Content-Type: multipart/mixed; boundary="-------------0109280336854" This is a multi-part message in MIME format. ---------------0109280336854 Content-Type: text/plain; name="01-341.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-341.comments" MSC-class: 81S99 ---------------0109280336854 Content-Type: text/plain; name="01-341.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-341.keywords" supersymmetry, zero modes ---------------0109280336854 Content-Type: application/x-tex; name="fig3.pstex_t" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fig3.pstex_t" \begin{picture}(0,0)% \epsfig{file=fig3.pstex}% \end{picture}% \setlength{\unitlength}{4144sp}% % \begingroup\makeatletter\ifx\SetFigFont\undefined% \gdef\SetFigFont#1#2#3#4#5{% \reset@font\fontsize{#1}{#2pt}% \fontfamily{#3}\fontseries{#4}\fontshape{#5}% \selectfont}% \fi\endgroup% \begin{picture}(2014,2079)(214,-1378) \put(505,266){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}\special{ps: gsave 0 0 0 setrgbcolor}$IV$\special{ps: grestore}}}} \put(1062,-212){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}\special{ps: gsave 0 0 0 setrgbcolor}$I$\special{ps: grestore}}}} \put(505,-1008){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}\special{ps: gsave 0 0 0 setrgbcolor}$IV$\special{ps: grestore}}}} \put(1858,-1008){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}\special{ps: gsave 0 0 0 setrgbcolor}$IV$\special{ps: grestore}}}} \put(266,-251){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}\special{ps: gsave 0 0 0 setrgbcolor}$III$\special{ps: grestore}}}} \put(2097,-251){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}\special{ps: gsave 0 0 0 setrgbcolor}$III$\special{ps: grestore}}}} \put(1858,266){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}\special{ps: gsave 0 0 0 setrgbcolor}$IV$\special{ps: grestore}}}} \put(1380,266){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}\special{ps: gsave 0 0 0 setrgbcolor}$II$\special{ps: grestore}}}} \put(2097,-530){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}\special{ps: gsave 0 0 0 setrgbcolor}$x$\special{ps: grestore}}}} \put(1340,545){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}\special{ps: gsave 0 0 0 setrgbcolor}$y$\special{ps: grestore}}}} \put(1301,-1167){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}\special{ps: gsave 0 0 0 setrgbcolor}$II$\special{ps: grestore}}}} \put(1659,-530){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}\special{ps: gsave 0 0 0 setrgbcolor}$M$\special{ps: grestore}}}} \put(1261,-132){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}\special{ps: gsave 0 0 0 setrgbcolor}$M$\special{ps: grestore}}}} \end{picture} ---------------0109280336854 Content-Type: application/x-tex; name="xy3.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="xy3.tex" \documentclass[12pt,fleqn]{article} \usepackage{epsfig} \usepackage{graphics} \usepackage{latexsym} \usepackage{amssymb} \usepackage{amsmath} \setlength{\textwidth}{16cm} \setlength{\textheight}{24cm} \setlength{\topmargin}{-1.5cm} \addtolength{\evensidemargin}{-1.5cm} \addtolength{\oddsidemargin}{-1.5cm} \def\i{\mathrm{i}} \def\e{\mathrm{e}} \def\Qm{Q_\mathrm{min}} \def\Qmc{\overline{Q}_\mathrm{min}} \def\Hm{H_\mathrm{min}} \def\Hmc{\overline{H}_\mathrm{min}} \def\slim{\mathop{\mathrm{s-}}\!\lim} \newcommand{\ben}{\begin{displaymath}} \newcommand{\een}{\end{displaymath}} \newtheorem{lemma}{Lemma} \newtheorem{theorem}[lemma]{Theorem} \newtheorem{remark}[lemma]{Remark} \newtheorem{proposition}[lemma]{Proposition} \setcounter{secnumdepth}{2} \title{No zero energy states for the supersymmetric $x^2y^2$ potential} \author{ G.M. Graf${}^{(a)}$, D. Hasler${}^{(a)}$, J. Hoppe${}^{(b)}$ \\ \vspace*{-0.05truein} \\ \normalsize\it ${}^{(a)}$ Theoretische Physik, ETH-H\"onggerberg, CH--8093 Z\"urich\\ \normalsize\it ${}^{(b)}$ QFT, HU-Berlin, Invalidenstrasse 110, D--10115 Berlin} \begin{document} \maketitle \vspace{0.4cm} \begin{abstract} We show that the positive supersymmetric matrix-valued differential operator $H={p_x}^2 + {p_y}^2 + x^2y^2 + x \sigma_3 + y \sigma_1$ has no zero modes, i.e., $H \psi = 0$ implies $\psi =0$. \end{abstract} %\vspace{0.4cm} \section{Introduction} \label{sec:intr} The Hamiltonian of the model is plainly given as \ben H={p_x}^2 + {p_y}^2 + x^2y^2 + x \sigma_3 + y \sigma_1 \ , \een acting on the Hilbert space $\mathcal H ={\mathrm L}^2(\mathbb R^2) \otimes \mathbb C^2$, where $\sigma_i$ are the Pauli matrices. The supercharge is \begin{equation} Q = p_x \sigma_3 - p_y \sigma_1 - xy \sigma_2\ . \label{q} \end{equation} Together with the reflection \ben ( P \psi)(x,y) = \frac{1}{\sqrt{2}}(\sigma_1 +\sigma_3) \psi (y,x) \ , \een the system ($H,P,Q$) exhibits supersymmetry: \ben H=Q^2\ ,\qquad P^2 = 1\ ,\qquad QP + PQ = 0 \ . \een It was shown in \cite{dewitetal}\footnote{The Pauli matrices used there differ from ours by an irrelevant unitary conjugation.} that the spectrum of $H$ is $\sigma(H)=[0,\infty)$. The question whether $0$ is an eigenvalue of $H$ has however so far eluded a definite answer --- despite some efforts in this direction such as \cite{ak}. The precise definition of the model is as follows: $Q$ is the closure of (\ref{q}) on $C_0^{\infty}\otimes \mathbb C^2$, where $\mathcal{C}_0^{\infty} \equiv \mathcal{C}_0^{\infty}(\mathbb{R}^2)$, and $H$ is the self-adjoint operator associated to the positive symmetric quadratic form $q(\varphi,\psi) = (Q \varphi, Q \psi)$ on $\mathcal{D}(Q) \times \mathcal{D}(Q)$. Hence $C_0^{\infty}\otimes \mathbb C^2$ is a form core for $q$. The main result is the following. \begin{theorem} \label{nozm} Let $\psi \in \mathcal{D}(H)$ with $H \psi = 0$. Then $\psi = 0$. \end{theorem} The proof relies on a simple commutator argument. Let $f = f(x,y)$ be real-valued. Since $H \psi = 0$ implies $Q \psi = 0 $ we then have \begin{eqnarray*} ( \psi, i[Q,f] \psi) & = & \i\{ (Q \psi,f\psi) - (f \psi , Q \psi ) \} = 0 \ ,\\ \i [Q,f] & = & \sigma_3 \partial_x f - \sigma_1 \partial_y f \end{eqnarray*} for `any' function $f$. In particular for $f = (x^2 - y^2)/2$ we get \ben (\psi, (x \sigma_3 + y \sigma_1 ) \psi ) = 0 \ . \een Subtracting this from $(\psi,H \psi) = 0$ we obtain \ben (\psi, (p_x^2 + p_y^2 + x^2 y^2) \psi ) = 0 \ , \een which is impossible unless $\psi = 0$, since $p_x^2 + p_y^2 + x^2 y^2 > 0$ \cite{bsimon1}. The trouble with this is that $\psi$ may fail to be in the domain of $f = (x^2 - y^2)/2$ or, what matters more, in that of $\i[Q,f] = x \sigma_3 + y \sigma_1$. The proof given in the next section will circumvent this difficulty. We stress that any natural realization of $Q$ and $H$ coincides with the one given above. This is the content of the following statement, which is however not needed for the main result. \begin{proposition} \label{char} \ \begin{description} \item[(a)] Q is self-adjoint, and its domain is \ben \mathcal{D}(Q) = \{ \psi \in \mathcal{H} \ | \ Q \psi \in \mathcal{H} \textrm{ (in the sense of distributions on $C_0^{\infty}\otimes \mathbb C^2$)} \} \ . \een \item[(b)] $\mathcal{D}(H) = \{ \psi \in \mathcal{H} \ | \ H \psi \in \mathcal{H} \textrm{ (in the sense of distributions on $C_0^{\infty}\otimes \mathbb C^2$)} \}$. \item[(c)] $C_0^{\infty}\otimes \mathbb C^2$ is an operator core for $H$. \item[(d)] $H=Q^2$, i.e., $\psi \in \mathcal{D}(H)$ iff $\psi \in \mathcal{D}(Q)$ and $Q \psi \in \mathcal{D}(Q)$, in which case $H \psi =Q^2\psi$. \end{description} \end{proposition} Before turning to the proofs we give a simple argument showing that a possible zero mode of $H$ cannot be unique, though this statement is superseded by Theorem \ref{nozm}. The operators on $\mathcal{H}$ \ben (P_1\psi)(x,y)=\sigma_1\psi(-x,y)\ ,\; (P_2\psi)(x,y)=\sigma_2\psi(-x,-y)\ ,\; (P_3\psi)(x,y)=\sigma_3\psi(x,-y) \een satisfy $P_i^2=1,\,[P_i,P_j]=2\i\varepsilon_{ijk}P_k$ and $[Q,P_i]=0$. Thus $Q\psi=0$ implies $QP_i\psi=0$ and, if uniqueness is assumed, $P_i\psi=s_i\psi$ with $s_i=\pm 1$. But this contradicts the commutation relations of the $P_i$. A related argument shows that the index $\textrm{tr}( P \Pi )$, where $\Pi$ is the ground state projection of $H$, vanishes. This is seen from $P P_2 = - P_2 P$ and \ben \mathrm{tr} ( P \Pi ) = \mathrm{tr} ( P \Pi P_2^2 ) = - \mathrm{tr} ( P \Pi P_2^2 ) \ , \een where one power of $P_2$ has been turned around the trace. Finally we remark that absence of zero energy states had been suggested by the asymptotic analysis of \cite{fghhy} and, for a slightly less elementary model, been proven in \cite{fh} by different means. \section{Proofs} \label{sec:pr} {\bf Proof of Theorem \ref{nozm}.} Let \begin{eqnarray*} h(x) = \left\{ \begin{array}{ll} -Mx -\frac1 2 M^2 & \ \ \ (x \le -M) \\ \frac1 2 x^2 & \ \ \ (-M \le x \le M) \\ Mx - \frac1 2 M^2 & \ \ \ (x \ge M) \end{array} \right. \ , \end{eqnarray*} whence \begin{eqnarray*} {h}'(x) & = & \left\{ \begin{array}{ll} -M & \ \ \ (x \le -M) \\ x & \ \ \ (-M \le x \le M) \\ M & \ \ \ (x \ge M) \end{array} \right. \\ & = & x - g(x) \ , \end{eqnarray*} where we have defined $g$ as \begin{eqnarray*} g(x) = \left\{ \begin{array}{ll} x + M & \ \ \ (x \le -M) \\ 0 & \ \ \ (-M \le x \le M) \\ x - M & \ \ \ (x \ge M) \end{array} \right. \ . \end{eqnarray*} Let furthermore $h_{\epsilon}(x) = h(x)\e^{-\epsilon \sqrt{1 + x^2}}, \, f_{\epsilon}(x,y) = h_{\epsilon}(x) - h_{\epsilon}(y)$. For $\varphi, \psi \in \mathcal{D}(Q)$ we have \begin{equation} \label{qf-fq} \i[ ( Q\varphi,f_{\epsilon} \psi) - ( f_{\epsilon} \varphi, Q \psi ) ] = (\varphi,( \sigma_3 \partial_x f_{\epsilon} - \sigma_1 \partial_y f_{\epsilon} ) \psi ) \ . \end{equation} This equality is straightforward for $\varphi,\psi \in \mathcal{C}_0^{\infty}\otimes \mathbb{C}^2$ and extends to $\mathcal{D}(Q)$, since the operator on the right side is bounded and $\mathcal{C}_0^{\infty}\otimes \mathbb{C}^2$ is an operator core for $Q$. By dominated convergence, \begin{equation} \label{slim} \slim_{\epsilon \downarrow 0} ( \sigma_3 \partial_x f_{\epsilon} - \sigma_1 \partial_y f_{\epsilon} ) = {h}'(x) \sigma_3 + {h}'(y) \sigma_1 \ . \end{equation} Subtracting the r.h.s. from $H$, we obtain \begin{eqnarray} H - {h}'(x) \sigma_3 - {h}'(y) \sigma_1 & = & p_x^2+ p_y^2 + x^2y^2 + g(x)\sigma_3 + g(y) \sigma_1 \nonumber \\ & \ge & p_x^2 + p_y^2 + x^2y^2 - |g(x)| - |g(y)| \equiv H_M \ . \label{lb} \end{eqnarray} The inequality is understood in the sense of forms \cite{rs4,k}, where $H_M$ is the self-adjoint operator associated to the corresponding form with form core $\mathcal{C}_0^{\infty}\otimes \mathbb{C}^2$. We claim that for $M$ large enough $H_M$ is positive, i.e., $H_M > 0$. Now let $\psi \in \mathcal{D}(H)$ with $H \psi = 0$. This implies $Q \psi = 0$ and, by (\ref{qf-fq}) and (\ref{slim}), \ben (\psi, ( H - {h}'(x) \sigma_3 - {h}'(y) \sigma_1 ) \psi ) = 0 \ . \een By (\ref{lb}) and the claim this is possible only if $\psi = 0$ and the theorem follows. To prove the claim, we consider the partition of $\mathbb{R}^2$ as in the figure \begin{figure}[h] \begin{center} \input{fig3.pstex_t} \caption{Partition of $\mathbb{R}^2$} \label{fig:GL1} \end{center} \end{figure} and introduce Neumann conditions along the boundaries. Then (\cite{rs4}, XIII.15, Proposition 4) \ben H_M \ge H_I + H_{II} + H_{III} + H_{IV} \ , \een where the forms \begin{eqnarray*} & H_{I} & = p_x^2 + p_y^2 + x^2 y^2 \\ & H_{II} & = p_x^2 + p_y^2 + x^2 y^2 - |y| + M \\ & H_{III} & = p_x^2 + p_y^2 + x^2y^2 - |x| + M \\ & H_{IV} & = p_x^2 + p_y^2 + x^2y^2 - |x| - |y| + 2M \end{eqnarray*} act on the corresponding regions. We show that $H_a > 0$ for $a = I,\ldots IV$ and $M$ large enough. $I$. We have $H_I \ge 0$, and $(\psi, H_I \psi) = 0$ implies $(\psi, x^2 y^2 \psi)=0$ and hence $\psi = 0$. $II$. The operator $p_x^2 + x^2$ on ${\mathrm L}^2(-a,a)$ with Neumann boundary conditions at $x=\pm a$ satisfies \ben p_x^2 + x^2 \ge 1 - Ca^{-2} \ , \een where $C$ denotes a generic constant. This can e.g. be seen by means of a partition of unity $j_1^2 +j_2^2 =1$ with $j_1=j_1(x/a)$ equal to $1$ near $x/a=0$ and to $0$ near $x/a=\pm 1$. Hence, by scaling, $p_x^2 + x^2y^2$ on ${\mathrm L}^2(-M,M)$ is estimated from below as \ben p_x^2 + x^2y^2 \ge |y| ( 1 - C(|y|^{1/2}M)^{-2}) = |y| - C M^{-2} \ . \een As a result, $H_{II} \ge M - CM^{-2}$, which is positive for $M$ large enough. $III$. Is analogous to case $II$. $IV$. There we have $x^2y^2 \ge M^3|x|,\,M^3|y|$ and hence \ben x^2y^2 - |x| - |y| \ge \Bigl( \frac1 2 M^3 - 1 \Bigr) (|x| + |y|) \ge \Bigl( \frac1 2 M^3 - 1 \Bigr) 2 M \ , \een which is again positive for $M$ large enough. \medskip \noindent \noindent {\bf Proof of Proposition \ref{char}.} (a) Let $\mathcal{D}(\Qm) = \mathcal{C}_0^{\infty} \otimes \mathbb{C}^2$, then $\Qmc = Q$ by definition of $Q$ and \[ \mathcal{D}(\Qm^*) = \{ \psi \in \mathcal{H} \ | \ Q \psi \in \mathcal{H} \ \textrm{(in the sense of distributions on $\mathcal{C}_0^{\infty}\otimes \mathbb{C}^2$)} \} \ . \] Since $\Qm$ is symmetric, $\Qmc\subset \Qm^*$. We will show that \begin{equation} \Qm^*\subset \Qmc \ . \label{QQ} \end{equation} This implies $\Qmc=\Qm^*=\Qmc^*$. It remains to prove (\ref{QQ}). We pick $f \in \mathcal{C}_0^{\infty}$ with $f(0) = 1$, such that $\slim_{n \to \infty}f_n = 1,\ \| \nabla f_n\|_{\infty} \to 0$ for $f_n(\vec{x}) = f(\vec{x}/n)$, and set $\tilde{f}_n (\vec{p})=f(\vec{p}/n^2)$. We approximate a given $\psi\in\mathcal{D}(\Qm)$ by $\psi_n= f_n \tilde{f}_nf_n\psi\in\mathcal{C}_0^{\infty}\otimes\mathbb{C}^2$. with $n\to\infty$. We have \ben Q \psi_n = f_n\tilde{f}_n f_nQ \psi + [ Q, f_n \tilde{f}_nf_n] \psi\ , \een as one checks by taking inner products with $\varphi\in \mathcal{C}_0^{\infty}\otimes\mathbb{C}^2$. Here \begin{eqnarray*} [Q,f_n \tilde{f}_nf_n]&=& [Q,f_n] \tilde{f}_nf_n + f_n [Q,\tilde{f}_n]f_n+f_n\tilde{f}_n[Q,f_n]\ , \\ {[Q,f_n] } & = & -\i(\partial_x f_n) \sigma_3 + \i (\partial_y f_n)\sigma_1\ , \\ {[Q,\tilde{f}_n]}& = & [xy,\tilde{f}_n] =\i\bigl( y(\partial_{p_x}\tilde{f}_n)+x(\partial_{p_y}\tilde{f}_n)\bigr) + \partial^2_{p_x p_y} \tilde{f}_n \end{eqnarray*} are bounded operators with $\|[Q,f_n]\|\to 0$ and, due to $|x|,\,|y| \le C n$ on supp$f_n$, \ben \| f_n [Q,\tilde{f}_n ] \| \le C n \cdot \frac{1}{n^2} + C \frac{1}{n^4} \to 0 \ . \een Hence $\psi_n\to \psi$ and $Q\psi_n \to Q\psi$, i.e., $\psi \in \mathcal{D}(\Qmc)$. (d) By definition of the operator $H$ associated to the form $q$ we have: $\psi\in \mathcal{D}(H)$ iff \begin{equation} \psi\in \mathcal{D}(Q) \quad\textrm{and}\quad\exists \varphi \in \mathcal{H} \ \forall\eta\in \mathcal{D}(Q):\ (Q\psi,Q\eta)=(\varphi,\eta)\ , \label{cond} \end{equation} in which case $H\psi=\varphi$. This condition is also equivalent to $Q\psi\in \mathcal{D}(Q^*)$, with $Q^*Q\psi=\varphi$ in case of validity. That proves (d). (b) In (\ref{cond}) one can replace $\mathcal{D}(Q)\ni\eta$ by the core $\mathcal{C}_0^{\infty} \otimes \mathbb{C}^2$, which proves (b) with $\mathcal{H}$ replaced by $\mathcal{D}(Q)$. Left to show is that if $\psi \in \mathcal{H}$ with $H \psi\, (= Q^2 \psi)\in\mathcal{H}$ (in the sense of distributions), then $Q \psi \in\mathcal{H}$ (in the same sense). By elliptic regularity, $f_n^2 Q \psi\in\mathcal{D}(Q)$ and \ben Qf_n^2 Q \psi=f_n^2Q^2\psi+2[Q,f_n]f_nQ\psi\ . \een This implies $\|f_n Q\psi\|^2\le C\|\psi\|(\|H\psi\|+\|f_nQ\psi\|)$ and hence that $\|f_nQ\psi \|$ is bounded. We conclude $Q\psi\in\mathcal{H}$ by monotone convergence. (c) Let $\mathcal{D}(\Hm) = \mathcal{C}_0^{\infty} \otimes \mathbb{C}^2$. By (b), $\Hm^*=H$ and thus $\Hmc = \Hm^{**}= H^*=H$. \medskip \noindent \noindent {\bf Acknowledgments.\/} We thank J. Fr\"ohlich for useful discussions. \begin{thebibliography}{99} \bibitem{dewitetal} B. de Wit, M. L\"uscher, H. Nicolai, The supermembrane is unstable, Nucl. Phys. B {\bf 320}, 135-159 (1989). \bibitem{ak} A. Koubek, Das Potential $x^2y^2$ und seine supersymmetrische Erweiterung, diploma thesis, Universit\"at Hamburg (1990). \bibitem{bsimon1} B. Simon, Some quantum operators with discrete spectrum but classically continuous spectrum, Ann. Phys. {\bf 146}, 209-220 (1983). \bibitem{fghhy} J. Fr\"ohlich, G.M. Graf, D. Hasler, J. Hoppe, S.-T. Yau, Asymptotic form of zero energy wave functions in supersymmetric matrix models, Nucl. Phys B {\bf 567}, 231-248 (2000). \bibitem{fh} J. Fr\"ohlich, J. Hoppe, On zero-mass ground states in super-membrane matrix models. Comm. Math. Phys. {\bf 191}, 613-626 (1998). \bibitem{rs4} M. Reed, B. Simon, Methods of Modern Mathematical Physics, IV. Analysis of Operators, Academic Press, New York (1978). \bibitem{k} T. Kato, Perturbation Theory for Linear Operators, Springer (1966). \end{thebibliography} \end{document} ---------------0109280336854 Content-Type: application/postscript; name="fig3.pstex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fig3.pstex" %!PS-Adobe-2.0 EPSF-2.0 %%Title: fig3.pstex %%Creator: fig2dev Version 3.2 Patchlevel 1 %%CreationDate: Wed Sep 19 16:57:38 2001 %%For: gmgraf@babi.ethz.ch (Gian Michele Graf) %%Orientation: Portrait %%BoundingBox: 0 0 143 132 %%Pages: 0 %%BeginSetup %%EndSetup %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -13.0 140.0 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 3217 m -1000 -1000 l 3465 -1000 l 3465 3217 l cp clip 0.06299 0.06299 sc 7.500 slw % Ellipse n 1220 812 27 27 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr % Ellipse n 1618 1210 27 27 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr % Polyline gs clippath 2109 1183 m 2215 1210 l 2109 1237 l 2230 1237 l 2230 1183 l cp clip n 225 1210 m 2215 1210 l gs col0 s gr gr % arrowhead n 2109 1183 m 2215 1210 l 2109 1237 l 2109 1210 l 2109 1183 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 1193 321 m 1220 215 l 1247 321 l 1247 200 l 1193 200 l cp clip n 1220 2205 m 1220 215 l gs col0 s gr gr % arrowhead n 1193 321 m 1220 215 l 1247 321 l 1220 321 l 1193 321 l cp gs 0.00 setgray ef gr col0 s % Polyline 15.000 slw n 424 812 m 2016 812 l gs col0 s gr % Polyline n 1618 414 m 1618 2006 l gs col0 s gr % Polyline 7.500 slw n 424 812 m 2016 812 l gs col0 s gr % Polyline 15.000 slw n 424 1608 m 2016 1608 l gs col0 s gr % Polyline n 822 414 m 822 2006 l gs col0 s gr % Polyline 7.500 slw n 822 2006 m 822 414 l gs col0 s gr % Polyline n 424 1608 m 2016 1608 l gs col0 s gr % Polyline n 1618 414 m 1618 2006 l gs col0 s gr $F2psEnd rs ---------------0109280336854--