\alpha}=\varphi+\alpha^{-1}(1-P)A\varphi+O(\alpha^{- 2})=\varphi+\alpha^{-1}\eta+O(\alpha^{-2}).$$ The first order term is standard perturbation theory where the reduced resolvent $(H_{0}-E)^{-1}(1-P)$ is just $-(1-P)$ since $H_0$ is $P$ is 0 on $\text{Ran}(1-P)$. Thus, with respect to $A+\alpha P=\alpha B_\alpha$, the measure $(1+\alpha^{2})\,d\mu_{\alpha}$ has a pole of weight $(1+\alpha^{2})$ at $E_{\alpha}=\alpha+(\varphi, A\varphi)+O(\alpha^{-1})$ plus the spectral measure of $\eta$ for the operator $A_\infty$ plus an error of order $\alpha^{-1}$. If $\nu>2$, the pole at $E_\alpha$ makes no asymptotic contribution to $\int\limits_{\Bbb R}\frac{d\rho_{\alpha} (x)}{|x-z|^\nu}$ as $\alpha\to\infty$ but for $\nu=2$, it makes a contribution of $(1+\alpha^{2})/E^{2}_{\alpha}\to 1=1/\|\varphi\|^{2}$. \bigpagebreak \flushpar{\bf \S5. A Further Example} Let $0<\gamma<1$. Let $d\mu_{0}(x)=\pi^{-1}|x|^{-\gamma}\sin(\pi\gamma)\, dx$ on $[0,\infty)$. Let $A$ be multiplication by $x$ on $L^{2}([0, \infty), d\mu_{0}(x))$ and $\varphi\equiv 1$. Then $\int\limits^{\infty}_{0} \frac{d\mu_{0}(x)}{|x|+1}<\infty$ so $\varphi\in\Cal H_{- 1}(A_{\infty})$. $$F(z)=\int\limits^{\infty}_{0}\frac{d\mu_{0}(x)}{x-z}=(-z)^{-\gamma}$$ (the easiest way to see this is to compute the imaginary part of $(- z)^{-\gamma}$ for $z=x+i\epsilon$ with $\epsilon\to 0$). Then, by Theorem 5, $$d\rho_{\infty}(x)=\pi^{-1}|x|^{\gamma}\sin(\pi\gamma)\,dx.$$ It follows that $\int\limits^{\infty}_{0} d\rho_{\infty}(x)/(|x|+1)^{k}<\infty$ only if $k>1+\gamma$. Thus, we cannot conclude in general that $\int\limits^{\infty}_{0} d\rho_{\infty}(x)/(|x|+1)^{k}<\infty$ for any $k<2$. \vskip 0.4in \subhead Acknowledgment \endsubhead F.G.~is indebted to the Department of Mathematics at Caltech for its hospitality and support during the summer of 1993 where some of this work was done. \vskip 0.5in \Refs \endRefs \item{[1]} E.A.~Coddington and N.~Levinson, {\it Theory of Ordinary Differential Equations}, Krieger, Malabar, 1985. \item{[2]} E.~Hille, {\it Lectures on Ordinary Differential Equations}, Addison-Wesley, New York, 1969. \item{[3]} T.~Kato, {\it Perturbation Theory for Linear Operators}, 2nd ed., Springer, Berlin, 1980. \item{[4]} M.G.~Krein, {\it Perturbation determinants and a formula for the traces of unitary and self-adjoint operators}, Sov.~Math.~Dokl. {\bf 3} (1962), 707--710. \item{[5]} B.M.~Levitan and I.S.~Sargsjan, {\it Introduction to Spectral Theory}, Amer.~Math.~Soc. Transl. 39, Providence, R.I., 1975. \item{[6]} B.~Simon, {\it A canonical decomposition for quadratic forms with applications to monotone convergence theorems}, J.~Funct.~Anal. {\bf 28} (1978), 377--385. \item{[7]} B.~Simon, {\it Spectral analysis of rank one perturbations and applications}, Lecture given at the 1993 Vancouver Summer School, preprint. \item{[8]} E.C.~Titchmarsh, {\it Eigenfunction Expansions}, 2nd ed., Oxford University Press, Oxford, 1962. \enddocument