Content-Type: multipart/mixed; boundary="-------------0102021104446" This is a multi-part message in MIME format. ---------------0102021104446 Content-Type: text/plain; name="01-52.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-52.keywords" Singular perturbations ---------------0102021104446 Content-Type: application/x-tex; name="koval.cls" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="koval.cls" %------------------------------------------------------------------------------ % Beginning of koval.cls %------------------------------------------------------------------------------ % \NeedsTeXFormat{LaTeX2e} \ProvidesClass{koval} [2000/09/01 Class for Kovalevski Conference] \DeclareOption*{\PassOptionsToClass{\CurrentOption}{amsproc}} \ProcessOptions \LoadClass[a4paper,reqno]{amsproc} \def\@maketitle{% \normalfont\normalsize \let\@makefnmark\relax \let\@thefnmark\relax \ifx\@empty\@subjclass\else \@footnotetext{\@setsubjclass}\fi \ifx\@empty\@keywords\else \@footnotetext{\@setkeywords}\fi \ifx\@empty\thankses\else \@footnotetext{% \def\par{\let\par\@par}\@setthanks}\fi \@mkboth{\@nx\shortauthors}{\@nx\shorttitle}% \global\topskip3pc\relax % 5pc to base of first title line \@settitle \ifx\@empty\authors \else \@setauthors \fi \ifx\@empty\@dedicatory \else \baselineskip26\p@ \vtop{\centering{\footnotesize\itshape\@dedicatory\@@par}% \global\dimen@i\prevdepth}\prevdepth\dimen@i \fi \@setabstract \normalsize \if@titlepage \newpage \else \dimen@34\p@ \advance\dimen@-\baselineskip \vskip\dimen@\relax \fi } % end \@maketitle %\newtheorem{theorem}{Theorem}[section] \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} %\numberwithin{equation}{section} \setlength{\hoffset}{-1truecm} % set margin \setlength{\textwidth}{14truecm} \setlength{\textheight}{20truecm} \endinput %------------------------------------------------------------------------------ % End of koval.cls %------------------------------------------------------------------------------ ---------------0102021104446 Content-Type: application/x-tex; name="koval.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="koval.tex" % Kovalevski conference, Stockholm 2000 % Use AMSLaTeX2e. % We have defined document class koval.cls which is a % modification of amsproc.cls % This file can be foubd at the conference web-page % http://www.matematik.su.se/events/koval-conf % % Incoming papers in LaTeX formats can be email to % pak@matematik.su.se % \documentclass[12pt]{koval} \usepackage{eucal} % Begin user definitions \def\D{\widehat D(A)} \def\DC{\widehat D(C)} \def\DB{{\mathcal D}(C)} \def\SC{{\mathcal D}'(\RE^n)} \def\SCT{\widetilde{\mathcal D}'(\RE^n)} \def\E{{\mathcal E}'(\RE^n)} \def\B{\mathcal B} \def\L{\mathsf L} \def\H{\mathcal H} \def\X{\mathcal X} \def\Y{\mathcal Y} \def\G{\mathcal G} \def\F{\mathcal F} \def\RE{\mathbb R} \def\C{{\mathbb C}} \def\LD{L^2(\RE^n)} \def\phireg{\phi_{\text{\rm reg}}} \def\vp{\varphi} \def\vpreg{\vp_{\text{\rm reg}}} \def\-{\,\text{\rm -}\,} \def\h{\text{\rm h}} \def\wtilde{\widetilde} \def\p{\par\noindent} % End user definitions \begin{document} \title[Boundary conditions for Singular Perturbations]{Boundary Conditions for Singular Perturbations of Self-Adjoint Operators} \author{Andrea Posilicano} \address{Dipartimento di Scienze, Universit\`a dell'Insubria, I-22100 Como, Italy} \email{posilicano@mat.unimi.it} \begin{abstract} Let $A:D(A)\subseteq\H\to\H$ be an injective self-adjoint operator and let $\tau:D(A)\to\X$, $\X$ a Banach space, be a surjective linear map such that $\|\tau\phi\|_\X\le c\,\|A\phi\|_\H$. Supposing that \text{\rm Range}$\,(\tau')\cap\H' =\left\{0\right\}$, we define a family $A^\tau_\Theta$ of self-adjoint operators which are extensions of the symmetric operator $A_{\left|\left\{\tau=0\right\}\right.}$. Any $\phi$ in the operator domain $D(A^\tau_\Theta)$ is characterized by a sort of boundary conditions on its univocally defined regular component $\phireg$, which belongs to the completion of $D(A)$ w.r.t. the norm $\|A\phi\|_\H$. These boundary conditions are written in terms of the map $\tau$, playing the role of a trace (restriction) operator, as $\tau\phireg=\Theta\, Q_\phi$, the extension parameter $\Theta$ being a self-adjoint operator from $\X'$ to $\X$. The self-adjoint extension is then simply defined by $A^\tau_\Theta\phi:=A\,\phireg$. The case in which $A\phi=T*\phi$ is a convolution operator on $\LD$, $T$ a distribution with compact support, is studied in detail. \end{abstract} \maketitle \section{Introduction} Let $$A:D(A)\subseteq \H\to\H$$ be a self-adjoint operator on the complex Hilbert space $\H$. As usual $D(A)$ inherits a Hilbert space structure by introducing the scalar product leading to the graph norm $$ \|\phi\|_A^2:=\langle\phi,\phi\rangle_\H+\langle A\phi,A\phi\rangle_\H\,. $$ Considering then a linear operator $$ \tau:D(A)\to\X \,,\qquad \tau\in \B(D(A),\X)\ , $$ $\X$ a complex Banach space, we are interested in describing the self-adjoint extensions of the symmetric operator $A_{\left|\{\tau=0\}\right.}$. In tipical situations $A$ is a (pseudo-)differential operator on $\LD$ and $\tau$ is a trace (restriction) operator along some null subset $F\subset\RE^n$ (see e.g. \cite{[AFHKL]}-\cite{[AK]}, \cite{[Br]}, \cite{[C]}, \cite{[KKO]}-\cite {[P]}, \cite {[Sh]}, \cite{[T]} and references therein). \par Denoting the resolvent set of $A$ by $\rho(A)$, we define $R(z)\in \B(\H,D(A))$, $z\in\rho(A)$, by $$ R(z):=(-A+z)^{-1} $$ and we then introduce, for any $z\in\rho(A)$, the operators $\breve G(z)\in \B(\H,\X)$ and $G(z)\in\wtilde \B(\X',\H)$ by $$ \breve G(z):=\tau\cdot R(z)\,,\qquad G(z):=C_\H^{-1}\cdot\breve G(z^*)' \ ,\eqno(1) $$ (we refer to section 2 below for definitions and notations). As an immediate consequence of the first resolvent identity for $R(z)$ we have (see [{\bf 18}, lemma 2.1]) $$ (z-w)\, R(w)\cdot G(z)=G(w)-G(z)\eqno(2) $$ and so $$ \forall\,w,z\in\rho(A),\qquad \text{\rm Range}\,(\,G(w)-G(z)\,)\subseteq D(A)\,.\eqno(3) $$ In [{\bf 18}, thm. 2.1], by means of a Kre\u\i n-like formula, and under the hypotheses $$\tau\,\text{ is surjective}\eqno (\h 1) $$ $$\text{\rm Range}\,(\tau')\cap\H' =\left\{0\right\}\eqno (\h2)$$ (the hypothesis (h1) could be weakened, see \cite{[P]}, but here we prefer to use a simpler framework), we constructed a family $A^\tau_\Theta$ of self-adjoint extension of $A_{\left|\left\{\tau=0\right\} \right.}$ by giving its resolvent family. The advantage of the formula given in \cite{[P]} over other approaches (see e.g. \cite{[S]}, \cite{[DM1]}-\cite{[GMT]} and references therein) is its relative simplicity, being expressed directly in terms of the map $\tau$; moreover the domain of definition of $A^\tau_\Theta$ can be described, interpreting the map $\tau$ as a trace (restriction) operator, in terms of a sort of boundary conditions (see [{\bf 18}, remark 2.10]). In the case $0\notin\sigma(A)$, $\sigma(A)$ denoting the spectrum of $A$, this description becomes particularly expressive since $A^\tau_\Theta\phi$ can be simply defined by the original operator applied to the regular component of $\phi$. Such a regular component $\phi_0\in D(A)$ is univocally determined by the natural decomposition which enter in the definition of $D(A_\Theta^\tau)$ and it has to satisfy the boundary condition $$\tau\phi_0=\Theta\,Q_\phi\,.$$ More precisely, by (h1), (h2) and by [{\bf 18}, lemma 2.2, thm. 2.1, prop. 2.1, remarks 2.10, 2.12], we have the following \begin{theorem} Let $A:D(A)\subseteq\H\to\H$ be self-adjoint with $0\notin\sigma(A)$, let $\tau:D(A)\to\X$ be continuous and satisfy {\rm (h1)} and {\rm (h2)}. If $\Theta\in\wtilde \L(\X',\X)$ is self-adjoint, $G:=G(0)$ and $$ D(A_\Theta^\tau):=\left\{\phi\in\H\, :\, \phi= \phi_0+GQ_\phi,\, \phi_0\in D(A),\, Q_\phi\in D(\Theta),\, \tau\phi_0=\Theta\,Q_\phi\right\}, $$ then the linear operator $$ A_\Theta^\tau:D(A_\Theta^\tau)\subseteq\H\to\H\,,\qquad A_\Theta^\tau\phi:=A\phi_0 $$ is self-adjoint and coincides with $A$ on the kernel of $\tau$; the decomposition entering in the definition of its domain is unique. Its resolvent is given by $$ R^\tau_\Theta(z):=R(z)+G(z)\cdot(\Theta+\Gamma(z))^{-1}\cdot\breve G(z)\,, \qquad z\in W^-_\Theta\cup W_\Theta^+\cup\C\backslash\RE\, , $$ where $$\Gamma(z):=\tau\cdot(G-G(z))$$ and $$ W^\pm_\Theta:=\left\{\,\lambda\in\RE\cap \rho(A)\ :\ \gamma(\pm\Gamma(\lambda))>-\gamma(\pm\Theta)\,\right\}\ . $$ \end{theorem} \begin{remark} By (h1) one has $\X\simeq D(A)/\text{\rm Kernel$\,(\tau)$} \simeq (\text{\rm Kernel$\,(\tau)$})^\perp$ and so $$D(A)\simeq \text{\rm Kernel$\,(\tau)$}\oplus\X\,.$$ This implies that $\X$ inherits a Hilbert space structure and we could then use the identification $\X'\simeq\X$. Even if this gives some advantage (see [{\bf18}, remarks 2.13-2.16, lemma 2.4]) here we prefer to use only the Banach space structure of $\X$. \end{remark} The purpose of the present paper is to extend the above theorem to the case in which $A$ is merely injective. Thus, denoting the pure point spectrum of $A$ by $\sigma_{\text{\rm }pp}(A)$, we require $0\notin\sigma_{\text{\rm }pp}(A)$ but we do not exclude the case $0\in\sigma(A)\backslash\sigma_{\text{\rm }pp}(A)$; this is a tipical situation when $A$ is a differential operator on $\LD$. In order to carry out this program we will suppose that the map $\tau$ has a continuos extension to $\D$, the completion of $D(A)$ with respect to the norm $\|A\phi\|_\H$ (note that $\D=D(A)$ when $0\notin\sigma(A)$). Such a further hypothesis allows then to perform the limit $\lim_{\epsilon\to 0}G(i\epsilon)-G(z)$ (see lemma 3); thus an analogue on the above theorem 1 is obtained (see theorem 5). Such an abstract construction is successively specialized to the case in which $A\phi=T*\phi$ is a convolution operator on $\LD$, where $T$ is a distribution with compact support (so that this comprises the case of differential-difference operators). In this situation the results obtained in theorem 5 can be made more appealing (see theorem 11). The case in which $A=\Delta:H^2(\RE^n)\to\LD$ and $\tau$ is the trace (restriction) operator along a $d$-set with a compact closure of zero Lebesgue measure, $00$ such that for any $\zeta\in\RE^n$ we can find a point $\xi\in\RE^n$ such that $$ |\zeta-\xi|\le k\log(1+|\zeta|)\,, $$ $$ |\F T(\xi)|\ge (k+|\xi|)^{-k}\,. $$ By [{\bf 5}, thm. 1] we know that $T*:\SC\to\SC$ is surjective if and only if $\F T$ is slowly decreasing. This is certainly true when $C$ is a differential operator, i.e. when $\F T$ is a polynomial. The hypotheses we made on $\F T$ permit us to state the following \begin{lemma} Given $T$ as above, one has the identification $$\DC\,\simeq\,\DB/ \sim\,, $$ where $$ \vp_1\sim\vp_2\quad\iff\quad T*\vp_1=T*\vp_2\,.$$ This identification is given by the isometric maps which to the equivalence class of Cauchy sequences $[\left\{\phi_n\right\}_1^\infty]\in \DC$ associates the equivalence class of distributions $[\vp]\in\DB/ \sim$ such that $$ L^2\-\lim_{n\uparrow\infty}\,T*\phi_n=T*\vp\,. $$ \end{lemma} \begin{proof} Given $[\left\{\phi_n\right\}_1^\infty]\in \DC$, the sequence $\left\{T*\phi_n\right\}_1^\infty$ is a Cauchy one in $\LD$ and so it converges to some $f\in\LD$. Then, by [{\bf 5}, thm. 1], there exists $\vp\in\DB$ such that $T*\vp=f$. Conversely let $\vp\in\DB$; since $C$ has a dense range there exists a (unique in $\DC$) sequence $\left\{\phi_n\right\}_1^\infty \subset D(C)$ such that $T*\phi_n$ converges in $\LD$ to $T*\vp$. \end{proof} Defining $$ \SCT:=\SC/\sim $$ and then the sum of $\phi\in\LD\subset\SC$ plus $\psi=[\vp]\in\DC\,\simeq\,\DB/ \sim\,\subseteq\SCT$ by $$ \phi+\psi:=[\phi+\vp]\in\SCT\,, $$ we can introduce the linear operator $$ G:\X'\to\SCT\,,\qquad G:=G(z)+K(z)\,. $$ According to lemma 4, theorem 5 and the definition of $G$, for any $\phi\in D(C^\tau_\Theta)$ we can give the unique decomposition $$ \phi=\phireg+GQ_\phi\,. $$ Thus we can define $D(C^\tau_\Theta)$ as the set of $\phi\in\LD$ for which there exists $Q_\phi\in D(\Theta)$ such that $$ \phi-GQ_\phi=:\phireg\in\DC $$ and $$ \tau\phireg=\Theta\, Q_\phi\,. $$ \begin{lemma} The definition of $G$ is $z$-independent and $$ \forall\,\ell\in\X'\,,\qquad G\ell=[G_*\ell]\,, $$ where $$ G_*:\X'\to \SC\,, $$ is any conjugate linear operator such that $$ -T*G_*\ell=\tau^*\ell\,.\eqno(9) $$ Here $\tau^*:\X'\to \SC$ is defined by $$ \tau^*\ell(\varphi):=(\,\ell(\tau\varphi^*)\,)^*\,, \qquad\varphi\in C_0^\infty(\RE^d)\,.$$ \end{lemma} \begin{proof} $z$-independence is an immediate consequence of (4). By the definition of $G$ and by (5) there follows $$ -T*G_*\ell=(-{T*}\,+z)G(z)\ell $$ and the proof is concluded by the relation $$ (-{T*}\,+z)\cdot G(z)=\tau^*\eqno(10) $$ which can be obtained proceeding as in [{\bf 18}, remark 2.4]. \end{proof} \begin{remark} By [{\bf 5}, thm. 1], as $\F T$ is slowly decreasing, the equation (9) is always resoluble; in particular, denoting the fundamental solution of $-T*$ by $\G$, when the convolution $\G*\tau^*\ell$ is well defined (e.g. when $\tau^*\ell\in\E$), one has $$ G_*:\X'\to\SC\,, \qquad G_*\ell=\G*\tau^*\ell\,. $$ Analogously, denoting the fundamental solution of $-{T*}\,+z$ by $\G_z$, one has $$ G(z):\X'\to\LD\,, \qquad G(z)\ell=\G_z*\tau^*\ell\,. $$ \end{remark} \begin{remark} Note that $C*\phireg=T*\vp$ and $\tau\phireg=\tau\vp$ for any $\vp\in\DB$ such that $\phireg=[\vp]$. Here we implicitly used the extension given in lemma 6 and the identification given in lemma 7. This also implies that $\Gamma(z)$ in lemma 3 can be re-written as $$\Gamma(z)=\tau\cdot\left(G_*-G(z)\right)\,.$$ By remark 9, when the convolution is well defined, one can also write $$\Gamma(z)\ell=\tau\left(\left(\G-\G_z\right)*\tau^*\ell\right)\,.$$ \end{remark} In conclusion, by making use of the previous lemmata and remarks, we can restate theorem 5 in the following way: \begin{theorem} Let $T\in\E$ with $\F T$ real-valued, slowly decreasing and having a null set of real zeroes, let $C:D(C)\subseteq\LD\to\LD$, $C\phi:=T*\phi$, let $\tau:D(C)\to\X$ satisfy {\rm (h1)-(h3)}. Given $\Theta\in\wtilde \L(\X',\X)$ self-adjoint, let $D(C^\tau_\Theta)$ be the set of $\phi\in\LD$ for which there exists $Q_\phi\in D(\Theta)$ such that $$ \phi-G_*Q_\phi=:\vpreg\in\DB $$ and $$ \tau\vpreg=\Theta\, Q_\phi\,. $$ Then $$ C^\tau_\Theta:D(C^\tau_\Theta)\subseteq\LD\to\LD\,,\qquad C^\tau_\Theta\,\phi:= T*\vpreg\,, $$ is a self-adjoint operator which coincides with $C$ on the kernel of $\tau$ and its resolvent is given by $$ R^\tau_\Theta(z):=R(z)+G(z)\cdot(\,\Theta+\Gamma(z)\,)^{-1}\cdot\breve G(z)\,, \qquad z\in W_\Theta^-\cup W_\Theta^+\cup\C\backslash\RE\,, $$ where $$ \Gamma(z):=\tau\cdot(G_*-G(z))\,. $$ \end{theorem} \begin{remark} The boundary conditions and the operators $\Gamma(z)$ and $C^\tau_\Theta$ appearing in the previous theorem are independent of the choice of the representative (see lemma 8) $G_*$ entering in the definition of $\vpreg$. Indeed any different choice will not change the equivalence class to which $\vpreg$ belongs, and both $\tau$ and $C$ do not depend on the representative in such a class (see remark 10). \end{remark} \begin{remark} Proceeding as in [{\bf 18}, remark 2.4] one can give the following alternative definition of $C_\Theta^\tau$ where only $Q_\phi\in D(\Theta)$ appears: $$ C^\tau_\Theta\,\phi:= T*\phi+\tau^*Q_\phi\,. $$ This is an immediate consequence of identity (10). \end{remark} \begin{example} Let us consider the case $A=\Delta:H^2(\RE^n)\to\LD$. Obviously $A$ is an injective convolution operator, thus we can apply to it the previous theorem. %Note that, by Sobolev embedding theorem, when $n>4$ one has %$$\widehat D(\Delta)=\left\{\phi\in L^{\frac{2n}{n-4}}(\RE^n)\,:\ %\Delta\phi\in\LD\right\}\,.$$ \par A Borel set $F\subset\RE^n$ is called a $d$-set, $d\in(0,n]$, if $$ \exists\, c_1,\,c_2>0\ :\ \forall\, x\in F,\ \forall\,r\in(0,1),\qquad c_1r^d\le\mu_d(B_r(x)\cap F)\le c_2r^d\ , $$ where $\mu_d$ is the $d$-dimensional Hausdorff measure and $B_r(x)$ is the closed $n$-dimensional ball of radius $r$ centered at the point $x$ (see [{\bf 10}, \S 1.1, chap. VIII]). Examples of $d$-sets are $d$-dimensional Lipschitz submanifolds and (when $d$ is not an integer) self-similar fractals of Hausdorff dimension $d$ (see [{\bf 10}, chap. II, example 2]). Moreover a finite union of $d$-sets which intersect on a set of zero $d$-dimensional Hausdorff measure is a $d$-set. \par In the case $01$ and $F$ is a generic $d$-set the functions $\phi_F^{(j)}\in L^2(F;\mu_F)$ are not uniquely determined by $\phi^{(0)}_F$; contrarily we may then identify $\{\phi_F^{(j)}\}_{|j|< \alpha}$ with the single function $\phi^{(0)}_F$. This is possible when $F$ preserves Markov's inequality (see [{\bf 10}, \S 2, chap. II]). Sets with such a property are closed $d$-sets with $d>n-1$ (see [{\bf 10}, thm. 3, \S 2.2, chap. II]), a concrete example being e.g. the boundary of von Koch's snowflake domain in $\RE^2$ (a $d$-set with $d=\log 4/\log 3$, see \cite{[W]}). If $F$ has some additional differential structure then $B^{2,2}_\alpha(F)\simeq H^{\alpha}(F)$, where $H^{\alpha}(F)$ denotes the usual (fractional) Sobolev-Slobodecki\u\i\ space. Some known cases where $B^{2,2}_\alpha(F)\simeq H^{\alpha}(F)$ (for any value of $\alpha>0$) are the following:\p - $F$ is the graph of a Lipschitz function $f:\RE^d\to \RE^{n-d}$ (see [{\bf 6}, \S 20]);\p - $F$ is a bounded manifold of class $C_\gamma$, $\gamma>\min(3,\max(1,\alpha))$, i.e. $F$ has an atlas where the transition maps are of class $C^k$, $k<\gamma\le k+1$, and have derivatives of order less or equal to $k$ which satisfy Lipschitz conditions of order $\gamma-k$ (see \cite{[J]} for the case $\gamma>\max(1,\alpha)$ and see [{\bf 6}, \S 24] for the case $\gamma>3$ );\p - $F$ is a connected complete Riemannian manifold with positive injectivity radius and bounded geometry, in particular a connected Lie group (see [{\bf 22}, \S 7.4.5, \S 7.6.1]).\par Supposing now that $F$ has a compact closure, let $B_1$ and $B_2$ be two $n$-dimensional balls such that $F\subset B_1\subset B_2$; given $\phi\in H^2(\RE^n)$, let $\tilde\phi\in H^2(\RE^n)$ coincide with $\phi$ on $B_1$, have support contained in $B_2$ and satisfy $\|\Delta\tilde\phi\|_{L^2}\le c\,\|\Delta\phi\|_{L^2}$. Then one has, by Poincar\'e inequality applied to $\tilde\phi\in H^2_0(B_2)$, $$ \|\tau\phi\|_{B^{2,2}_\alpha(F)}=\|\tau\tilde\phi\|_{B^{2,2}_\alpha(F)}\le c\,\|\tilde\phi\|_{H^2(\RE^n)} \le c\,\|\Delta\tilde\phi\|_{L^2(\RE^n)}\le c\,\|\Delta\phi\|_{L^2(\RE^n)}\,, $$ and so $\tau_F$ satisfies (h3). Moreover, since $$ {\mathcal D}(\Delta)=\left\{\vp\in\SC\,:\,\Delta\phi\in\LD\right\}\subseteq H^2_{\text{\rm loc}}(\RE^n) $$ and $\bar F$ is supposed to be compact, the extension of $\tau_F$ to ${\mathcal D}(\Delta)$ is again defined by (11). Denoting the dual of $B^{2,2}_\alpha(F)$ by $B^{2,2}_{-\alpha}(F)$ (the space $B^{2,2}_{-\alpha}(F)$ can be explicitly characterized in the case $0<\alpha<1$ or when $F$ preserves Markov's inequality, see \cite{[JW2]}), hypothesis (h2) is equivalent to $\tau'_F\ell\notin L^2(\RE^n)$ for any $\ell\in B^{2,2}_{-\alpha}(F)\backslash\left\{0\right\}$, where $\tau'_F\ell\in H^{-2}(\RE^n)$ is defined by $$ \tau'_F\ell(\phi):=\ell(\tau_F\phi)\,. $$ Therefore, as the support of $\tau'_F\ell$ is given by $\bar F$, (h2) is certainly verified when $\bar F$ has zero Lebesgue measure. Considering the fundamental solution of $-\Delta$, given by $$ \G(x)=\cases \frac{1}{2\pi}\,\log \frac{1}{|x|},&\text{for $n=2$}\\ \frac{1}{(n-2)\sigma_n}\,\frac{1}{|x|^{n-2}},&\text{for $n>2$}\,, \endcases $$ $\sigma_n$ the measure of the unitary sphere in $\RE^n$, the convolution $\G*\tau'_F\ell$ is well distribution defined as $\tau'_F$ is in $\E$. Therefore, by lemma 8 we can choose $G_*$ to be the map $$ G_*:B^{2,2}_{-\alpha}(F)\to\SC\,,\qquad G_*\ell:=\G*\tau^*_F\ell\,. $$ Thus, by the previous theorem (and remark 13), supposing that the $d$-set $F$ has a compact closure of zero Lebesgue measure, given any self-adjoint operator $\Theta\in\wtilde \L(B^{2,2}_{-\alpha}(F),B^{2,2}_\alpha(F))$, $\Theta\in\L(B^{2,2}_\alpha(F))$ if one uses the identification $B^{2,2}_{-\alpha}(F)\simeq B^{2,2}_\alpha(F)$, we have then the self-adjoint operator $$\Delta^F_\Theta\,\phi:=\Delta\vpreg\equiv \Delta\phi+\tau_F^*Q_\phi\,,$$ where $$ \phi=\vpreg+\G*\tau^*_FQ_\phi \,,\qquad\vpreg\in{\mathcal D}(\Delta)\,,\quad Q_\phi\in D(\Theta)\,,$$ and $$ \left\{\lim_{r\downarrow 0}\,\frac{1}{\lambda_n(r)}\int_{B_r(x)}dy\, D^j\left(\phi -\G*\tau^*_FQ_\phi\right)(y)\right\}_{|j|< \alpha}=\Theta\,Q_\phi(x)\,, $$ $|j|=0$ if $F$ preserves Markov's inequality or $B^{2,2}_\alpha(F)\simeq H^\alpha(F)$. When $F=M$, $M$ a connected complete Riemannian manifold with positive injectivity radius and bounded geometry, a natural choice for $\Theta$ is given by $\Theta=(-\Delta_M+\lambda^2)^{-\alpha}$ ($\lambda=0$ if $M$ is compact), where $\Delta_M$ denotes the Laplace-Beltrami operator. A particular case of this situation, when $A=(\Delta-\epsilon):H^2_0(\Omega)\to L^2(\Omega)$, $\epsilon>0$, $\Omega=\RE^2\times [0,\pi]$, (note that here $0\notin\sigma(A)$, thus theorem 1 applies), and $F=[0,\pi]$, is treated, without giving boundary conditions, in \cite{[KKO]} (also see [{\bf 18}, example 3.2] for connections with Birman-Kre\u\i n-Vishik theory). \end{example} \begin{thebibliography}{99} \bibitem{[AFHKL]} S. Albeverio, J.R. Fenstad, R. H\o egh-Krohn, W. Karwowski, T. Lindstr\o m: Schr\"odinger Operators with Potentials Supported by Null Sets. Published in {\it Ideas and Methods in Mathematical Analysis, Vol. II}. New-York, Cambridge: Cambridge Univ. Press 1991 \bibitem{[AGHH]} S. 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