%\input amstex %revised 12/5/01 by ml (after ag) \documentstyle{amsppt} \magnification=1000 \NoBlackBoxes %\define\newtext{} \define\newtext{\bf} \define\a{\alpha} \define\be{\beta} \def\C{\Bbb C} \define\ga{\gamma} \define\la{\lambda} \define\k{\kappa} \define\eps{\varepsilon} \define\ep{\epsilon} \def\R{\Bbb R} \define\Rn{{\Bbb R^n}} \define\Rd{{\Bbb R^d}} \define\sgn{\text{sign }} \define\rank{\text{rank }} \define\coker{\text{\rm Coker }} \define\im{\text{\rm Im }} \redefine\Ker{\text{\rm Ker }} \define\loc{{\text{\rm loc}}} \define\comp{{\text{\rm comp}}} \define\supp{\text{supp }} \def\singsupp{\text{sing--supp }} \define\sing{\text{sing }} \define\dist{\text{dist }} \redefine\Re{\roman{Re} \,} \define\vphi{\varphi} \define\id{\text{Id}} \redefine\exp{{\text{\rm exp}}} \def\acs{adapted{ }coordinate{ }system} \def\cinf{C^\infty} \def\d{\delta} \def\fp{\flushpar} \def\fs{f^\sharp} \def\g{\gamma} \def\lm{\tau^2-|\xi|^2} \def\L{\Lambda} \def\o{\omega} \def\O{\Omega} %\def\pt{\tilde\pi_{T^*M}} \def\pw{\delta(t-x\cdot\o} \def\rta{\rightarrow} \def\xs{\xi^{\sharp}} \def\b{\backslash} \def\bb{\Bbb B} \def\br{\Bbb R} \def\a{\alpha} \def\l{\lambda} \def\ltlt{L^2\rta L^2} \def\pl{\pi_L} \def\po{\partial\O} \def\pr{\pi_R} \def\P{\Phi} \def\rsn{\br^{n+1}\times\sn} \def\s#1{\sigma^{(#1)}} \def\si{\square^{-1}} \def\sn{\Bbb S^{n-1}} \def\rts{\R^{n+1}\times\sn} \def\ts{\tilde{\sigma}} \def\tsd{\dot{\ts}} \def\tsdd{\ddot{\ts}} \def\tx{\tilde x} \def\tz{\tilde z} \def\X{\Cal X} \def\zs{0_{T^*\br^n}} \def\ns{\nu_H} \def\supp{\text{supp}} \def\D{\Delta} \def\Dr{\D_\rho} \def\bs{\square_*} \def\bsi{\bs^{-1}} \def\Ls{\L_\square} \def\ut{u^{(2)}} \def\pt{\psi^{(2)}} \def\hra{\hookrightarrow} \def\bsfp{\bigskip\fp} \def\vz{\Cal V^0} \def\rtv{\br^{n+1}\times\vz} \def\back{\backslash} \def\csig{C_{\Sigma}} \def\fpp#1{\frac{\partial}{\partial #1}} \def\dv{\vec{\delta}} \def\imd{I^{m,\dv}_{N_x,N_\theta}} \def\smd{S^{m,\dv}_{N_x,N_\theta}} \def\N{\Bbb N} \def\px{\partial_x} \def\path{\partial_\theta} \def\opt{(1+|\theta|)} \def\od{\O^\delta} \def\ud{\Cal U^\delta} \def\dz{\delta_0} \def\tdz{T^\dz} \def\wt{\widetilde} \def\CD{\Cal C\Cal D} \def\xpr{X^{p,r}} \def\codim{\text{ codim}} \def\p{\partial} \NoRunningHeads \topmatter \title The Calder\'on problem for conormal potentials, I: Global uniqueness and reconstruction \endtitle \author Allan Greenleaf, Matti Lassas and Gunther Uhlmann \endauthor %\date{December 5, 2001}\enddate \thanks This work was supported in part by National Science Foundation grants DMS-9877101, DMS-0070488 and Finnish Academy project 172434. The third author was also supported by a John Simon Guggenheim Fellowship. The second and third author are grateful for the hospitality of the Mathematical Sciences Research Institute in Berkeley, CA. \endthanks \address Department of Mathematics, University of Rochester, Rochester, NY 14627 \endaddress \address Rolf Nevanlinna Institute, P.O. Box 4, 00014 University of Helsinki, Finland \endaddress \address Department of Mathematics, University of Washington, Seattle, WA 98195 \endaddress \endtopmatter \document {\bf Abstract.} We consider inverse problems for Schr\"odinger operator in the case of potential having co-normal singularities in dimensions $n\geq 3$. For instance, in the case where singularity is on hypersurface $H\subset \R^n$, we allow singularities of type $dist(x,H)^{-1+\alpha},$ $\alpha>0$. Moreover, by considering Schr\"odinger equation in generalized sense (as solution of minimization problem), we give certain counter examples to uniqueness of inverse problem. \medskip \head {\bf 1. Introduction} \endhead The goal of this paper is to establish global uniqueness and obtain reconstruction, in dimensions $n\ge3$, for the Calder\'on problem in the class of potentials conormal to a smooth submanifold $H$ in $\R^n$. In the case of hypersurfaces, the potentials considered here may have any singularity weaker than that of the delta function $\delta_H$ on the hypersurface $H$; in general, these potentials correspond to conductivities which are in $C^{1+\epsilon}$ and thus fail to be covered by previously known results. Let $\O\subset\R^n$ be a bounded Lipschitz domain, $H\subset\O$ a smooth submanifold of codimension $k$, and $q\in I^{\mu}(H)$ a real conormal distribution of order $\mu$ with $\mu<1-k$. Thus, if $H=\{x: F_j(x)=0, 1\le j\le k\}$ is a local representation of $H$ by means of defining functions with $\{\nabla F_j: 1\le j\le k\}$ linearly independent on $H$, then locally $q(x)$ has the Fourier integral representation $$q(x)=\int_{\R^k} e^{i\sum_j F_j(x)\cdot\theta_j} a(x,\theta) d\theta,\quad a\in S^{\mu}_{1,0},\tag1.1$$ where $S^{\mu}_{1,0}$ denotes the standard class of symbols of order $\mu$ and type $(1,0)$ on $\R^n\times (\R^k\back 0)$. A general element $q\in I^{\mu}(H)$ is a locally finite sum of such expressions. We assume throughout that $\supp(q)$ is compact in $\O$. If $-k<\mu<0$, then $q$ satisfies $|q(x)|\le C\cdot\dist(x,H)^{-k-\mu}$, so that $q\in L^{\frac{k}{k+\mu}-\epsilon}(\O),\forall\epsilon>0$, and no better in general; in particular, a general element of $I^\mu(H)$ is unbounded. For comparison, surface measure $\delta_H\in I^0(H)$ and, in the hypersurface case, a Heaviside discontinuity across $H$ belongs to $I^{-1}(H)$. Rather than working with the Dirichlet-to-Neumann map, $\L_q$, we state our main results in terms of the Cauchy data, $\CD_q$, of sufficiently regular solutions of the Schr\"odinger equation $$(\D+q(x))u(x)=0\hbox{ on }\O.\tag1.2$$ This is more flexible, since $\CD_q$ can be defined for potentials $q$ for which $\L_q$ is either not defined (for example, if $\lambda=0$ is a Dirichlet eigenvalue) or is not known to be defined (due to the low regularity of $q(x)$); it is perhaps more natural as well. \medskip It will be convenient to write $\mu=\nu-k$. Assume that $\nu_0(k)<\nu<1$, where $\nu_0(k)\buildrel{def}\over{=}max(\frac23,1-\frac{k}4)$. Fix $p$ and $r$ satisfying $$2\le r<\frac{k}{2(1-\nu)}\buildrel{def}\over{=} r_0(k,\nu)\nu_0(k)$.) Fixing a smooth function $\psi\in C^\infty(\R^n), \psi\equiv 1$ near $\partial\O$ and $int\{\psi=0\}\cap\O\ne \phi$, define the norm $$||f||_{\xpr} =||f||_{L^p(\O)}+||\D f||_{L^{p'}(\O)}+||\psi f||_{W^{2,r}(\O)},\tag1.4$$ where $p'$ is the dual exponent to $p$ and $W^{2,r}$ is the standard Sobolev space of $f\in\Cal D'(\O)$ having two derivatives in $L^r(\O)$. Set $$\xpr(\O)=\{f\in\Cal D'(\O): ||f||_{\xpr} <\infty\}\tag1.5$$ and note that the Schr\"odinger operator $\D+q$ maps $\xpr(\O)\rta L^{p'}(\O)$ continuously. We denote throughout this paper by $n$ the unit outer normal to $\O$. \medskip \proclaim{Definition} For a potential $q\in I^\mu(H)$ with $H\cap\supp(q)\subset int\{\psi=0\}$, the Cauchy data of the Schr\"odinger operator $\D+q$ relative to $\xpr(\O)$ is $$\CD_q=\CD_q^{p,r}= \Big\{(u|_{\po},\frac{\partial u}{\partial n}|_{\po}): u\in\xpr(\O), (\D+q)u=0\hbox{ on }\O\Bigr\}.\tag1.6$$ \endproclaim \medskip By Sobolev embedding, $\CD_q$ is a subspace of $W^{2-\frac1r,r}(\po)\times W^{1-\frac1r,r}(\po)$. Observe that if the Dirichlet-to-Neumann map $\L_q$ {\it is} defined (on $W^{2-\frac1r,r}(\po)$, say), then $\CD_q$ is simply its graph. We will construct certain nontrivial exponentially growing solutions $u\in \xpr(\O)$, so that, for the potentials considered, $\CD_q$ is in fact nontrivial. We can now state our first result. \bigskip \proclaim{Theorem 1} Suppose that for $j=1,2$, $H_j\subset\subset\O$ are submanifolds of codimension $k_j$. Suppose further that $q_j\in I^{\mu_j}(H_j)$ are real potentials with $\nu_0(k_j)-k_j<\mu_j<1-k_j$ and $\supp(q_j)\subset\subset\O$. Let $p,r$ satisfy $2\le r2$, proved in [3] using the $\overline\partial$ technique of [1,25,26]. Global uniqueness for piecewise-analytic conductivities was proven [20], and special types of jump discontinuities were treated in [17]. \bigskip Here, we will follow the general argument of [32], although employing a different integral identity so as to avoid difficulties when applying Green's Theorem. It is this identity that makes $\xpr$ a convenient space for the problem; indeed, both sides of $\int_\O \D u\cdot v- u\cdot\D v dx=\int_{\po} \partial_n u\cdot v-u\cdot\partial_n v d\sigma$ are continuous with respect to $||\cdot ||_{\xpr}$ and thus Green's Theorem holds for $u,v\in\xpr(\O)$. \bigskip We now start the proof of Thm. 1, so as to motivate the technicalities that follow. Given a submanifold $H$ of codimension $k$ and a potential $q\in I^{\mu}(H)$ with $\mu<1-k$, we will construct exponentially growing solutions of (1.2) belonging to $\xpr(\O)$, of the form $v(x)= e^{\rho\cdot x}(1+\psi(x,\rho))$, with $\rho\in\Bbb C^n$ satisfying $\rho\cdot\rho=0$. Let $v_1\in \xpr(\O)$ be a solution of $(\D+q_1)v_1=0$. By the hypothesis of Thm. 1, there is a solution $v_2\in\xpr(\O)$ of $(\D+q_2)v_2=0$ with $$v_2|_{\po}=v_1|_{\po}\quad\hbox{ and }\quad \frac{\partial v_2}{\partial n}|_{\po}= \frac{\partial v_1}{\partial n}|_{\po}.\tag1.7$$ Let $w_2\in\xpr(\O)$ be any other solution to $(\D+q_2)w_2=0$. Then, $$(\D+q_2)(v_1-v_2)=(\D+q_2)v_1=(\D+q_1+(q_2-q_1))v_1=(q_2-q_1)v_1,$$ so that $$\eqalign{\int_\O(q_2-q_1)v_1w_2 dx=&\int_\O (\D+q_2)(v_1-v_2)\cdot w_2 dx\cr =&\int_\O (\D+q_2)(v_1-v_2)\cdot w_2-(v_1-v_2)\cdot (\D+q_2)w_2 dx\cr =&\int_\O \D(v_1-v_2)\cdot w_2 - (v_1-v_2)\cdot\D w_2 dx\cr =& \int_{\po} \frac{\partial}{\partial n}(v_1-v_2)\cdot w_2 - (v_1-v_2)\cdot \frac{\partial}{\partial n} w_2 d\sigma\cr =& 0,}\tag1.8$$ where the application of Green's Theorem is valid since $v_1-v_2$ and $ w_2\in \xpr$ and the last equality holds by (1.7). If we carry this out for the solutions $v_1$ and $w_2$ constructed below for complex frequencies $\rho_1$ and $\rho_2$ satisfying $\rho_1+\rho_2=-i\xi$, with $\xi\in\R^n\back 0$, then we have, as in [32], $$\eqalign{0=& \int_\O (q_1-q_2) e^{(\rho_1+\rho_2)\cdot x} \Bigl(1+\psi_1(x,\rho_1)\Bigr)\Bigl(1+\psi_2(x,\rho_2)\Bigr) dx\cr =&\widehat{(q_1-q_2)}(\xi) +\int_\O e^{-i\xi\cdot x} (q_1-q_2)(\psi_1+\psi_2+\psi_1\psi_2)dx}.\tag1.9$$ If one can do this for pairs $(\rho_1,\rho_2)$ with $|\rho_j|\rta\infty$ and show that the last integral $\rta 0$ as $|\rho|\rta\infty$, then $\hat{q_1}(\xi)=\hat{q_2}(\xi)$; doing this for all $\xi\in\R^n$ will finish the proof of Thm. 1. As is well known, $v(x)= e^{\rho\cdot x} (1+\psi(x))$ is a solution of the Schr\"odinger equation iff $\psi$ is a solution of $$(\D_\rho+q)\psi=-q(x)\hbox{ where } \D_\rho=\D+2\rho\cdot\nabla.\tag1.10$$ We will show in Prop. 2.6 that (1.10) is uniquely solvable, with some decay in $|\rho|$, in a Banach space of finite-regularity conormal distributions associated with $H$, yielding exponentially growing solutions $v_j\in\xpr$ to (1.2) which allow the argument above to be carried out. In \S3, this result is extended to a hybrid global space, and this is applied to obtain reconstruction of the potential from the Cauchy data, following the general argument of [25]. Finally, in \S4 we show that uniqueness can fail in a weak formulation of the problem for potentials with very strong singularities on a hypersurface, with blow-up rates corresponding to those of distributions conormal of order greater than 1 for $H$. We would like to thank Steve McDowall for valuable discussions. \bigskip \head {\bf 2. Uniqueness for conormal potentials} \endhead As described in the Introduction, to prove Thm.1, it suffices to construct exponentially growing solutions to (1.2) of the form $v(x)=e^{\rho\cdot x}(1+\psi(x,\rho))$ for $\rho\cdot\rho=0,|\rho|\rta\infty$ so that the second integral in (1.9) tends to 0 as $|\rho|\rta\infty$. To do this for potentials $q\in I^\mu(H)$, the standard space of (infinite-regularity) conormal distributions of order $\mu$ associated with the codimension $k$ submanifold $H$, we will also need to formulate Banach spaces of finite-regularity conormal distributions in $\R^n$ associated with $H$. Rather than working in unnecessary generality, we will restrict ourselves to the spaces needed here; unlike [23],[22], where several other types of finite-regularity conormal spaces are defined using iterated regularity with respect to Lie algebras of tangent vector fields, we impose the finite-regularity assumption directly on the symbols in the oscillatory representations of the distributions. Fix a compact set $K$ with $\Omega\subset\subset K$; all of the spaces will involve distributions supported in $K$. Also fix an order $m\in\R$, a sequence $\dv=(\delta_1,\delta_2,...)$ of numbers $0\le \delta_j\le 1$, and $N_x,N_\theta\in\N$. \bigskip \proclaim{Definition 2.1} (i) $$\smd=\Bigl\{a(x,\theta): |\px^\beta \path^\alpha a(x,\theta)|\le C(1+|\theta|)^{m-\sum_{j=1}^{|\alpha|} \delta_j}, \forall |\beta|\le N_x, |\alpha|\le N_\theta\Bigr\}$$ and $$||a||_{\smd}=\max_{\buildrel{0\le |\beta|\le N_x}\over{0\le\alpha\le N_\theta}}\sup_{x\in K,\theta\in\R^k} (1+|\theta|)^{ -m+\sum_{j=1}^{|\alpha|} \delta_j} |\px^\beta \path^\alpha a(x,\theta)|.$$ (ii) If $H=\{x: F_j(x)=0, 1\le j\le k\}$ locally, with $\{\nabla F_j\}$ linearly independent on $H$, then $$\imd(H)=\Bigl\{ u(x)=\int_{\br^k} e^{i\sum_{j=1}^k F_j(x)\cdot\theta_j} a(x,\theta) d\theta: a\in\smd\Bigr\}$$ locally and in general consists of all locally finite sums of such expressions. \endproclaim \fp{\bf Remarks}. (1) $\smd$ is a Banach space with respect to the norm defined in (i), and $\imd(H)$ inherits this structure. \fp (2) $\path: S^{m,(\delta_1,\delta_2,...)}_{N_x,N_\theta}\rta S^{m-\delta_1,(\delta_2,\delta_3,...)}_{N_x,N_\theta -1}$ and $\partial_x: S^{m,(\delta_1,\delta_2,...)}_{N_x,N_\theta}\rta S^{m,(\delta_1,\delta_2,...)}_{N_x-1,N_\theta}$ continuously. \fp (3) $\imd(H)$ is well-defined, since changing the defining function of $H$ locally corresponds to a change of variable in $x$, which leaves the symbol class invariant. \fp (4) The usual (infinite-regularity) conormal space has a continuous inclusion with respect to its Fr\'echet structure: $ I^m(H)\hra \imd(H)$ for any $\dv, N_x,N_\theta$. \bigskip \proclaim{Proposition 2.2}Let $H\subset \Omega\subset \R^n$, with $\codim(H)=k$. (i) For $-k\le m<-\frac{k}2,N_x\ge 1, N_\theta\ge1, \delta_1>0$ and $m-\sum_{j=1}^{N_\theta} \delta_j<-k$, $$I^{m,\dv}_{N_x,N_\theta}(H)\hra L^p( \O)\hbox{\text{ continuously for all} } 1\le p <\frac{k}{m+k}.\tag2.1$$ \fp (ii) If $N_x\ge2$, $N_\theta\ge2$, $\delta_1>1-\frac{k}2$ and $\sum_{j=1}^{N_x}\delta_j>1$, then for each smooth function $F$ vanishing on $H$, $$v\in I^{-k,\dv}_{N_x,N_\theta}(H)\implies F\cdot v\in W^{1,p}(\Omega)\hbox{ for }1\le p<\frac{k}{1-\delta_1}\tag2.2$$ and thus, for $G\subset\subset\overline\O\back H$, $I^{-k,\dv}_{N_x,N_\theta}(H)\hra W^{1,p}(G)$ continuously. \fp (iii) Suppose $\nu_0(k)<\nu<1$, $N_x\ge 3$ and $N_{\theta}\ge 6$. Set $$\dv_0=(\nu,1-\nu,2\nu-1,1-\nu,1-\nu,3\nu-2,\dots).$$ Then, for any $F$ vanishing on $H$, $$v\in I^{-k,\dv_0}_{N_x,N_\theta}(H)\implies F^3\cdot v\in W^{2,r}(\Omega)\hbox{ for }1\le r<\frac{k}{2(1-\nu)}\tag2.3$$ and thus, for $G\subset\subset\overline\O\back H$, $I^{-k,\dv_0}_{N_x,N_\theta}(H)\hra W^{2,r}(G)$ continuously. \endproclaim \fp Proof. (i) Since $L^p$ and $\imd(H)$ are diffeomorphism-invariant, it suffices to assume that, with respect to coordinates $x=(x',x'')\in\R^{n-k}\times\R^k$, $$H=\{x''=0\}\quad\hbox{\text and }\quad u(x)=\int_{\R^k} e^{ix''\cdot\theta} a(x,\theta) d\theta,\quad a\in S^{m,\dv}_{1,1},$$ with $S^{m,\dv}_{1,1}$ as in Def.2.1(i). We then have $$u(x',x'')=\int e^{ix''\cdot\theta}[a(x',0,\theta)+\sum_{0<|\alpha''|-\frac{k}{m}$ and $-\frac{k}{m}<2$, the Hausdorff-Young inequality implies that $\int e^{ix''\cdot\theta} a(x',0,\theta) d\theta\in L^{q'}(\R^k_{ x''}),\forall 2\le q'<\frac{k}{m+k}$ , uniformly in $x'$, and thus belongs to $L^p(\R^n)$ for all $2\le p<\frac{k}{m+k}$; since it is compactly supported, the range is in fact $1\le p<\frac{k}{m+k}$. For the second term, note that for each function $a_{\alpha''}(x',\theta)\buildrel{def}\over{=}\frac1{(\alpha'')!}\partial^{\alpha''}a(x',0,\theta)$, $$\int e^{ix''\cdot\theta} a_{\alpha''}(x',\theta)( x'')^{\alpha''} d\theta = \int ( \frac1{i}\partial_\theta)^{\alpha''}(e^{ix''\cdot\theta}) a_{\alpha''}(x,\theta) d\theta = $$ $$= \int e^{i x''\cdot\theta} (\frac1{i}\partial_\theta)^{\alpha''} (a_{\alpha''}(x,\theta)) d\theta,$$ whose amplitude is $\le C(1+|\theta|)^{m-\sum_{j=1}^{|\alpha''|}\delta_j}$, and hence is treated by the same argument as for $a_0=a(x',0,\theta)$. In the final term, we integrate by parts: $$\int e^{ix''\cdot\theta} b_{\alpha''}(x,\theta) (x'')^{\alpha''} d\theta = \int e^{i x''\cdot\theta} (i\partial_{\theta})^{\alpha''} b_{\alpha''}(x,\theta) d\theta,$$ and since $|\partial_{\theta}^{\alpha''} b(x,\theta)|\le C(1+|\theta|)^{m-\sum_{j=1}^{N_\theta}\delta_j}\in L^1(\R_\theta)$, uniformly in $x'$, this yields a bounded function of $x$. \medskip \fp (ii) If $v\in I^{-k,\dv}_{N_x,N_\theta}(H)$, then $v\in L^p,\forall p<\infty$, by part (i). From $$v(x)=\int e^{i\sum_{j=n-k+1}^n x_j\theta_j} a(x,\theta) d\theta,\quad a\in S^{-k,\dv}_{N_x,N_\theta},$$ one finds $$\eqalign{\partial_{x_l} v(x)=&\int e^{i\sum_{j=n-k+1}^n x_j\theta_j} (i\theta_l a+\partial_{x_l}a) d\theta\cr \in& I^{1-k,\dv}_{N_x,N_\theta}(H) + I^{-k,\dv}_{N_x-1,N_\theta}(H).}$$ The second term is covered by part (i) and hence is in $L^p,\forall p<\infty$. If the first term is multiplied by some $x_{j_0}, n-k+1\le j_0\le n$ and integrated by parts, it becomes an element of $I^{1-k-\delta_1,(\delta_2,\delta_3,...)}_{N_x,N_\theta-1}(H)$, which by (2.1) is in $L^p, 1\le p<\frac{k}{1-\delta_1}$. Since any $F$ vanishing on $H$ can be represented as a linear combination of $x_{j_0}$'s with smooth coefficients, $F(x)\nabla v\in L^p$ and so $v\in W^{1,p}(G)$ for any set $G$ on which $|F|$ is bounded below. \medskip \fp (iii) By (i) and (ii) above, both $v$ and $F\cdot\nabla v\in L^r$. Now, arguing as in (ii), $$\eqalign{\partial^2_{x_lx_{l'}}v=&\int e^{i \sum_{j=n-k+1}^nx_j\theta_j}\Bigl(-\theta_l\theta_{l'}a+i(\theta_l a_{x_{l'}}+\theta_{x_{l'}}a_{x_l})+a_{x_lx_{l'}}\Bigr) d\theta\cr \in&I^{2-k,\dv_0}_{N_x,N_\theta}(H)+I^{1-k,\dv_0}_{N_x-1,N_\theta}(H) +I^{-k,\dv_0}_{N_x-2,N_\theta}(H)}$$ By (i), the last term is in $L^r,\forall r<\infty$, if $N_x\ge 3,N_\theta\ge 1$. Integrating by parts and using (ii), $x_{j_0}$ times the second term is in $L^r, 1\le r<\frac1{1-\delta_1}=\frac1{1-\nu}$ if $N_x\ge 2,N_\theta \ge 2$. As for the first term, $x_{j_0}x_{j_1}x_{j_2}$ times it is seen, after integrating by parts three times, to be in $I^{2-k-(\delta_1+\delta_2+\delta_3),(\delta_4,\dots)}_{N_x,N_\theta-3}(H) =I^{2-k-2\nu,(1-\nu,1-\nu,3\nu-2,\dots)}_{N_x,N_\theta-3}(H)$, which, by (i), $\hra L^r(\O),\forall 1\le r<\frac{k}{2(1-\nu)}$, if $N_x\ge 2,N_\theta\ge 6$, since $-k\le 2-k-2\nu<-\frac{k}2$ if $1-\frac{k}4<\nu\le 1$ and $2-k-\sum_{j=1}^6\delta_j=2-3\nu-k\frac23$. Thus, for any $F$ vanishing on $H$, $F^3\cdot I^{-k,\dv_0}_{N_x,N_\theta}(H)\hra W^{2,r}(\O)$ if $1\le r<\frac{k}{2(1-\nu)},N_x\ge 3$ and $N_\theta\ge 6$. \qed \bigskip We also have \bigskip \proclaim{Proposition 2.3} If $A\in\Psi^r(\R^n)$ is properly supported, then for any $m\in\R,\dv, N_x\ge 2n$ and $N_\theta\ge 0$, $$A:I^{m,\dv}_{N_x,N_\theta}(H)\rta I^{m+r,\dv}_{N_x,N_\theta}(H)\tag2.4$$ and $||A||$ is bounded by a finite number of semi-norms of $\sigma(A)$ in $S^r_{1,0}(\R^n\times(\R^n\back 0))$. \endproclaim \bigskip \fp{\bf Proof.} Write $$Af(x)=\int e^{i(x-y)\cdot\xi} a(x,y,\xi) f(y) d\xi dy,\quad a\in S^r$$ and $$u(y)=\int e^{iF(y)\theta} b(y,\theta) d\theta,\quad b\in\smd.$$ Then $$\eqalign{Au(x)=&\int e^{i[(x-y)\cdot\xi+F(y) \theta]} a(x,y,\xi) b(y,\theta) d\theta d\xi dy\cr =&\int e^{iF(x)\theta}\Bigl[\int e^{i[(x-y)\cdot\xi+ (F(y)-F(x))\theta]} a(x,y,\xi) b(y,\theta) d\xi dy\Bigr] d\theta\cr =&\int e^{iF(x)\theta}c(x,\theta) d\theta.}$$ Writing $$\eqalign{F(y)-F(x)=&\int_0^1 \frac{d}{ds}(F(x+s(y-x))ds\cr =&(\int_0^1\nabla F(x+s(y-x))ds)\cdot (y-x)\cr\buildrel{def}\over{=}&G(x,y)\cdot (y-x),}$$ and making the substitution $z=y-x,\zeta=\xi-\theta G(x,y)$, we obtain $$c(x,\theta)=\int e^{-iz\cdot\zeta} a(x,x+z,\zeta+\theta G(x,y)) b(x+z,\theta) d\zeta dz.$$ An examination of standard proofs of stationary phase, ({\it cf.},[31,Lem.1.1.8]) shows that it suffices for the amplitude to have two derivatives for each dimension in order to obtain the estimate $|c(x,\theta)|\le C(1+|\theta|)^{m+r}$. Noting that $\theta$ plays the role of a parameter here, since the stationary phase is in $z,\zeta$, one sees that the derivatives in $x$ and $\theta$ satisfy Def. 2.1(i) for order $m+r$. One concludes that $c\in S^{m+r,\dv}_{N_x,N_\theta}$.\qed To obtain the boundedness of $M_q$ on some of the finite-regularity conormal spaces, we will need the following elementary result, which we state without proof. \bigskip \proclaim{Lemma 2.4} Let $a,b\in C(\R^k)$ satisfy $$|a(\theta)|\le C(1+|\theta|)^m\quad\hbox{ and }\quad |b(\theta)|\le C(1+|\theta|)^{m'}\tag2.5$$ with $m+m'<-k$. Then the convolution $a*b$ satisfies $$|a*b(\theta)|\le C(1+|\theta|)^{m''},\tag2.6$$ with $m''=\max\bigl( (m+k)_{\tilde+}+m',m+(m'+k)_{\tilde+},m+m'+k)$. Here, $t_{\tilde+}=t_+=\max(t,0)$ if $t\ne 0$ and $t_{\tilde+}=\epsilon$ for $t=0$, with $\epsilon>0$ arbitrarily small. \endproclaim \bigskip Using this, we show \bigskip \proclaim{Proposition 2.5} Let $H\subset\R^N$ be codimension $k$. For $q\in I^\mu(H)$, let $M_q$ denote multiplication by $q(x)$. Suppose $\frac23-k\le \mu<1-k$. Then, for any $N_x\in\N$, $$M_q: I^{-k,\dv_0}_{N_x,6}(H)\rta I^{\tilde{\mu},\dv_0}_{N_x,6}(H)\quad\hbox{continuously}\tag2.7$$ where, if we write $\mu=\nu-k$ with $\frac23\le\nu<1$, $$\dv_0=(\nu,1-\nu,2\nu-1,1-\nu,1-\nu,3\nu-2,\dots)\hbox{ and }\tilde\mu=\mu+\epsilon\tag2.8$$ for any $0<\epsilon<-\mu$. \endproclaim \bigskip \fp{\bf Remark.} Since $N_\theta=6$, we only need specify $\delta_1,\dots,\delta_6$. Note also that (2.8) is compatible with the $\dv_0$ in Prop. 2.2(iii). \bigskip \fp{\bf Proof.} As in the proof of Prop. 2.2, we can assume that $H=\{x''=0\}$, $$q(x)=\int_{\R^k} e^{ix''\cdot\theta} a(x,\theta) d\theta,\quad a\in S^\mu_{1,0}$$ and $$u(x)=\int_{\R^k} e^{i x''\cdot\theta} b(x,\theta) d\theta,\quad b\in S^{-k,\dv_0}_{N_x,6}.$$ Hence, $M_qu(x)=\int e^{ix''\cdot\theta} (a*b)(x,\theta) d\theta$, where $*$ denotes the $k$-dimensional convolution in the $\theta$ variable. By Lemma 2.4, $$|a*b(x,\theta)|\le C(1+|\theta|)^{\max(\mu,\mu+\epsilon,\mu)}=C\opt^{\mu+\epsilon},\forall \epsilon>0.$$ Since $\path a\in S^{\mu-1}$ and $\mu-1<-k$, $$|\path(a*b)(x,\theta)|=|(\path a)*b (x,\theta)|\le C\opt^{\max(-k,\mu-1+\epsilon,\mu-1)}=C\opt^{-k}$$ (for $\epsilon<-\mu$), which gives a gain of $\ge \mu-(-k)=\nu$, consistent with $\delta_1=\nu$. (The additional gain of $\epsilon$ we choose to ignore.) Noting that $\path b\in S^{-k-\nu,(1-\nu,2\nu-1,...)}_{N_x,3}$ by Remark 4 above, we have $$|\path^2(a*b)|=|(\path a)*(\path b)|\le C\opt^{\max(-k-\nu,\nu-k-1,-k-1)}=C\opt^{\nu-k-1}$$ since $\nu\ge\frac12$, which is a gain of $\delta_2=-k-(\nu-k-1)=1-\nu$. Since $\path^2a\in S^{\mu-2}$ and $\path b$ is as noted above, $$|\path^3(a*b)|=|(\path^2a)*(\path b)|\le C\opt^{\max(-k-\nu,\nu-k-2)}=C\opt^{-k-\nu},$$ which gives a gain of $\delta_3=\nu-k-1-(-k-\nu)=2\nu-1$. Since $\path^2a\in S^{\mu-2}$ and $\path^2 b\in S^{-k-1,(2\nu-1,1-\nu,...)}_{N_x,4}$, $$|\path^4(a*b)|=|(\path^2a)*(\path^2 b)|\le C\opt^{\max(-k-1,\nu-k-2,2\nu-k-3)}=C\opt^{-k-1},$$ which gives a gain of $\delta_4=-k-\nu-(-k-1)=1-\nu$. Continuing in this fashion, we may estimate $$|\path^5(a*b)|=|(\path^2a)*(\path^3b)|\le C(1+|\theta|)^{\nu-k-2}\implies \delta_5=1-\nu\hbox{ if }\nu\ge \frac23,$$ and $$|\path^6(a*b)|=|(\path^3a)*(\path^3b)|\le C(1+|\theta|)^{-2\nu-k}\implies \delta_6=3\nu-2.$$ The $x$ derivatives, not involving any gain in $|\theta|$, are handled in the obvious way. Hence, $a*b\in S^{\mu+\epsilon,\dv_0}_{N_x,6}$. \qed \bigskip Now recall some facts concerning the Faddeev Green's function [10], $G_\rho$. As is well-known (see,e.g., [32], where this is used implicitly), the families $\{|\rho|G_\rho:\rho\cdot\rho=0\}$ and $\{ G_\rho: \rho\cdot\rho=0\}$ are uniformly bounded in $\Psi^0(K)$ and $\Psi ^{-1}(K)$, respectively, for $K\subset\subset\R^n$ and hence interpolation implies $$\Bigl\{ |\rho|^{1-t} G_\rho:\rho\cdot\rho=0\Bigr\}\subset \Psi^{-t}(K)\hbox{ is bounded },\forall t\in [0,1].\tag2.9$$ \bigskip We can now state a local analogue for the finite-regularity conormal spaces of the result of [32] concerning solvability of inhomogeneous equations involving $\Dr+q(x)$ in weighted $L^2$ spaces. \bigskip \proclaim{Proposition 2.6} If $q\in I^{\mu}(H)$ with $\frac23-k\le\mu<1-k$ and $0\le\sigma\le 1$, then for $\dv_0$ as in (2.8), the inhomogeneous equation $$(\Dr+q)w=g\in I^{1-k-\sigma,\dv_0}_{2n,6}(H)\tag2.10$$ has, for $|\rho|$ large, a unique solution $w\in I^{-k,\dv_0}_{2n,6}(H)$, with $\|w\|\le \frac{C}{|\rho|^\sigma}\|g\|$.\endproclaim \bigskip \fp{\bf Proof.} Applying $G_\rho$ to both sides of (2.10), using (2.9) for $t=1-\sigma$ and Prop. 2.3, we are reduced to showing that $$(I+G_\rho M_q)w=G_{\rho}g\in I^{-k,\dv_0}_{2n,6}(H)$$ has a unique solution for $|\rho|$ sufficiently large, with $\|w\|\le C\|G_\rho g\|$. By Prop. 2.5, $M_q: I^{-k,\dv_0}_{2n,6}(H)\rta I^{\mu+\epsilon,\dv_0}_{2n,6}(H)$ for any $0<\epsilon<-\mu$. Note that $t=\mu+\epsilon-(-k)<1$, so we can use Prop. 2.5 and (2.9) with this value of $t$ to obtain \medskip $$I^{-k,\dv_0}_{2n,6}(H)\buildrel{M_q}\over{\rta} I^{\mu+\epsilon,\dv_0}_{2n,6}(H) \buildrel{G_{\rho}}\over{\rta} I^{-k,\dv_0}_{2n,6}(H)$$ with norm $\le C(q)|\rho|^{t-1}\rta 0$ as $|\rho|\rta\infty$. Hence, for $|\rho|$ sufficiently large, $||G_\rho M_q||<\frac12$ and $I+G_\rho M_q$ is invertible on $I^{-k,\dv_0}_{2n,6}(H)$.\qed \bigskip We may now complete the proof of Thm. 1 as described in \S1. Construct two solutions $\psi_j$, $j=1,2$, to (1.10) for potentials $q_j\in I^{\mu_j}(H_j)$. Note that $-q_j$, the right hand side of (1.10), belongs to $I^{\mu_j}(H)\hra I^{1-k_j-\sigma_j,\dv}_{2n,6}(H)$, with $\sigma_j>0$ since $\mu_j<1-k_j$. Thus, we may apply Prop. 2.6 with $g=-q_j, j=1,2$, and then form as above the corresponding solutions, $v_1(x,\rho_1)=e^{\rho_1\cdot x}(1+\psi_1(x,\rho_1))$ of $(\D+q_1)v_1=0$ and $w_2(x,\rho_2)=e^{\rho_2\cdot x}(1+\psi_2(x,\rho_2))$ of $(\D+q_2)w_2=0$, with $\|\psi_j\|_{I^{-k_j,\dv}_{2n,6}}\le C|\rho|^{-\sigma_j}$. The solutions $v_1$ and $w_2$ belong to $\xpr(\O)$, with $p,r$ as in (1.3). In fact, each is in $L^p(\O)$, and in $W^{2,r}(\O)$ away from $H$ by Prop. 2.2(i) and (iii), resp., since $r<\frac1{2(1-\nu)}$. Furthermore, since $q_1\in L^{\frac{k_1}{\nu_1}-\epsilon},\forall\epsilon>0$, we have $\D v_1=-q_1v_1\in \Bigl(L^{\frac{k_1}{\nu_1}-\epsilon}\Bigr)\times (L^s\Bigr),\forall\epsilon>0,\forall s<\infty$. Since $p>\frac{k_1}{k_1-\nu_1}\implies p'<\frac{k_1}{\nu_1}$, we thus have $\D v_1\in L^{p'}(\O)$, and similarly for $w_2$. These solutions are constructed for all large $|\rho|$. Since $n\ge 3$, for any $\xi\in\R^n\back 0$ and $\lambda\ge c|\xi|$, one can find $\rho_1,\rho_2\in\{\rho\cdot\rho=0\}$ with $|\rho_1|\simeq|\rho_2|\ge\lambda$ and $\rho_1+\rho_2=-i\xi$. By the assumption that $\CD_{q_2}=\CD_{q_1}$, there exists a $v_2\in\xpr(\O)$ such that (1.7) holds. Applying (1.8) and (1.9), it suffices to show that $$\int_\O e^{-i\xi\cdot x} (q_1-q_2)(\psi_1+\psi_2+\psi_1\psi_2)dx\rta 0\hbox{ as }\l\rta \infty.\tag2.11$$ Since each $q_j\in L^{\frac{k_j}{\nu_j}-\epsilon}(\O),\forall\epsilon>0$, and, by (1.3), $p>(\frac{k_j}{\nu_j})', j=1,2$, we have $$\int_{\O} |q_1-q_2|\cdot |\psi_j| dx\le\frac{c}{\lambda^{\epsilon_j}},\quad\epsilon_j>0$$ as $\l\rta \infty$ by H\"older's inequality. In fact, $p>2(\frac{k_j}{\nu_j})'$ by (1.3), so we have $$\int_{\O} |q_1-q_2|\cdot |\psi_1|\cdot |\psi_2| dx\le\frac{c}{\lambda^{\epsilon_{12}}},\quad \epsilon_{12}>0,$$ as well. Thus, we have shown that $\CD_{q_1}=\CD_{q_2}$ implies that $\widehat{q_1-q_2}(\xi)=0,\forall\xi$, and hence $q_1=q_2$, concluding the proof of Thm.1. \head {\bf 3. Reconstruction of the potential} \endhead In this section we prove that the potential $q$ can be obtained constructively from the Cauchy data of $\D+q$. We follow here the general method of [25]; see also [26] and [19]. However, there are some additional difficulties in our case because we deal with the set of Cauchy data instead of the Dirichlet-to-Neumann map. Moreover, we work with more complicated function spaces due to the singularities of the potential. We will show: \bigskip \proclaim{Theorem 2} Suppose that $\O$ is Lipschitz and $H\subset\subset\O$ is a submanifold of codimension $k$. Suppose further that $q\in I^{\mu}(H)$ is a real potential with $\nu_0(k)-k<\mu<1-k$ and $\supp(q)\subset\subset\O$. Let $p,r$ satisfy $2\le r0.\tag3.4$$ Arguing as in the proof of Prop. 2.6, but replacing the local finite-regularity spaces with the global spaces $Y^{m,s}_\delta(\R^n)$, we obtain the following result: \bigskip \proclaim{Proposition 3.1} If $q\in I^\mu(H)$ with $\frac23-k<\mu<1-k, s\le -2, -1<\delta<0$ and $0\le\sigma\le 1$, then the inhomogeneous equation $$(\Dr+q)w=g\in Y^{1-k-\sigma,s}_{\delta+1}(\R^n)\tag3.5$$ has, for $|\rho|$ large, a unique solution $w\in Y^{-k,s+1-\sigma}_\delta(\R^n)$ with $||w||\le\frac{C}{|\rho|^\sigma} ||g||$. \endproclaim \bigskip Next we will construct the boundary values of the exponentially growing solutions on $\partial \Omega$. For this purpose we use the Green's function $G_\rho^q(x,y)$ defined by $$ (\Delta+q)G_\rho^q(\cdotp,y)=\delta_y \quad\hbox{in}\quad \R^n, \tag3.6 $$ $$ e_\rho(\cdotp) G_\rho^q(\cdotp,y)\in Y^{-k,s}_{\delta}(\R^n), $$ where $y\in \R^n\setminus \overline\Omega$, $e_\rho(x)=\exp(-\rho\cdotp x)$ and $s<-n/2$. When $|\rho|$ is large enough, the equation (3.6) has a unique solution by Prop. 3.1. Next we consider the case when $\rho$ is fixed and sufficiently large. As $\supp(q)\subset\subset \O$, we see that $\p\O$ has a neighborhood $V$ such that in $V\times V$ Green's function $G_\rho^q(x,y)$ has the same singularities as the Green's function $G^0_0$ (for the zero potential and $\rho=0$), that is, $G_\rho^q(x,y)-G_0^0(x,y)\in C^\infty(V\times V)$. Using the Green's function (3.6) we define the corresponding single and double layer potentials $$ S_{q}\phi(y)=\int_{\p\O}G_\rho^q(x,y)\phi(x)dS_x,\quad K_{q}\phi(y)=\int_{\p\O} (\frac \p{\p n(x)} G_\rho^q(x,y))\phi(x)dS_x,\quad y\not\in \p\O $$ which define continuous operators $S_{q}:W^{1-\frac 1r,r}(\p\O)\to X|_\O \oplus W^{2,r}_{loc}(\R^n\setminus\O)$ and $K_{q}:W^{2-\frac 1r,r}(\p\O)\to X|_\O \oplus W^{2,r}_{loc}(\R^n\setminus\O)$. Here $X\subset {\Cal D}'(\R^n)$ is the space with the norm $||\chi_0\cdot f ||_{I^{m,\dv_0}_{2n,6}(H)} + ||\chi_\infty\cdot f||_{W^{2,r}_\delta(\R^n)}.$ These layer potentials can be considered as operators on the boundary $\p\O$, defined in principal value sense. Since $\p\O$ is Lipschitz, it follows from the results of [7] that these operators are continuous, $S^{\p\O}_{q}:W^{1-\frac 1r,r}(\p\O)\to W^{2-\frac 1r,r}(\p\O)$ and $K^{\p\O} _{q}:W^{2-\frac1r,r}(\p\O)\to W^{2-\frac 1r,r}(\p\O)$. % [{\bf I changed the order on the first $W$ space for %$K^{\p\O}$ from 2 to $2-\frac1r$ - please check.}] Similarly, on $\p\O$ we define normal derivatives of the layer potentials, $$ \p_n S^{\p\O}_{q}\phi(y)=\hbox{p.v.}\int_{\p\O} (\frac \p{\p n(y)} G_\rho^q(x,y)\phi(x)dS_x, $$ $$ \p_n K_{q}^{\p\O}\phi(y)=\hbox{p.v.}\int_{\p\O} (\frac \p{\p n(y)} \frac \p{\p n(x)} G_\rho^q(x,y))\phi(x)dS_x $$ which are continuous operators $\p_n S^{\p\O}_{q}:W^{1-\frac 1r,r}(\p\O)\to W^{1-\frac 1r,r}(\p\O)$ and $\p_n K^{\p\O}_{q}:W^{2-\frac 1r,r}(\p\O)\to W^{1-\frac 1r,r}(\p\O)$. Next we consider the Calder\'on projector [5]. We start with the operator $$ A_q( \phi,\psi)= \left(- S^{\p\O}_q\phi+(-\frac 12+K^{\p\O}_q)\psi, -(\frac 12+\p_n S_q^{\p\O})\phi+\p_n K_q^{\p\O}\psi\right) $$ \bigskip \proclaim{Proposition 3.2} Let $Z=W^{2-\frac 1r,r}(\p\O)\times W^{1-\frac 1r,r}(\p\O)$. Then the operator $$ A_q:Z/ \hbox{Ker}(A_q)\to Z $$ is semi-Fredholm. Moreover, $-A_q:Z\to Z$ is a projection operator with range $\CD_q$ and kernel independent of $q$. In particular, this shows that $\CD_q$ is a closed subspace of $Z$. \endproclaim \bigskip \fp Proof. First we show that kernel of $A_q$ does not depend on $q$. Assume that $(\phi,\psi)\in Z$. We consider the function $$ u_{\phi,\psi}(y)=-S_q(\phi)+K_q(\psi),\quad y\in \R^n\setminus \p \O $$ and the trace operators $$ T_+:W^{2,r}(\R^n\setminus \O)\to W^{2-\frac 1r,r}(\p\O)\times W^{1-\frac 1r,r}(\p\O), $$ $$ T_-:W^{2,r}(\O)\to W^{2-\frac 1r,r}(\p\O)\times W^{1-\frac 1r,r}(\p\O) $$ defined by $T_\pm u=(u|_{\p\O},\p_n u|_{\p\O})$. As the Green's functions $G^q_\rho(x,y)$ have the same singularities near $\p \O\times \p\O$ as the standard Green's function of $\R^n$, we can use the standard jump relations for layer potentials (see e.g. [8]). We conclude that $$ T_- u_{\phi,\psi}=A_q(\phi,\psi),\quad T_+ u_{\phi,\psi}=(\phi,\psi)+A_q(\phi,\psi). $$ Thus we get that $u=u_{\phi,\psi}\in (e_\rho)^{-1}Y_\delta^{-k,s}$ and it is the unique solution of $$ (\Delta+q)u=g_{\phi,\psi}=\psi\delta_{\p \O}+ \nabla\cdotp (n\phi \delta_{\p \O}) \quad\hbox{in}\quad \R^n, \tag3.7 $$ satisfying $ e_\rho(\cdotp) u\in Y^{-k,s}_{\delta}(\R^n).$ Now, if $(\phi,\psi)\in\hbox{Ker}(A_q)$ we have that $$ (\phi,\psi)=(\phi,\psi)+T_- u_{\phi,\psi}=T_+ u_{\phi,\psi}. $$ Thus, $v=u_{\phi,\psi}$ is the solution of the scattering problem $$ \Delta v=0 \quad\hbox{in}\quad \R^n\setminus\O, \tag3.8 $$ $$ T_+ v=(\phi,\psi),\quad e_\rho(\cdotp) v\in W^{s,2}_{\delta}(\R^n\setminus \O). $$ On other hand, assume that (3.8) has a solution, and let $v_0$ be the zero-continuation of $v$, that is $v_0|_{\R^n\setminus \O}=v,$ $v|_\O=0$. Then we conclude that $v_0$ is a solution of the problem (3.7), and as this solution is unique, we see that $v_0=u_{\phi,\psi}$. This shows that $(\phi,\psi)\in\hbox{Ker}(A_q)$ if and only if the problem (3.8) has a solution. This is obviously independent of $q$ and thus we see that $$ \hbox{Ker}(A_q)=\hbox{Ker}(A_0).\tag3.9 $$ Next we consider range of $A_q$. A standard application of Green's formula (see, e.g., [8, Th. 3.1]) shows that if $v\in X^{p,r}$ satisfies $$ (\Delta+q)v=0 \quad\hbox{in}\quad \O, $$ and $(\phi,\psi)=T_-v$, then $v=-u_{\phi,\psi}.$ (Observe the negative sign which is due to the fact that we use exterior normal vector $n$.) Also, by for $(\phi,\psi)\in Z$ we have $\chi_0u_{\phi,\psi}\in I^{-k,\dv_0}_{2n,6}(H)\subset L^{t_1}(\O)$ for any $t_1<\infty$ by Prop. 2.2. As $q\in I^{\mu}(H)\subset L^{t_2}(\O)$ for $1-\frac k\mu$. Hence $u_{\phi,\psi}\in X^{p,r}$. Thus the set of all solutions of the Schr\"odinger equation in $X^{p,r}$ equals to the set of solutions $u_{\phi,\psi},$ $(\phi,\psi)\in Z$. As $T_-u_{\phi,\psi}=A_q(\phi,\psi)$, we obtain that the range of $A_q$ equals to $\CD_q$. Now, when the potential is equal to zero, the Dirichlet-to-Neumann operator $\Lambda_0:u|_{\p\O}=\p_n u|_{\p\O}$ is well defined, $\Lambda_0:W^{2-\frac 1r,r}(\p\O)\to W^{1-\frac 1r,r}(\p\O)$. The Cauchy data $\CD_0$ is the graph of the operator $\Lambda_0$ and is thus closed. Thus we see that the range of $A_0$ is a closed subspace, and therefore the operator $$ A_0:Z/\hbox{Ker}(A_0)\to Z $$ has zero kernel and closed range. Thus it is a semi-Fredholm operator. Now, consider the operator $A_q-A_0$. Using (3.9) we know that the operator $$ A_q-A_0:Z/\hbox{Ker}(A_0)\to Z $$ is well defined and compact. As compact perturbations of semi-Fredholm operators are also semi-Fredholm, we conclude that $A_q$ is also semi-Fredholm. It remains to show that $-A_q$ is a projection. This can be seen similarly to the smooth case. Indeed, if $(\phi,\psi)\in \hbox{Ran}(A_q)$, $(\phi,\psi)=A_q(\tilde \phi,\tilde \psi)$ we see that the solution $u_{\tilde \phi,\tilde \psi}$ has the trace $T_-u_{\tilde \phi,\tilde \psi}=(\phi,\psi).$ Hence Green's formula gives $$ u_{\tilde \phi,\tilde \psi}=-(-S_q\phi+K_q\psi)\quad\hbox{in}\quad \O. $$ Taking trace $T_-$ from both sides we obtain that $(\phi,\psi)=-A_q(\phi,\psi)$, i.e. $(-A_q)^2=-A_q$. Thus, Prop. 3.2 is proven.\qed \bigskip Now we can construct the boundary values of the exponentially growing solutions from the Cauchy data. As we are given $\CD_q=\hbox{Ran}(A_q)$, and we know $\hbox{Ker}(A_q)=\hbox{Ker}(A_0)$, we can construct the projection $-A_q$. Next, let $(\phi_\rho,\psi_\rho)= T_+e_\rho$ be the boundary values of the incoming plane wave. Consider the solution $v(x)= e^{\rho\cdot x}(1+\psi(x,\rho))=e^{\rho\cdot x}+u_0$ and let $(\phi,\psi)=T_+u_0$. Then $(\phi,\psi)\in \hbox{Ker}(A_q)$ and $u_0=u_{\phi,\psi}$ in $\R^n\setminus \O$. Moreover, as $v$ is solution of Schr\"odinger equation inside $\O$, we have $$ T_+(e_\rho+u_{\phi,\psi})=(\phi_0+\phi,\psi_0+\psi)+ A_q(\phi,\psi)\in \hbox{Ran}(A_q).\tag3.9 $$ By applying with projection $I+A_q$ to (3.9) and using $A_q(\phi,\psi)=0$, we see that $$ 0=(1+A_q)(\phi_0+\phi,\psi_0+\psi)= (1+A_q)(\phi_0,\psi_0)+(\phi,\psi). $$ As $A_q$ and $(\phi_0,\psi_0)$ are known, we find $(\phi,\psi)$ and the Cauchy data of $v(x)$ on $\p \O$. So far, we have constructed the Cauchy data of the solutions $v_{\rho_1}(x)= e^{\rho_1\cdot x}(1+\psi(x,\rho_1))$ for all sufficiently large $\rho_1$. Thus if we consider complex frequencies $\rho_1$ and $\rho_2$ satisfying $\rho_1+\rho_2=-i\xi$, with $\xi\in\R^n\back 0$, an application of Green's formula yields $$ \eqalign{{\hat q}(\xi)&=\lim_{|\rho_1|\to \infty} \int_\O q(x) e^{\rho_1\cdot x} \Bigl(1+\psi_1(x,\rho_1)\Bigr)\cdotp e^{\rho_2\cdot x}\,dx \cr =&\lim_{|\rho_1|\to \infty} \int_{\p \O}\Bigl( v_{\rho_1}\cdotp \p_ne^{\rho_2\cdot x} - \p_n v_{\rho_1}\cdotp e^{\rho_2\cdot x} \Bigl)\,dx \cr} $$ This proves Theorem 2.\qed \head {\bf 4. Non-uniqueness for highly singular potentials} \endhead We next discuss how very strong singularities of the potential can cause non-uniqueness in a closely related inverse problem. Due to the strength of the singularities, the Schr\"odinger equation has to be interpreted in a weak sense. Let us consider the boundary value problem $$ (\Delta+q+E)u=0\quad \hbox{in}\quad \Omega,\tag4.1 $$ $$ u|_{\p\O}=f $$ with the potential $q$ having the form $$ q(x)=-\dist(x,H)^{\mu}c_0(x),\tag 4.2 $$ near $H$, where $\dist$ is the Euclidean distance, $H$ is a closed hypersurface bounding a region $\O_0\subset\subset\O$, $\mu<-2$, and $c_0(x)$ is a smooth function, satisfying $$ c_0(x)>C_0>0 $$ in some neighborhood $V$ of $H$. Elements of $ I^{-1-\mu}(H)$ satisfy the pointwise estimate $|q(x)|\le C \dist (x,H)^\mu$, but a $q$ satisfying (4.2) is not even locally integrable and thus need not define a distribution. Hence, the solutions of (4.1) cannot be formulated in the usual sense of distributions. Instead, we define the solution of (4.1) (if it exists) to be the solution of the following convex minimization problem: Find $u$ such that $$ G(u)=\inf G(v)\tag4.3 $$ where $G=G_{q+E}:\{v\in H^1(\O):\ v|_{\p\O}=f\}\to \R\cup\{\infty\}$ is the convex functional $$ G_{q+E}(v)=\int_\O (|\nabla v(x)|^2-(q(x)+E) |v(x)|^2)dx $$ Here, since the function $-q(x)$ is bounded from below, we define $G(u)=\infty$ when $q|v|^2$ is not in $L^1(\O)$. \bigskip \proclaim{Proposition 4.1} The Cauchy data $$ \CD_{q+E}= \Big\{(u|_{\po},\frac{\partial u}{\partial n}|_{\po}): u\in H^1(\O),\ u\hbox{ is a minimizer of }G_{q+E}\}. $$ does not depend on $q$ in $\O_0$. In particular, if the solution of (4.3) is unique, $u$ vanishes identically in $\O_0$. \endproclaim \bigskip {\bf Remark.} We note that potentials having singularities similar to (4.2) as above has been used to produce counterexamples to strong unique continuation, e.g. potentials $q(x)=c/\vert x\vert \sp {2+\varepsilon}$ in [11]. Recently, counterexamples have been found for weak unique continuation for $L^1$-potentials [18], but here we need to construct potentials for which {\it all} solutions vanish inside $H$. Finally, we wish to emphasize that since the solutions of (4.1) considered here are not defined in the usual sense of distributions, but rather as solutions of a convex minimization problem, the solutions we construct do not give new counterexamples for the unique continuation problem. \bigskip \fp{\bf Proof.} Obviously we can assume that $q(x)\leq 0$ everywhere. We start first with the case where $E=0$ and $f\in C^\infty(\p \O)$. As the potential $q$ is not in the Kato class ([6,p.62]), consider instead a decreasing sequence of smooth functions $q_n\in C^\infty(\O)$, $q_{n+1}(x)\leq q_n(x)$, for which $q_n(x)=q(x)$ when $d(x,H)>\frac 1n$ and in some neighborhood $V$ of $H$ $$ q_n(x)\leq \max(-c_1 n^{-\mu},q(x))\tag4.4 $$ where $00$ we have $$ -\int_\O q_{n_0}(x) |\tilde u(x)|^2dx =\lim_{n\to\infty} \int_\O (-q_{n_0}(x)) |u_n(x)|^2dx\leq $$ $$ \leq \lim_{n\to\infty} \int_\O (-q_{n}(x)) |u_n(x)|^2dx\leq C_2. $$ Since this is valid for any $n_0$ we have by monotone convergence theorem $$ -\int_\O q(x) |\tilde u(x)|^2dx \leq C_2. \tag4.7 $$ As $G\leq G_n$, $$ \inf G(u)\leq \lim_{n\to\infty} \min G_n(u)=C_1+C_2. $$ Now, by (4.6) and (4.7) we have $$ G(\tilde u)\leq C_1+C_2 $$ and thus $\tilde u$ is a minimizer of $G$. Now, as $G=G_n+(G-G_n)$ where $G_n$ is a strictly convex functional and $G-G_n$ is a convex functional, $G$ is strictly convex. Thus the minimizer is unique. Hence we see that, for every $f$, the solution $\tilde u$ of the minimization problem (4.3) exists, is unique, and is given as the $L^2$-limit of the functions $u_n$. We note that the above analysis was based to the fact that the minimization problems for the $G_n$ epi-converge to the minimization problem for $G$ [30]. \bigskip Recalling that $\O_0$ is the region bounded by $H$, consider functions $u_n$ restricted to $\O_0$. Let $t\mapsto B_t$ be the Brownian motion in $\R^n$ starting from $x$ at time $t=0$, i.e., $B_0=x$. As the $q_n$ are strictly negative smooth functions, they are in the Kato class and the pair $(\O,q_n)$ is gaugeable (see [6], sect. 4.3 and Th. 4.19). By [6], Th. 4.7, the solution $u_n$ can be represented by the Feynman-Kac formula $$ u_n(x)=E(\exp\left(\int_0^\tau q_n(B_t)dt\right)f(B_\tau)) $$ where $\tau=\tau_{\p\O}$ is the first time when the process hits the boundary, i.e., $B_t\in \p \O$. Here, we assume $B_t$ is a version of Brownian motion for which all realizations are continuous curves (see [21] or [6], Th. 1.4). If $x\in \O_0$, the realizations of Brownian motion have to hit $H$ prior to hitting $\p \O$. Denote the first hitting time for $H$ by $\tau_H$; thus the first hitting point is $B_{\tau_H}$, and $\tau_H<\tau_{\p\O}$. (The stopping time $\tau_H$ is measurable function in the probability space, see [6,Prop. 1.15]). Let us now denote by $p(\rho,\eta)$ the probability that the Brownian motion sent from origin at time $t=0$ leaves the origin centered ball with radius $\rho$ before time $\eta$. Because of the scale-invariance of Brownian motion, $p(s\rho,s^2\eta)=p(\rho,\eta)$ for $s>0$. (Indeed, let us consider reparametrized Brownian motion $\tilde B_t=sB_{s^{-2}t}$. As the probability densities of $(\tilde B_{t_1},\tilde B_{t_2},\dots,\tilde B_{t_m})$ coincide to those of $(B_{t_1},B_{t_2},\dots,B_{t_m})$ we see that we see that $\tilde B_t$ is Brownian motion, too.) Let $A_{\rho,\eta}=\{|B_t-B_{\tau_H}|<\rho\hbox{ for }\tau_H\leq t<\tau_H+\eta\}$. %[{\bf I changed $\eta$ to $\rho$ in the definition of the %set - please check.}] This set is measurable in the probability space and the probability of $A_{\rho,\eta}$ is $P(A_{\rho,\eta})=1-p(\rho,\eta)$. Let $m>1$ and $\eta=\eta(m)$ be such that $p(1,\eta)\geq \frac {m-1}m$. Now, $q$ is non-positive and by (4.4) $q_n(x)<\max(-c_1n^{-\mu},q(x))$ in some neighborhood $V$ of $H$. When $s$ is so small that the $s$-neighborhood of $H$ is in $V$, we have by (4.2) that $$\eqalign{ |E(\exp\left(\int_0^\tau q(B_t)dt\right)f(B_\tau))|\leq& E(\exp\left(\int_{\tau_H}^{\tau_H+s^2\eta} q(B_t)dt\right)||f||_{L^\infty})\cr \leq& (1-P(A_{s\rho,s^2\eta(m)}))||f||_{L^\infty} \cr +& P(A_{s\rho,s^2\eta(m)}) \exp\bigg(-s^2\eta(m) \min(C_0s^\mu,c_1n^{-\mu})\bigg)||f||_{L^\infty}.} $$ Thus, choosing $s=n^{2/\mu}$ we see that for sufficiently large $n$ $$ ||u_n||_{L^\infty(\O_0)}\leq (\frac 1m+\frac {m-1}m \exp\left(-\eta(m)c_1n^{2(2+\mu)/\mu}\right)||f||_{L^\infty}. \tag4.8 $$ As $u_n\to \tilde u$ in norm in $L^2(\O_0)$, $$ ||\tilde{u}||_{L^2(\O_0)}\leq \frac 1m ||f||_{L^\infty}\,\hbox{vol}(\O_0)^{1/2}. $$ %[{\bf I changed $u$ to $\tilde{u}$ above -please check.}] for any $m$. Thus we see that $\tilde u=0$ in $\O_0$. Next we consider the case when $E\in \R$ and $f\in H^{1/2}(\p\O)$. First, let $H_r=\{x\in \O:\ \dist(x,H)