Content-Type: multipart/mixed; boundary="-------------9811071409760" This is a multi-part message in MIME format. ---------------9811071409760 Content-Type: text/plain; name="98-700.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-700.comments" To appear in the proceedings of the conference on Rigorous Results in Quantum Mechanics, Prague, June 1998 ---------------9811071409760 Content-Type: text/plain; name="98-700.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-700.keywords" Quantum dots, mesoscopic systems, quantum Hall effect ---------------9811071409760 Content-Type: application/x-tex; name="Prag_proc.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Prag_proc.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[twoside]{article} %\pagestyle{empty} \usepackage{epsf} \usepackage{rotating} \setlength{\textwidth}{125mm} \setlength{\textheight}{185mm} \setlength{\parindent}{8mm} \frenchspacing \setlength{\oddsidemargin}{0pt} \setlength{\evensidemargin}{0pt} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\sr{{\cal R}} \def\1{{\bf 1}} \def\d{{\rm d}} \def\A{{\bf A}} \def\Tr{{\rm Tr}} \def\C{{\bf C}} \def\R{{\bf R}} \def\E{{\cal E}} \def\TF{{\rm TF}} \def\MTF{{\rm MTF}} \def\mfr#1/#2{\hbox{${{#1} \over {#2}}$}} \def\const.{{\rm const.}} \def\beq{\begin{equation}} \def\eeq{\end{equation}} \def\beqa{\begin{eqnarray}} \def\eeqa{\end{eqnarray}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}{Lemma}[section] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \setlength{\unitlength}{1.0cm} \title{\bf Quantum dots\\ {\Large \bf A survey of rigorous results} \footnotetext{\noindent {To appear in the proceeedings of the conference on {\it Rigorous Results in Quantum Mechanics}, Prague, June 1998}}} \author{Jakob Yngvason\\Institut f\"ur Theoretische Physik, Universit\"at Wien\\ Boltzmanngasse 5, A 1090 Vienna, Austria\\} \date{} \maketitle \section{Introduction} Modern semiconductor technology has in recent years made it possible to fabricate ultrasmall structures that confine electrons on scales comparable to their de Broglie wavelength. If the confinement is only in one spatial direction such systems are called {\em quantum wells}. In {\em quantum wires} the electrons can move freely in one dimension but are restricted in the other two. Structures that restrict the motion of the electrons in all directions are called {\em quantum dots}. The number of electrons, $N$, in a quantum dot can range from zero to several thousand. The confinement length scales $R_{1}$, $R_{2}$, $R_{3}$ can be different in the three spatial dimensions, but typically $R_{3}\ll R_{1}\approx R_{2}\approx$ 100 nm. In models of such dots $R_{3}$ is often taken to be strictly zero and the confinement in the other two dimensions is described by a potential $V$ with $V(x)\to\infty$ for $|x|\to\infty$, $x=(x^1,x^2)\in\R^2$. A parabolic potential, $V=\mfr1/2\omega |x|^2$, is often used as a realistic and at the same time computationally convenient approximation. Quantum dots have potential applications in microelectronics and have been extensively studied both experimentally and theoretically. Apart from possible practical uses they are of great interest for basic quantum physics. Their parameters (strength and shape of the confining potential, magnetic field strength, number of electrons) can be varied in a controlled way and their properties can be studied by clever experimental techniques. This offers many possibilities to confront theoretical predictions with experimental findings. There exist by now many excellent reviews on the physics of quantum dots, e.g.\ \cite{Chak92}--\cite{Jac98}. In the present contribution the focus will be on some theoretical aspects that are only partly covered by these reviews, in particular on rigorous limit theorems \cite{LSY95}, \cite{LSYBi94} which apply to dots in high magnetic fields and/or with high electron density. A quantum dot with $N$ electrons is usually modeled by a Hamiltonian of the following form, acting on the Hilbert space \beq {\cal H}_{N}=\bigwedge\limits^N_1 L^2(\R^2; \C^2)\label{space}\eeq appropriate for two dimensional Fermions of spin $1/2$: \beq H_N = \sum \limits^N_{j=1} H^{(j)}_1 + \sum \limits_{1 \leq i < j \leq N} W( x_i - x_j), \label{HN}\eeq where $x_i \in \R^2$, $i=1,\dots,N$ and $H^{(j)}_1=1\otimes\cdots \otimes H_1\otimes \cdots\otimes 1$ ($H_{1}$ in the $j$-th place) with the one-body hamiltonian \beq H_1 = {\hbar^2 \over 2m_*} \left( {\rm i} \nabla - {e \over \hbar c} A(x) \right)^2 + V (x)+ g_* \left( {\hbar e \over 2m_{{\rm e}}c} \right) S_{3}B -C B.\label{H1} \eeq Here $A(x) =\mfr1/2(-Bx^2, Bx^1)$ is the vector potential of a homogeneous magnetic field of strength $B$ in the $x^3$-direction, $V$ is the confining potential, assumed to be continuous with $V(x)\to\infty$ for $|x|\to\infty$, and $S_{3}$ is the spin operator in $x^3$-direction. The parameters $m_*$ and $g_*$ are respectively the effective mass and the effective $g$-factor of the electrons, while $m_{{\rm e}}$ and $e$ are the bare values of the electron mass and electric charge, and $\hbar$ and $c$ have their usual meanings. The constant \beq C=\left( {\hbar e \over 2m_{{\rm e}} c} \right) \left({m_{{\rm e}}\over m_*} - {\vert g_*\vert \over 2} \right) \eeq has been introduced in (3) for convenience: Subtraction of $CB$ has the effect that the spectrum of the kinetic energy operator (including spin) $H^{\rm kin}_{1}=H_1-V$ starts at zero for all $B$, even if $m_{*}\neq m_{{\rm e}}$ and $g_{*}\neq 2$. The interaction potential $W$ represents the Coulomb repulsion between the electrons, modified by the properties of the surrounding medium. Usually it is simply taken to be \beq W(x_{i}-x_{j})=e_*^2\vert x_{i}-x_{j}\vert^{-1}\label{coul}\eeq where $e_*=e/\sqrt{\epsilon}$ with $\epsilon$ the dielectric constant, but some regularization of the bare Coulomb potential, e.g., \beq W(x_{i}-x_{j})=e_*^2\left[(|x_{i}-x_{j}|^2+\delta_{+}^2)^{-1/2} -(|x_{i}-x_{j}|^2+\delta_{-}^2)^{-1/2}\right] \label{modcoul}\eeq with $\delta_{-}>\delta_{+}>0$ \cite{EFK92}, or even a potential that depends not just on the differences $x_{i}-x_{j}$, may sometimes fit the effective interaction better. For the proof of some of the theorems below the important property of (\ref{coul}) is that $W$ is repulsive, of positive type and tends to zero at infinity; these features are shared by (\ref{modcoul}). Writing the Hamiltonian in the above form is, of course, an approximation, because the effect of the medium on the electrons is only taken into account through the modification of the parameters from their bare values. The size of a quantum dot ($\approx 100$ nm) is however usually much larger than the lattice constant of the medium where it resides ($<0.5$ nm), so this approximation is usually a good one. Quantum dots, especially such with few electrons, are sometimes referred to as {\em artificial atoms} with $V$ playing the role of the attractive nuclear potential in real atoms. The analogy is not perfect, however, because $V$ is regular around the origin in contrast to the potential from an atomic nucleus, and also because the electron interaction is the three dimensional Coulomb potential (\ref{coul}) (or modified Coulomb potential (\ref{modcoul})), while the motion is (essentially) restricted to two dimensions. But in many respects quantum dots can indeed be regarded as artificial atoms, with an important additional aspect: The effective parameters are to a certain extent tunable and may differ appreciably form their counterparts in real atoms. In a quantum dot the natural atomic unit of length is $a_*=\epsilon\hbar^2/(m_*e^2)$. Compared with the usual Bohr radius, $a_0=\hbar^2/(m_{{\rm e}}e^2)=0.53\times 10^{-1}$ nm, the length $a_*$ is typically large, e.g., $a_*\approx 185\ a_0\approx 10$ nm in GaAs. The natural energy unit is $E_* = e_*^2/a_*=e^4_* m_*/\hbar^2$, and in GaAs $E_*\approx 12$ meV, which should be compared with $E_{0}=e^2/a_0= e^4 m_{{\rm e}}/\hbar^2=27.2$ eV, i.e., $E_*\approx 4\times 10^{-4} E_{0}$. The natural unit, $B_*$, for magnetic field strength is the field at which the magnetic length $\ell_B=\hbar e/(B^{1/2}c)$ equals $a_*$, or equivalently, at which the cyclotron energy $\hbar eB_{*}/m_{*}c$ equals $E_{*}$. Hence $B_*=(a_0/a_*)^2B_0$, where $B_0=e^3m_{{\rm e}}^2c/\hbar^3=2.35\times 10^5$ T is the value corresponding to free electrons. If $a_0/a_*$ is small, $B_*$ can be much smaller than $B_0$. Thus $B_*\approx 7$ T in GaAs. This opens the very interesting possibility to study in the laboratory magnetic effects, whose analog for real atoms require field strengths prevailing only on neutron stars. On the experimental side the main techniques for studying quantum dots are {\em charge transport and capacitance spectroscopy} (\cite{Be91}, \cite{As96}, \cite{Kouw97}) and {\em optical far infrared spectroscopy} (\cite{Me93}, \cite{Heit97}). The former is in particular suited for measuring the $N$ dependence of ground state energies, but excited states can be investigated as well. The applicability of optical spectroscopy is to a certain extent limited by Kohn's theorem, to be discussed below, but refined techniques allow also to infer many properties by this method. Altogether it is fair to say that the energy spectrum of quantum dots and even some aspects of the corresponding wave functions are experimentally accessible with considerable precision. See also \cite{Zhin97} for recently discovered effects in charge transport spectroscopy that wait for an adequate explanation. On the theoretical side the spectral properties of the Hamiltonian (\ref{HN}) have been studied by a variety of methods, which can be roughly divided into the following categories: \begin{itemize} \item Exact analytic solutions \item Rigorous limit theorems \item Numerical diagonalizations \item Hartree and Hartree-Fock approximations \item Variational calculations \item Density functional methods \end {itemize} In the condensed matter literature the last four categories are by far the most prominent, but the present survey is mostly concerned with the first two, which lie within the realm of mathematical physics. It is impossible here to do any justice to the extensive physics literature on the theory of quantum dots, but an annotated bibliography of some representative references will be given in the last section. \section{Exact solutions} \subsection{The Fock-Darwin spectrum} From now on units are chosen such that $\hbar=e_{*}=m_{*}=B_{*}=1$. The Hamiltonian (\ref{H1}) can then be written \beq H_1 = {1\over 2} \left({\rm i} \nabla - A(x) \right)^2 + V (x)+ \gamma S_{3}B -\mfr1/2(1-|\gamma|) B\label{H11} \eeq with $\gamma=g_{*}m_{*}/(2m_{{\rm e}})$. For a confining potential of the harmonic oscillator form \beq V(x)=\mfr1/2 \omega |x|^2, \label{quadr}\eeq the eigenvalues and eigenfunctions of \beq H_{1}^{\rm orb}=\mfr1/2 \left( {\rm i} \nabla -A(x) \right)^2 + \mfr1/2 \omega^2 |x|^2\label{H12}\eeq were determined by Fock already in 1928 \cite{F28}, and also by Darwin in 1930 \cite{D30}. The Fock-Darwin spectrum of (\ref{H12}) consists of the eigenvalues \beq \varepsilon_{k,l}^{\rm FD}=(2k+|l|+1)\Omega-\mfr1/2 l B\eeq with \beq \Omega=\left(\mfr1/4 B^2+\omega^2\right)^{1/2},\eeq $k=0,\,1,\,2\dots$, $l=0,\,\pm1,\,\pm2,\dots$. The corresponding eigenfunctions are \beq \psi_{k,l}^{\rm FD}(x)={\rm(const.)}\exp({\rm i}l\varphi)r^{|l|}\exp(-\Omega r^2/2) L_k^{|l|}(\Omega r^2) \eeq where $x=(r\,\cos\varphi,r\,\sin \varphi)$ and $L_k^{|l|}$ is an associated Laguerre polynomial. These functions are also eigenfunctions of the angular momentum $L_{3}$ in the $x^3$-direction with eigenvalue $l$. Eigenfunctions with the same value of \beq n=k+\mfr1/2(|l|-l)\eeq are grouped together in a {\em Fock-Darwin level} (FDL). In the limit $\omega/B\to 0$ the eigenvalues in a FDL coalesce and a FDL becomes identical to a {\em Landau level} (LL) with the eigenvalues \beq \varepsilon_{n}^{\rm L}=(n+\mfr1/2)B\eeq and eigenfunctions \beq \psi_{k,l}^{\rm L}(x)={\rm(const.)}\exp({\rm i}l\varphi)r^{|l|} \exp(-B r^2) L_k^{|l|}(2Br^2)\label{Landau} \eeq The degeneracy of a LL per unit area is $B/(2\pi)$. If the interaction $W$ between the electrons is ignored the FD spectrum together with the Pauli principle, taking spin into account, completely solves the eigenvalue problem for $H_{N}$ in the case of the quadratic potential (\ref{quadr}). This approximation even fits some experimental data quite well \cite{Chak92}, \cite{Ka93}. When the interaction $W$ is taken into account this picture has to be modified, of course. For a quadratic confining potential, however, the FD spectrum continues to apply to the motion of the center of mass, independently of the interaction. This simple, but important fact \cite{Kohn61}, \cite{GoCho90} goes under the heading {\em Kohn's theorem}. The proof is essentially contained in the identity \beq N\sum_{j=1}^{N}x_{j}^2=\left(\sum_{j=1}^{N}x_{j}\right)^2 +\sum_{i1$} For $N=2$ the Hamiltonian for the relative coordinate $x=x_{1}-x_{2}$ is \beq H^{\rm rel}_{2}=({\rm i}\nabla-\mfr1/2A(x))^2+\mfr1/4\omega^2 |x|^2+W(x)\label{H2rel}\eeq In the case of a pure Coulomb interaction, $W(x)=1/|x|$, explicit formulas for eigenfunctions and eigenvalues of (\ref{H2rel}) have been found by Taut \cite{Tau95}. His approach is based on an ansatz for the wave functions of the form \beq\psi(x)=\exp({\rm i}m\varphi)\exp(-\rho^2/2)\rho^{|m|}P(\rho) \label{Tautans}\eeq where $P$ is a polynomial in $\rho=(\Omega/2)^{1/2}r$, with $\Omega=\left(\mfr1/4 B^2+\omega^2\right)^{1/2}$, $\varphi$ is the angular variable and $m\in{\bf Z}$. Solutions of the form (\ref{Tautans}) with $P$ a polynomial do not exist for arbitrary values of $B$, but Taut's method produces at least eigenfunctions and eigenvalues for a countable infinity of values of $\Omega$ which accumulate at 0. In fact, an ansatz like (\ref{Tautans}) in the eigenvalue equation $H^{\rm rel}_{2}\psi=E\psi$ with $P$ a power series, \beq P(\rho)=\sum_{\nu=0}a_{\nu}\rho^\nu,\eeq leads to \beq a_{\nu}=F({|m|},\nu,E')a_{0}\eeq with a certain recursively computable function $F$ and where $E'$ is related to $E$ by \beq E=\mfr1/4\Omega E'-\mfr1/2 mB.\label{EE'}\eeq The condition that $a_{\nu}$ vanishes for all $\nu$ larger than some positive integer $n$ is equivalent to the two equations \beq F({|m|},n,E',\Omega)=0\label{T1}\eeq and \beq E'=2({|m|}+n).\label{T2}\eeq For given $n$ and $m$ this gives one or more acceptable values for $\Omega$ and corresponding energy values \beq E=\mfr1/2(n+|m|)\Omega-\mfr1/2 mB.\eeq The solutions found in this way are not necessarily ground states of (\ref{H2rel}) but the position of $E$ in the spectrum can be inferred from the number of nodes of the corresponding wave function. The solutions of Taut seem so far to be the only known exact solutions for $N=2$ and the Coulomb interaction (\ref{coul}). They are limited to the special values of $\Omega$ defined by (\ref{T1}) and (\ref{T2}). For $W$ of the inverse square form \beq W(x)=\alpha\vert x\vert^{-2}\label{invsq}\eeq on the other hand, the Hamiltonian (\ref{H2rel}) can be exactly diagonalized fo all $B$ \cite{QuirJoh93}. In fact, addition of (\ref{invsq}) merely modifies the centrifugal term in the radial part of the FD Hamiltonian (\ref{H12}) and we obtain as eigenvalues of (\ref{H2rel}) \beq E=[2n+\mu+1]\Omega-\mfr1/2 mB\eeq with $\mu=(m^2+\alpha)^{1/2}$ not necessarily an integer, and the eigenfunctions \beq\psi(x)=\exp({\rm i}m\varphi)\exp(-\rho^2/2)\rho^{\mu}L_{n}^\mu(\rho^2). \label{Tautmod}\eeq It should be noted that the inverse square form (\ref{invsq}) for the effective interaction between the electrons is not necessarily less realistic than the pure Coulomb repulsion (\ref{coul}). In fact, the form (\ref{modcoul}) of the interaction, that is motivated by the situation in real dots \cite{EFK92}, has an inverse square decrease for large separation. At small separation, on the other hand, the effective interaction may be less singular than (\ref{coul}). It is therefore not entirely academic to consider also a harmonic interaction \cite{JohPay91} of the form \beq W(x_{i}-x_{j})=2W_{0}-\mfr1/2\beta|x_{i}-x_{j}|^2\label{harm}\eeq with positive parameters $W_{0}$ and $\beta$. For this case one can even solve the problem for {\em all} $N$ exactly, using (\ref{decoupl}) to decouple the oscillators. The result for the ground state energy $E^{\rm Q}(N,B)$ of $H_{N}$ with $N\geq 2$ and $\gamma=0$ (for simplicity) is \cite{JohPay91} \beq E^{\rm Q}(N,B)=\Omega+\mfr1/2(N-1)(N-2)\Omega_{0}(N)- \mfr1/4N(N+1)B+N(N-1)W_0\eeq with $\Omega_{0}(N)=(\Omega^2-N\beta^2)^{1/2}$. It is assumed that $\omega\geq N^{1/2}\beta$, so $\Omega_{0}(N)\in\R_{+}$ for all $B$. The corresponding wave function for the relative motion is \beq\psi(x)=\prod_{i2$, it is possible to analyze at least the ground state properties for large large $B$ or large $N$ by minimizing simple functionals of the electron density. In this analysis, which implies a drastic reduction of the quantum mechanical $N$-body problem, it is not necessary that the confining potential $V$ has the quadratic form (\ref{quadr}). From now on $V$ will stand for an arbitrary continuous function of $x\in\R^2$ tending to $\infty$ for $|x|\to \infty$; when additional properties are required these will be explicitly stated. It is no restriction to assume that $V\geq 0$. To be able to consider variations of the strength of the potential at fixed shape, we write \beq V(x)=Kv(x)\eeq with a coupling constant $K$. The interaction $W$ will for definiteness be assumed to be pure Coulomb with $e_{*}=1$, i.e., \beq W(x-y)=|x-y|^{-1},\eeq but other repulsive potentials of positive type could be treated similarly. The quantum mechanical ground state energy is \beq E^{\rm Q}(N,B,K)=\langle \Psi_{0},H_{N}\Psi\rangle,\eeq where $\Psi_{0}$ is a normalized ground state of $H_{N}$. The corresponding ground state electron density is \beq\rho^{\rm Q}_{N,B,K}(x)=\sum_{{\rm spins}\,\sigma_{k}=\pm1/2} \int|\Psi_{0}(x,\sigma_{1};x_{2},\sigma_{2};\dots, x_{N},\sigma_{N})|^2dx_{2}\cdots dx_{N}.\eeq We are concerned with the asymptotics of these quantities when one or more of the parameters $N$, $K$ and $B$ tends to $\infty$ with $v$ fixed. The large $B$ limit at fixed $N$ and $K$ is easiest and will be considered first. \subsection{High field limit} In the lowest Landau level (i.e., for $n=0$) the wave functions (\ref{Landau}) are essentially localized on scale $B^{-1/2}$, and the quantum mechanical kinetic energy vanishes after the spin contribution and the subtraction of $\mfr1/2(1-|\gamma|)$ in (\ref{H11}) have been taken into account. In the limit $B\to\infty$ it can therefore be expected that a classical model of $N$ point particles with the energy function \beq\E^{\rm P} [x_1,\dots,x_N]= \sum_{i=1}^N V(x_i)+\sum_{i0 $. The TF functional (\ref{TF}) and the classical functional (\ref{class}) are both limiting cases of the MTF functional (\ref{MTF}), for $B\to 0$ and $\to \infty$ respectively. More precisely, \beqa\lim_{B\to 0} E^{\MTF}(N,B,K)&=&E^{\rm TF}(N,K)\label{first}\\ \lim_{B\to 0}\rho^{\rm MTF}_{N,B,K}&=&\rho^{\rm TF}_{N,K}\eeqa and \beqa\lim_{B\to \infty} E^{\MTF}(N,B,K)&=&E^{\rm C}(N,K)\\ \lim_{B\to \infty}\rho^{\rm MTF}_{N,B,K}&=&\rho^{\rm C}_{N,K}.\label{last}\eeqa The limit for the densities should be understood in the weak $L_{1}$ sense, but for special $V$ much stronger convergence may hold. For instance, if $V$ is monotonically increasing with $|x|$, $\rho^{\rm C}_{N,K}$ is a bounded function, and $\rho^{\rm C}_{N,K}= \rho^{\rm MTF}_{N,B,K}$ for sufficiently large $B$, because $j_{B}(\rho^{\rm C}_{N,K})=0$ for $B>2\pi\Vert \rho^{\rm C}\Vert_{\infty}$. The MTF theory has two nontrivial parameters because of the {\bf scaling relations} \beqa E^{\MTF}(N,B,K)&=&N^2E^{\MTF}(1,B/N,K/N)\\ \rho^{\MTF}_{N,B,K}(x)&=&N\rho^{\MTF}_{1,B/N,K/N}(x).\eeqa Corresponding relations (without $B$) hold for the TF theory and the classical theory, and also for $E^{\rm P}$. A further important property of the densities is their compact support: For fixed $K/N$ the minimizers of $\E^{\TF}$, $\E^{\MTF}$, $\E^{\rm C}$ and also of $\E^{\rm P}$ have support in a disc whose radius is uniformly bounded in $N$ and $B$ (Lemma A.1 in \cite {LSY95}). Each minimizer satisfies a variational equation, which in the case of the MTF theory has an unusual form, since it consists really of inequalities. To state it compactly it is convenient to modify the definition (\ref{step}) slightly and regard $j'_{B}$ as an {\em interval valued} function if $2\pi\rho/B$ is an integer, namely, if $2\pi\rho/B=n$, then $j'_{B}$ is the closed interval $[(n-1)B,nB]$. The {\bf MTF equation} that is satisfied by $\rho^{\MTF}$ can then be written \beq \mu-V(x)-\rho*\vert x\vert^{-1} \left\{ \begin{array} {r@{\quad \hbox{\rm if}\quad} l} \in j'_B(\rho(x))& \rho(x)>0\\ \leq 0 & \rho(x)=0\end{array}\right.\eeq with a unique $\mu=\mu(N,B,K)$. Such generalized variational equations have been studied by Lieb and Loss \cite{LL}. If the potential is quadratic, $V(x)=K|x|^2$, there is an explicit formula (\cite{Shi91}, \cite{LSY95}) for the minimizer for $\E^{\rm C}$, which is equal to $\rho^{\rm MTF}$ for $B$ sufficiently large: \beq \rho^{\rm C}_{N,K}(x) = \left\{ \begin{array} {r@{\quad \hbox{\rm if}\quad} l} {3 \over 2 \pi} N\lambda \sqrt{1 - \lambda \vert x \vert^2} & \vert x \vert \leq \lambda^{-1} \\ 0 & \vert x \vert > \lambda^{-1}\end{array}\right.\label{rhoc}\eeq with $\lambda = (8K/3\pi N)^{2/3}$. The density profile has the shape of a half ellipsoid with a maximum at $x=0$. Note the difference between the two dimensional case considered here, and three dimensional electrostatics: In three dimensions the density would be homogeneously distributed in a ball. The criterion for $\rho^{\rm MTF}=\rho^{\rm C}$ is that $j_{B}(\rho^{\rm C}(0))=0$, which holds if \beq B\geq (6/3^{2/3}\pi^{5/3})K^{2/3}N^{1/3}.\label{cond}\eeq Numerically computed profiles of the minimizers $\rho^{\MTF}$ and the corresponding effective potentials \beq V_{\rm eff}(x)=V(x)+\rho^{\MTF}*\vert x\vert^{-1}\eeq with $V(x)=K|x|^2$ are shown in Fig.\ 1. The computations were carried out by Kristinn Johnsen. At the highest value of the field (Fig.\ 1(a)) condition (\ref{cond}) is fulfilled and $\rho^{\rm MTF}$ has the form (\ref{rhoc}). On the support of $\rho^{\MTF}=\rho^{\rm C}$ we have $V_{\rm eff}(x)=$constant$=\mu$. \begin{figure} \begin{center} \begin{picture}(12.5,12.5) % % Make rule to place ojects after % %\multiput(0,0)(0.1,0){150}{\line(0,1){0.1}} %\multiput(0,0)(0.5,0){30}{\line(0,1){0.2}} %\multiput(0,0)(1.0,0){16}{\line(0,1){0.3}} %\multiput(0,0)(0,0.1){110}{\line(1,0){0.1}} %\multiput(0,0)(0,0.5){22}{\line(1,0){0.2}} %\multiput(0,0)(0,1.0){12}{\line(1,0){0.3}} \epsfysize=3.0cm \put(-0.3,0.5){\epsfbox{ve2.eps}} \epsfysize=3.0cm \put(-0.3,3.0){\epsfbox{rho2.eps}} \put(0.0,1.3){\rotatebox{90}{$V_{{\rm eff}}$ $({\rm meV})$}} \put(0.0,3.7){\rotatebox{90}{$\rho$ $(10^{14}\mbox{m}^{-2})$}} \put(2.7,.1){$r$ (\mbox{nm})} \put(4.7,5.2){(c)} % \epsfysize=3.0cm \put(5.95,0.5){\epsfbox{ve0.eps}} \epsfysize=3.0cm \put(5.95,3.0){\epsfbox{rho0.eps}} \put(6.25,1.3){\rotatebox{90}{$V_{{\rm eff}}$ $({\rm meV})$}} \put(6.25,3.7){\rotatebox{90}{$\rho$ $(10^{14}\mbox{m}^{-2})$}} \put(8.95,.1){$r$ (\mbox{nm})} \put(10.95,5.2){(d)} % \epsfysize=3.0cm \put(5.95,6.9){\epsfbox{ve7.eps}} \epsfysize=3.0cm \put(5.95,9.4){\epsfbox{rho7.eps}} \put(6.25,7.7){\rotatebox{90}{$V_{{\rm eff}}$ $({\rm meV})$}} \put(6.25,10.1){\rotatebox{90}{$\rho$ $(10^{14}\mbox{m}^{-2})$}} \put(8.95,6.5){$r$ (\mbox{nm})} \put(10.95,11.6){(b)} % \epsfysize=3.0cm \put(-0.3,6.9){\epsfbox{ve8.eps}} \epsfysize=3.0cm \put(-0.3,9.4){\epsfbox{rho8.eps}} \put(0.0,7.7){\rotatebox{90}{$V_{{\rm eff}}$ $({\rm meV})$}} \put(0.0,10.1){\rotatebox{90}{$\rho$ $(10^{14}\mbox{m}^{-2})$}} \put(2.7,6.5){$r$ (\mbox{nm})} \put(4.7,11.6){(a)} % \end{picture} \end{center} \caption{ Density profiles and effective potentials for the MTF theory at different magnetic field strengths, calculated for $N=50$ and $V(x)=K|x|^2$ with $K=1,7$ meV and the material parameters of GaAs. (a) $B=8$ T, (b) $B=7$ T, (c) $B=2$ T, (d) $B=0$ T.} \end{figure} When the field is gradually turned down the maximal density $B/(2\pi)$ of electrons that can be accommodated in the lowest Landau level goes down also. Condition (\ref{cond}) no longer holds, i.e., the density $\rho^{\rm C}$ near the center is higher than $B/(2\pi)$ and charges have to be moved into other states in phase space. If $B$ is only slightly smaller than the value given by (\ref{cond}) (Fig.\ 1(b)) it would cost too much energy to bring the electrons near the origin into the next Landau level and it pays to move them spatially away from the center, because the potential energy increase is less than $B$. Hence in a certain range of $B$ values, the density near the center is locked at the value $B/(2\pi)$ ( ``incompressible'' domain). The effective potential is no longer constant in this domain. In the complementary ``compressible'' domain, on the other hand, the density is below the critical value $B/(2\pi)$, and tends to zero in such a way that the effective potential stays constant. Reducing the field strength further brings more Landau levels into play (Fig.\ 1(c)). Incompressible domains, where the density is an integer multiple of $B/(2\pi)$, alternate with compressible domains, where the effective potential has a constant value. When $B\to 0$ the profile becomes indistinguishable from the smooth profile of $\rho^{\rm TF}$ (Fig.\ 1(d)). It is interesting to note that the alternation of compressible and incompressible domains in moderate magnetic fields may account for some fine structure in the charge transport spectroscopy of quantum dots with a large number of electrons \cite{VRK94}. The basic {\bf limit theorem} \cite{LSY95} that relates the energy functionals (\ref{TF})-(\ref{class}) to the quantum mechanical ground state of $H_{N}$ is as follows: \begin{theorem}[High density limit.] Let $N\to\infty$ with $K/N$ fixed. Then, uniformly in $B/N$, \beq E^Q(N,B,K)/E^{\MTF}(N,B,K)\rightarrow 1\label{elim}\eeq and \beq N^{-1}\rho^Q_{N,B,K}(x)\rightarrow \rho^{\MTF}_{1,B/N,K/N}(x)\eeq in weak $L^1$ sense. Moreover, if $B/N\to 0$, then \beq E^Q(N,B,K)/E^{\TF}(N,B,K)\rightarrow 1\eeq \beq N^{-1}\rho^Q_{N,B,K}(x)\rightarrow \rho^{\TF}_{1,B/N,K/N}(x),\eeq and if $B/N\to\infty$, then \beq E^Q(N,B,K)/E^{\rm C}(N,B,K)\rightarrow 1\eeq \beq N^{-1}\rho^Q_{N,B,K}(x)\rightarrow \rho^{\rm C}_{1,B/N,K/N}(x).\eeq \end{theorem} According to this theorem there are thus three asymptotic regimes for quantum dots as $N$ and $K$ tend to $\infty$ with $K/N$ fixed: $B\ll N$, $B\sim N$ and $N\ll B$. This should be compared with the more complex situation for three dimensional natural atoms in strong magnetic field, where there are five regimes \cite{LSY94a}, \cite{LSY94b} for $N\to\infty$ with $Z/N$ fixed ($Z$ = nuclear charge): $B\ll N^{4/3}$, $B\sim N^{4/3}$, $N^{4/3}\ll B\ll N^3$, $B\sim N^3$, $ N^{3}\ll B$. \smallskip For homogeneous potentials a stronger asymptotic result holds, for $K/N$ may tend to zero as $N\to\infty$. \begin{theorem} [Homogeneous potentials.] Assume that $V$ is homogeneous of degree $s\geq1$, i.e., \beq V(\lambda x)=\lambda^sV(x). \eeq Then \beq \lim_{N\to\infty} E^{\rm Q}(N,B,K)/E^\MTF(N,B,K)=1\eeq uniformly in $B$ and in $K$ as long as $K/N$ is bounded above. Moreover, if $K/N\to 0$ as $N\to\infty$, then \beq \lim_{N\to\infty} E^{\rm Q}(N,B,K)/E^{\rm C}(N,K)=1\eeq uniformly in $B$. \end{theorem} We shall now discuss briefly the main techniques used for the proof of these theorems. As usual it is sufficient to prove the limit theorems for the energy, because the corresponding results for the density can be obtained by variation with respect to the potential $V$. The basic result is thus Eq.\ (\ref{elim}); the other limit theorems follow by (\ref{first})--(\ref{last}). One has to prove upper and lower bounds for the quantum mechanical energy $E^{\rm Q}$ in terms of the energy $E^{\MTF}$, with controllable errors. The upper bound is obtained, using the variational principle of \cite{Lvar}, by testing $H_N$ with a suitable one particle density operator. Its kernel in the space and spin variables $x,\sigma$ has the form \beq {\cal K}(x,\sigma;x',\sigma')=\sum_\nu f_\nu(u)\Pi_{\nu u}(x,\sigma;x',\sigma')d^2u\eeq where the sum is over all Landau levels and $f_\nu(u)$ is the filling factor of the $\nu$-th Landau level at point $u$ when the density is $\rho^{\MTF}(u)$. The kernel $\Pi_{\nu u}(x,\sigma;x',\sigma')$ is obtained from the kernel $\Pi_\nu(x,\sigma;x',\sigma')$ of the projector on the $\nu$-th Landau level by localizing around $u$ with a smooth function $g$ of compact support, i.e., \beq \Pi_{\nu u}(x,\sigma;x',\sigma')=g(x-u)\Pi_\nu(x,\sigma;x',\sigma')g(x'-u).\eeq This operator is positive and approximately a projector, localizing simultaneously in space, i.e., around $u$, and in the Landau level index $\nu$. By letting the support of $g$ shrink with $N$ more slowly that the average electron spacing $N^{-1/2}$, the error terms in the estimate above for $E^{\rm Q}-E^{\MTF}$ are of lower order than $N^2$, which is the order of $E^{\MTF}$. The lower bound for $E^{\rm Q}$ is proved separately for large $B$ and for small $B$. For large $B$, i.e., $B\gg N$, one starts with the obvious estimate $E^{\rm Q}\geq E^{\rm P}$. One then has to compare $E^{\rm P}$ with $E^{\rm C}$, i.e., the energy of point charges with those of smeared charges. Since the electron distance is $\sim N^{-1/2}$ the self energy of a smeared unit charge is $\sim N^{1/2}$. Hence an estimate \beq E^{\rm P}(N,K)\geq E^{\rm C}(N,K)-bN^{3/2}\eeq with $b$ depending only on $K/N$ is to be expected, and this can indeed be proved, using an electrostatic lemma of Lieb and Yau \cite{LY88}. The lower bound for small $B$, i.e., $B\ll N$ or $B\sim N$, requires an estimate on the indirect Coulomb energy, that is derived in essentially the same way as a corresponding inequality in \cite{L79}, using the positive definiteness of the Coulomb interaction \ref{coul}, cf. also \cite{Ba92}. \begin{lemma}[Exchange inequality in 2 dimensions.] \beq \sum_{\rm spins}\,\, \int \limits_{\R^{2N}} \vert \Psi \vert^2 \sum \limits_{i < j } \vert x_i - x_j \vert^{-1} \geq D(\rho_\Psi,\rho_\Psi) - 192 (2\pi)^{1/2}\int \limits_{\R^2} \rho_\Psi^{3/2}.\label{exchange} \eeq \end{lemma} In order to control negative term $\sim \int \rho_\Psi^{3/2}$ on the right side of (\ref{exchange}) a lower bound on the kinetic energy is needed. This in turn is derived from a two dimensional magnetic Lieb-Thirring inequality, which has to be proved in a slightly different way from the corresponding inequality in \cite{LSY94b}, because there is no kinetic energy associated with a motion in the $x^3$ direction. The following inequality is adequate for the present purpose, but sharper Lieb-Thirring type inequalities that hold even for inhomogeneous fields have been derived by Erd\H os and Solovej \cite{ESa}, \cite{ESb}. \begin{theorem} [Lieb-Thirring inequality in 2 dimensions.] Let $U$ be locally integrable and let $e_1(U), e_2(U), \ldots$ denote the negative eigenvalues (if any) of the Hamiltonian ${1\over 2}({\rm i}\nabla-{\bf A})^2+S_{3}B-U$. Define $\vert U \vert_+ (x) = \mfr1/2 [\vert U(x) \vert + U(x)]$. For all $0<\lambda<1$ we have the estimate \beq \sum_j |e_j(U)|\leq \lambda^{-1}{B \over 2\pi} \int_{\R^2} |U|_+ (x) dx+ \mfr3/4(1-\lambda)^{-2} \int_{\R^2} |U|_+^2 (x) dx.\label{LT} \eeq\end{theorem} By a Legendre transformation with respect to $U$ it follows from (\ref{LT}) that for all $0<\lambda<1$ the kinetic energy $T_{\Psi}$ of a state $\Psi$ is bounded below by $\mfr1/3(1-\lambda)^2\int [\rho_{\Psi}-\lambda^{-1}B/2\pi]_{+}^2$. It is then possible, for $B/N$ smaller than a certain critical value depending on $v$, to chose an $N$ dependent $\varepsilon>0$ in such a way that $\varepsilon\to 0$ as $N\to\infty$, but $\varepsilon T_{\psi}- \hbox{\rm (const)\, }\int \rho_\Psi^{3/2}\geq 0$ for all $N$-particle states $\Psi$. \section{Other approaches} The rigorous results presented above concern mainly (but not exclusively) the extreme cases of very few ($N=1$ or $N=2$) or very many ($N\to\infty$) electrons. These cases play a similar role as the hydrogen atom and the Thomas-Fermi atom do in ordinary atomic physics, i.e., they set a standard that can be used as a starting point of various approximation schemes, or as a test for such schemes whose connection with the original Hamiltonian (\ref{HN}) may not be entirely clear. The physics literature on quantum dots is by now quite extensive, and many approaches have been used for gaining insight where rigorous results are not yet available. Here it is only possible to mention the main methods and give a list of some references that are representative for the approaches of condensed matter physicists to these problems and from it further sources can be traced. See also \cite{Jac98} for a more comprehensive list. For small $N$ a direct numerical diagonalization of $H_{N}$ is possible and has been carried out, e.g., in \cite{Yang93}, \cite{Wagn92}, \cite{Pfann94}, \cite{Meza97}, \cite{Eto97}, for various values of $N\leq 8$. It is, of course, necessary to restrict $H_{N}$ to a finite dimensional subspace of the full Hilbert space, and the error made in this step is seldom estimated rigorously. By the mini-max principle, however, the computed values give at least upper bounds to the true eigenvalues. One of the features studied by this method is the orbital angular momentum and spin of the ground state, and analogs of Hund's rules from atomic physics, as well as oscillations between triplet and singlet states for $N=2$ as $B$ is varied have been seen in the calculations \cite{Chak92}, \cite{Wagn92}. For $N>10$ numerical diagonalization of the Hamiltonian is at present hardly feasible and resource is taken to other techniques of many body theory like Hartree and Hartree-Fock approximations (e.g., \cite{Pfann94}, \cite{Pfann95}, \cite{MueKoon96}, \cite{Heit97}), perturbation theory \cite{Anis98}, variational methods (e.g. \cite{DiNaz97}), quantum Monte Carlo methods (\cite{HarNie98a}, \cite{HarNie98b}), and (current) density functional theory (\cite{Fer94}, \cite{Fer97}, \cite{Pi98}, \cite{Hein98}). In quantum dots with a moderate electron number correlations play a much greater role than in natural atoms because the confining potential is usually quite shallow around the origin and the density may be low. (By contrast, one of the main steps in the proof of Theorem 3.2 is to show that exchange and correlation effects vanish in the high density limit for arbitrary magnetic fields.) 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setlinewidth } def /PL { stroke gnulinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /P { stroke [] 0 setdash currentlinewidth 2 div sub M 0 currentlinewidth V stroke } def /D { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke P } def /A { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /B { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke P } def /C { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /T { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke P } def /S { 2 copy A C} def /F { stroke [] 0 setdash exch hpt sub exch vpt add moveto 0 vpt2 neg rlineto hpt2 0 rlineto 0 vpt2 rlineto hpt2 neg 0 rlineto closepath fill } def /I { stroke [] 0 setdash vpt 1.12 mul add moveto hpt neg vpt -1.62 mul rlineto hpt 2 mul 0 rlineto hpt neg vpt 1.62 mul rlineto closepath fill } def end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray /Times-Roman findfont 360 scalefont setfont newpath LTa 1728 541 M 4977 0 V -4977 0 R 0 2806 V LTb 1728 541 M 63 0 V 4914 0 R -63 0 V stroke 1512 541 M 0 (0) stringwidth pop add neg -120.000000 rmoveto (0) show 1728 1102 M 63 0 V 4914 0 R -63 0 V stroke 1512 1102 M 0 (4) stringwidth pop add neg -120.000000 rmoveto (4) show 1728 1663 M 63 0 V 4914 0 R -63 0 V stroke 1512 1663 M 0 (8) stringwidth pop add neg -120.000000 rmoveto (8) show 1728 2225 M 63 0 V 4914 0 R -63 0 V stroke 1512 2225 M 0 (12) stringwidth pop add neg -120.000000 rmoveto (12) show 1728 2786 M 63 0 V 4914 0 R -63 0 V stroke 1512 2786 M 0 (16) stringwidth pop add neg -120.000000 rmoveto (16) show 1728 3347 M 63 0 V 4914 0 R -63 0 V stroke 1512 3347 M 0 (20) stringwidth pop add neg -120.000000 rmoveto (20) show 1728 541 M 0 63 V 0 2743 R 0 -63 V 2439 541 M 0 63 V 0 2743 R 0 -63 V 3150 541 M 0 63 V 0 2743 R 0 -63 V 3861 541 M 0 63 V 0 2743 R 0 -63 V 4572 541 M 0 63 V 0 2743 R 0 -63 V 5283 541 M 0 63 V 0 2743 R 0 -63 V 5994 541 M 0 63 V 0 2743 R 0 -63 V 6705 541 M 0 63 V 0 2743 R 0 -63 V 1728 541 M 4977 0 V 0 2806 V -4977 0 V 0 -2806 V LT0 50 setlinewidth 1728 3087 M 45 3 V 46 -2 V 45 1 V 45 0 V 46 -2 V 45 -2 V 45 -3 V 46 -4 V 45 -4 V 46 -4 V 45 -5 V 45 -5 V 46 -6 V 45 -7 V 45 -7 V 46 -7 V 45 -8 V 45 -9 V 46 -9 V 45 -10 V 45 -10 V 46 -11 V 45 -12 V 45 -12 V 46 -13 V 45 -14 V 45 -15 V 46 -16 V 45 -17 V 46 -18 V 45 -20 V 45 -21 V 46 -24 V 45 -26 V 45 -31 V 46 -42 V 45 -58 V 45 -59 V 46 -1 V 45 0 V 45 0 V 46 -1 V 45 0 V 45 0 V 46 0 V 45 0 V 46 -1 V 45 0 V 45 0 V 46 0 V 45 0 V 45 0 V 46 0 V 45 -1 V 45 0 V 46 0 V 45 0 V 45 -1 V 46 0 V 45 -1 V 45 -89 V 46 -103 V 45 -80 V 46 -79 V 45 -91 V 45 -139 V 46 -91 V 45 -1 V 45 -1 V 46 0 V 45 0 V 45 0 V 46 -1 V 45 0 V 45 0 V 46 0 V 45 -1 V 45 0 V 46 0 V 45 -1 V 45 -66 V 46 -345 V 45 -262 V 46 -1 V 45 -1 V 45 0 V 46 -1 V 45 0 V 45 0 V 46 -1 V 45 0 V 45 -2 V 46 -675 V 45 0 V 45 0 V 46 0 V 45 0 V 45 0 V 46 0 V 45 0 V stroke grestore end showpage %%Trailer ---------------9811071409760 Content-Type: application/postscript; name="rho7.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="rho7.eps" %!PS-Adobe-2.0 EPSF-2.0 %%Creator: gnuplot %%DocumentFonts: Times-Roman %%BoundingBox: 50 50 410 226 %%EndComments /gnudict 40 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /vshift -120 def /dl {10 mul} def /hpt 31.5 def /vpt 31.5 def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /PL { stroke gnulinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /P { stroke [] 0 setdash currentlinewidth 2 div sub M 0 currentlinewidth V stroke } def /D { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke P } def /A { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /B { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke P } def /C { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /T { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke P } def /S { 2 copy A C} def /F { stroke [] 0 setdash exch hpt sub exch vpt add moveto 0 vpt2 neg rlineto hpt2 0 rlineto 0 vpt2 rlineto hpt2 neg 0 rlineto closepath fill } def /I { stroke [] 0 setdash vpt 1.12 mul add moveto hpt neg vpt -1.62 mul rlineto hpt 2 mul 0 rlineto hpt neg vpt 1.62 mul rlineto closepath fill } def end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray /Times-Roman findfont 360 scalefont setfont newpath LTa 1728 541 M 4977 0 V -4977 0 R 0 2806 V LTb 1728 541 M 63 0 V 4914 0 R -63 0 V stroke 1512 541 M 0 (0) stringwidth pop add neg -120.000000 rmoveto (0) show 1728 1102 M 63 0 V 4914 0 R -63 0 V stroke 1512 1102 M 0 (4) stringwidth pop add neg -120.000000 rmoveto (4) show 1728 1663 M 63 0 V 4914 0 R -63 0 V stroke 1512 1663 M 0 (8) stringwidth pop add neg -120.000000 rmoveto (8) show 1728 2225 M 63 0 V 4914 0 R -63 0 V stroke 1512 2225 M 0 (12) stringwidth pop add neg -120.000000 rmoveto (12) show 1728 2786 M 63 0 V 4914 0 R -63 0 V stroke 1512 2786 M 0 (16) stringwidth pop add neg -120.000000 rmoveto (16) show 1728 3347 M 63 0 V 4914 0 R -63 0 V stroke 1512 3347 M 0 (20) stringwidth pop add neg -120.000000 rmoveto (20) show 1728 541 M 0 63 V 0 2743 R 0 -63 V 2439 541 M 0 63 V 0 2743 R 0 -63 V 3150 541 M 0 63 V 0 2743 R 0 -63 V 3861 541 M 0 63 V 0 2743 R 0 -63 V 4572 541 M 0 63 V 0 2743 R 0 -63 V 5283 541 M 0 63 V 0 2743 R 0 -63 V 5994 541 M 0 63 V 0 2743 R 0 -63 V 6705 541 M 0 63 V 0 2743 R 0 -63 V 1728 541 M 4977 0 V 0 2806 V -4977 0 V 0 -2806 V LT0 50 setlinewidth 1728 2913 M 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 -1 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 46 -1 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 -1 V 46 0 V 46 0 V 46 0 V 46 0 V 46 -1 V 46 0 V 46 0 V 47 -1 V 46 0 V 46 -1 V 46 0 V 46 -1 V 46 -1 V 46 -1 V 46 -1 V 46 -2 V 46 -18 V 46 -78 V 46 -48 V 46 -39 V 46 -36 V 46 -35 V 46 -34 V 47 -34 V 46 -34 V 46 -34 V 46 -35 V 46 -36 V 46 -36 V 46 -38 V 46 -38 V 46 -40 V 46 -40 V 46 -43 V 46 -44 V 46 -45 V 46 -47 V 46 -50 V 46 -51 V 46 -55 V 47 -57 V 46 -60 V 46 -64 V 46 -69 V 46 -74 V 46 -81 V 46 -88 V 46 -101 V 46 -117 V 46 -148 V 46 -222 V 46 -291 V 46 0 V 46 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V stroke grestore end showpage %%Trailer ---------------9811071409760 Content-Type: application/postscript; name="rho8.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="rho8.eps" %!PS-Adobe-2.0 EPSF-2.0 %%Creator: gnuplot %%DocumentFonts: Times-Roman %%BoundingBox: 50 50 410 226 %%EndComments /gnudict 40 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /vshift -120 def /dl {10 mul} def /hpt 31.5 def /vpt 31.5 def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /PL { stroke gnulinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /P { stroke [] 0 setdash currentlinewidth 2 div sub M 0 currentlinewidth V stroke } def /D { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke P } def /A { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /B { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke P } def /C { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /T { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke P } def /S { 2 copy A C} def /F { stroke [] 0 setdash exch hpt sub exch vpt add moveto 0 vpt2 neg rlineto hpt2 0 rlineto 0 vpt2 rlineto hpt2 neg 0 rlineto closepath fill } def /I { stroke [] 0 setdash vpt 1.12 mul add moveto hpt neg vpt -1.62 mul rlineto hpt 2 mul 0 rlineto hpt neg vpt 1.62 mul rlineto closepath fill } def end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray /Times-Roman findfont 360 scalefont setfont newpath LTa 1728 541 M 4977 0 V -4977 0 R 0 2806 V LTb 1728 541 M 63 0 V 4914 0 R -63 0 V stroke 1512 541 M 0 (0) stringwidth pop add neg -120.000000 rmoveto (0) show 1728 1102 M 63 0 V 4914 0 R -63 0 V stroke 1512 1102 M 0 (4) stringwidth pop add neg -120.000000 rmoveto (4) show 1728 1663 M 63 0 V 4914 0 R -63 0 V stroke 1512 1663 M 0 (8) stringwidth pop add neg -120.000000 rmoveto (8) show 1728 2225 M 63 0 V 4914 0 R -63 0 V stroke 1512 2225 M 0 (12) stringwidth pop add neg -120.000000 rmoveto (12) show 1728 2786 M 63 0 V 4914 0 R -63 0 V stroke 1512 2786 M 0 (16) stringwidth pop add neg -120.000000 rmoveto (16) show 1728 3347 M 63 0 V 4914 0 R -63 0 V stroke 1512 3347 M 0 (20) stringwidth pop add neg -120.000000 rmoveto (20) show 1728 541 M 0 63 V 0 2743 R 0 -63 V 2439 541 M 0 63 V 0 2743 R 0 -63 V 3150 541 M 0 63 V 0 2743 R 0 -63 V 3861 541 M 0 63 V 0 2743 R 0 -63 V 4572 541 M 0 63 V 0 2743 R 0 -63 V 5283 541 M 0 63 V 0 2743 R 0 -63 V 5994 541 M 0 63 V 0 2743 R 0 -63 V 6705 541 M 0 63 V 0 2743 R 0 -63 V 1728 541 M 4977 0 V 0 2806 V -4977 0 V 0 -2806 V LT0 50 setlinewidth 1728 3237 M 46 0 V 47 0 V 46 0 V 46 0 V 46 0 V 47 0 V 46 -1 V 46 -1 V 47 0 V 46 -2 V 46 -1 V 46 -3 V 47 -6 V 46 -7 V 46 -6 V 47 -6 V 46 -7 V 46 -7 V 46 -7 V 47 -8 V 46 -8 V 46 -9 V 47 -8 V 46 -10 V 46 -10 V 46 -10 V 47 -10 V 46 -11 V 46 -12 V 47 -12 V 46 -12 V 46 -13 V 46 -13 V 47 -14 V 46 -14 V 46 -15 V 47 -16 V 46 -15 V 46 -17 V 46 -17 V 47 -17 V 46 -18 V 46 -19 V 47 -19 V 46 -19 V 46 -21 V 46 -21 V 47 -21 V 46 -23 V 46 -23 V 47 -23 V 46 -25 V 46 -25 V 46 -26 V 47 -27 V 46 -27 V 46 -29 V 47 -29 V 46 -31 V 46 -31 V 46 -33 V 47 -33 V 46 -35 V 46 -36 V 47 -38 V 46 -38 V 46 -41 V 46 -42 V 47 -43 V 46 -46 V 46 -48 V 47 -50 V 46 -52 V 46 -55 V 46 -59 V 47 -62 V 46 -67 V 46 -71 V 47 -78 V 46 -85 V 46 -95 V 46 -110 V 47 -133 V 46 -184 V 46 -363 V 47 -17 V 46 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 47 0 V stroke grestore end showpage %%Trailer ---------------9811071409760 Content-Type: application/postscript; name="ve0.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ve0.eps" %!PS-Adobe-2.0 EPSF-2.0 %%Creator: gnuplot %%DocumentFonts: Times-Roman %%BoundingBox: 50 50 410 226 %%EndComments /gnudict 40 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /vshift -120 def /dl {10 mul} def /hpt 31.5 def /vpt 31.5 def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /PL { stroke gnulinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /P { stroke [] 0 setdash currentlinewidth 2 div sub M 0 currentlinewidth V stroke } def /D { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke P } def /A { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /B { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke P } def /C { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /T { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke P } def /S { 2 copy A C} def /F { stroke [] 0 setdash exch hpt sub exch vpt add moveto 0 vpt2 neg rlineto hpt2 0 rlineto 0 vpt2 rlineto hpt2 neg 0 rlineto closepath fill } def /I { stroke [] 0 setdash vpt 1.12 mul add moveto hpt neg vpt -1.62 mul rlineto hpt 2 mul 0 rlineto hpt neg vpt 1.62 mul rlineto closepath fill } def end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray /Times-Roman findfont 360 scalefont setfont newpath LTa 1728 541 M 0 2806 V LTb 1728 541 M 63 0 V 4914 0 R -63 0 V stroke 1512 541 M 0 (110) stringwidth pop add neg -120.000000 rmoveto (110) show 1728 1128 M 63 0 V 4914 0 R -63 0 V stroke 1512 1128 M 0 (115) stringwidth pop add neg -120.000000 rmoveto (115) show 1728 1715 M 63 0 V 4914 0 R -63 0 V stroke 1512 1715 M 0 (120) stringwidth pop add neg -120.000000 rmoveto (120) show 1728 2302 M 63 0 V 4914 0 R -63 0 V stroke 1512 2302 M 0 (125) stringwidth pop add neg -120.000000 rmoveto (125) show 1728 2889 M 63 0 V 4914 0 R -63 0 V stroke 1512 2889 M 0 (130) stringwidth pop add neg -120.000000 rmoveto (130) show 1728 541 M 0 63 V 0 2743 R 0 -63 V stroke 1728 181 M 0 (0) stringwidth pop add 2 div neg -120.000000 rmoveto (0) show 2439 541 M 0 63 V 0 2743 R 0 -63 V stroke 2439 181 M 0 (20) stringwidth pop add 2 div neg -120.000000 rmoveto (20) show 3150 541 M 0 63 V 0 2743 R 0 -63 V stroke 3150 181 M 0 (40) stringwidth pop add 2 div neg -120.000000 rmoveto (40) show 3861 541 M 0 63 V 0 2743 R 0 -63 V stroke 3861 181 M 0 (60) stringwidth pop add 2 div neg -120.000000 rmoveto (60) show 4572 541 M 0 63 V 0 2743 R 0 -63 V stroke 4572 181 M 0 (80) stringwidth pop add 2 div neg -120.000000 rmoveto (80) show 5283 541 M 0 63 V 0 2743 R 0 -63 V stroke 5283 181 M 0 (100) stringwidth pop add 2 div neg -120.000000 rmoveto (100) show 5994 541 M 0 63 V 0 2743 R 0 -63 V stroke 5994 181 M 0 (120) stringwidth pop add 2 div neg -120.000000 rmoveto (120) show 6705 541 M 0 63 V 0 2743 R 0 -63 V stroke 6705 181 M 0 (140) stringwidth pop add 2 div neg -120.000000 rmoveto (140) show 1728 541 M 4977 0 V 0 2806 V -4977 0 V 0 -2806 V LT0 50 setlinewidth 1773 1391 M 46 0 V 45 0 V 45 1 V 45 1 V 45 1 V 45 1 V 45 1 V 45 2 V 45 2 V 45 2 V 45 2 V 45 3 V 45 2 V 45 3 V 45 3 V 45 3 V 45 4 V 45 4 V 45 3 V 45 5 V 45 4 V 45 4 V 45 5 V 45 5 V 46 5 V 45 6 V 45 5 V 45 6 V 45 6 V 45 7 V 45 6 V 45 7 V 45 7 V 45 7 V 45 8 V 45 7 V 45 8 V 45 9 V 45 8 V 45 9 V 45 9 V 45 9 V 45 10 V 45 10 V 45 10 V 45 10 V 45 11 V 45 11 V 45 11 V 46 12 V 45 12 V 45 12 V 45 12 V 45 13 V 45 14 V 45 13 V 45 14 V 45 15 V 45 14 V 45 16 V 45 15 V 45 16 V 45 16 V 45 17 V 45 18 V 45 18 V 45 18 V 45 19 V 45 19 V 45 20 V 45 21 V 45 21 V 45 21 V 45 23 V 46 23 V 45 24 V 45 25 V 45 25 V 45 27 V 45 27 V 45 28 V 45 30 V 45 30 V 45 32 V 45 33 V 45 35 V 45 36 V 45 38 V 45 40 V 45 42 V 45 45 V 45 48 V 45 51 V 45 55 V 45 61 V 45 70 V 45 81 V 45 90 V stroke grestore end showpage %%Trailer ---------------9811071409760 Content-Type: application/postscript; name="ve2.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ve2.eps" %!PS-Adobe-2.0 EPSF-2.0 %%Creator: gnuplot %%DocumentFonts: Times-Roman %%BoundingBox: 50 50 410 226 %%EndComments /gnudict 40 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /vshift -120 def /dl {10 mul} def /hpt 31.5 def /vpt 31.5 def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /PL { stroke gnulinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /P { stroke [] 0 setdash currentlinewidth 2 div sub M 0 currentlinewidth V stroke } def /D { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke P } def /A { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /B { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke P } def /C { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /T { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke P } def /S { 2 copy A C} def /F { stroke [] 0 setdash exch hpt sub exch vpt add moveto 0 vpt2 neg rlineto hpt2 0 rlineto 0 vpt2 rlineto hpt2 neg 0 rlineto closepath fill } def /I { stroke [] 0 setdash vpt 1.12 mul add moveto hpt neg vpt -1.62 mul rlineto hpt 2 mul 0 rlineto hpt neg vpt 1.62 mul rlineto closepath fill } def end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray /Times-Roman findfont 360 scalefont setfont newpath LTa 1728 541 M 0 2806 V LTb 1728 541 M 63 0 V 4914 0 R -63 0 V stroke 1512 541 M 0 (110) stringwidth pop add neg -120.000000 rmoveto (110) show 1728 1128 M 63 0 V 4914 0 R -63 0 V stroke 1512 1128 M 0 (115) stringwidth pop add neg -120.000000 rmoveto (115) show 1728 1715 M 63 0 V 4914 0 R -63 0 V stroke 1512 1715 M 0 (120) stringwidth pop add neg -120.000000 rmoveto (120) show 1728 2302 M 63 0 V 4914 0 R -63 0 V stroke 1512 2302 M 0 (125) stringwidth pop add neg -120.000000 rmoveto (125) show 1728 2889 M 63 0 V 4914 0 R -63 0 V stroke 1512 2889 M 0 (130) stringwidth pop add neg -120.000000 rmoveto (130) show 1728 541 M 0 63 V 0 2743 R 0 -63 V stroke 1728 181 M 0 (0) stringwidth pop add 2 div neg -120.000000 rmoveto (0) show 2439 541 M 0 63 V 0 2743 R 0 -63 V stroke 2439 181 M 0 (20) stringwidth pop add 2 div neg -120.000000 rmoveto (20) show 3150 541 M 0 63 V 0 2743 R 0 -63 V stroke 3150 181 M 0 (40) stringwidth pop add 2 div neg -120.000000 rmoveto (40) show 3861 541 M 0 63 V 0 2743 R 0 -63 V stroke 3861 181 M 0 (60) stringwidth pop add 2 div neg -120.000000 rmoveto (60) show 4572 541 M 0 63 V 0 2743 R 0 -63 V stroke 4572 181 M 0 (80) stringwidth pop add 2 div neg -120.000000 rmoveto (80) show 5283 541 M 0 63 V 0 2743 R 0 -63 V stroke 5283 181 M 0 (100) stringwidth pop add 2 div neg -120.000000 rmoveto (100) show 5994 541 M 0 63 V 0 2743 R 0 -63 V stroke 5994 181 M 0 (120) stringwidth pop add 2 div neg -120.000000 rmoveto (120) show 6705 541 M 0 63 V 0 2743 R 0 -63 V stroke 6705 181 M 0 (140) stringwidth pop add 2 div neg -120.000000 rmoveto (140) show 1728 541 M 4977 0 V 0 2806 V -4977 0 V 0 -2806 V LT0 50 setlinewidth 1774 1462 M 45 -1 V 46 0 V 45 0 V 45 0 V 46 0 V 45 0 V 45 0 V 46 0 V 45 0 V 45 0 V 46 0 V 45 0 V 45 0 V 46 0 V 45 0 V 45 0 V 46 0 V 45 0 V 45 0 V 46 0 V 45 0 V 46 0 V 45 0 V 45 0 V 46 0 V 45 0 V 45 0 V 46 0 V 45 0 V 45 0 V 46 0 V 45 0 V 45 0 V 46 0 V 45 0 V 45 -1 V 46 2 V 45 5 V 46 10 V 45 12 V 45 15 V 46 15 V 45 18 V 45 18 V 46 20 V 45 20 V 45 21 V 46 22 V 45 22 V 45 22 V 46 22 V 45 23 V 45 22 V 46 21 V 45 21 V 45 20 V 46 18 V 45 16 V 46 14 V 45 7 V 45 2 V 46 -1 V 45 0 V 45 0 V 46 0 V 45 3 V 45 12 V 46 20 V 45 24 V 45 27 V 46 30 V 45 32 V 45 33 V 46 35 V 45 34 V 46 35 V 45 33 V 45 32 V 46 29 V 45 21 V 45 7 V 46 3 V 45 18 V 45 32 V 46 40 V 45 45 V 45 49 V 46 50 V 45 52 V 45 51 V 46 47 V 45 27 V 46 33 V 45 60 V 45 73 V 46 81 V 45 90 V 45 96 V stroke grestore end showpage %%Trailer ---------------9811071409760 Content-Type: application/postscript; name="ve7.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ve7.eps" %!PS-Adobe-2.0 EPSF-2.0 %%Creator: gnuplot %%DocumentFonts: Times-Roman %%BoundingBox: 50 50 410 226 %%EndComments /gnudict 40 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /vshift -120 def /dl {10 mul} def /hpt 31.5 def /vpt 31.5 def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /PL { stroke gnulinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /P { stroke [] 0 setdash currentlinewidth 2 div sub M 0 currentlinewidth V stroke } def /D { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke P } def /A { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /B { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke P } def /C { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /T { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke P } def /S { 2 copy A C} def /F { stroke [] 0 setdash exch hpt sub exch vpt add moveto 0 vpt2 neg rlineto hpt2 0 rlineto 0 vpt2 rlineto hpt2 neg 0 rlineto closepath fill } def /I { stroke [] 0 setdash vpt 1.12 mul add moveto hpt neg vpt -1.62 mul rlineto hpt 2 mul 0 rlineto hpt neg vpt 1.62 mul rlineto closepath fill } def end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray /Times-Roman findfont 360 scalefont setfont newpath LTa 1728 541 M 0 2806 V LTb 1728 541 M 63 0 V 4914 0 R -63 0 V stroke 1512 541 M 0 (110) stringwidth pop add neg -120.000000 rmoveto (110) show 1728 1128 M 63 0 V 4914 0 R -63 0 V stroke 1512 1128 M 0 (115) stringwidth pop add neg -120.000000 rmoveto (115) show 1728 1715 M 63 0 V 4914 0 R -63 0 V stroke 1512 1715 M 0 (120) stringwidth pop add neg -120.000000 rmoveto (120) show 1728 2302 M 63 0 V 4914 0 R -63 0 V stroke 1512 2302 M 0 (125) stringwidth pop add neg -120.000000 rmoveto (125) show 1728 2889 M 63 0 V 4914 0 R -63 0 V stroke 1512 2889 M 0 (130) stringwidth pop add neg -120.000000 rmoveto (130) show 1728 541 M 0 63 V 0 2743 R 0 -63 V stroke 1728 181 M 0 (0) stringwidth pop add 2 div neg -120.000000 rmoveto (0) show 2439 541 M 0 63 V 0 2743 R 0 -63 V stroke 2439 181 M 0 (20) stringwidth pop add 2 div neg -120.000000 rmoveto (20) show 3150 541 M 0 63 V 0 2743 R 0 -63 V stroke 3150 181 M 0 (40) stringwidth pop add 2 div neg -120.000000 rmoveto (40) show 3861 541 M 0 63 V 0 2743 R 0 -63 V stroke 3861 181 M 0 (60) stringwidth pop add 2 div neg -120.000000 rmoveto (60) show 4572 541 M 0 63 V 0 2743 R 0 -63 V stroke 4572 181 M 0 (80) stringwidth pop add 2 div neg -120.000000 rmoveto (80) show 5283 541 M 0 63 V 0 2743 R 0 -63 V stroke 5283 181 M 0 (100) stringwidth pop add 2 div neg -120.000000 rmoveto (100) show 5994 541 M 0 63 V 0 2743 R 0 -63 V stroke 5994 181 M 0 (120) stringwidth pop add 2 div neg -120.000000 rmoveto (120) show 6705 541 M 0 63 V 0 2743 R 0 -63 V stroke 6705 181 M 0 (140) stringwidth pop add 2 div neg -120.000000 rmoveto (140) show 1728 541 M 4977 0 V 0 2806 V -4977 0 V 0 -2806 V LT0 50 setlinewidth 1775 1545 M 46 0 V 46 2 V 46 2 V 46 3 V 46 3 V 46 4 V 46 5 V 46 5 V 46 7 V 46 6 V 46 8 V 46 8 V 46 8 V 46 10 V 46 10 V 46 10 V 47 11 V 46 12 V 46 12 V 46 12 V 46 13 V 46 14 V 46 14 V 46 15 V 46 15 V 46 15 V 46 16 V 46 16 V 46 17 V 46 17 V 46 17 V 46 17 V 47 18 V 46 17 V 46 18 V 46 18 V 46 18 V 46 18 V 46 18 V 46 17 V 46 18 V 46 17 V 46 16 V 46 16 V 46 15 V 46 15 V 46 13 V 46 12 V 46 9 V 47 7 V 46 3 V 46 -1 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 46 0 V 47 0 V 46 -1 V 46 3 V 46 19 V 46 40 V 46 51 V 46 62 V 46 69 V 46 77 V 46 83 V 46 90 V 46 95 V 46 101 V 46 106 V 46 111 V 46 116 V stroke grestore end showpage %%Trailer ---------------9811071409760 Content-Type: application/postscript; name="ve8.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ve8.eps" %!PS-Adobe-2.0 EPSF-2.0 %%Creator: gnuplot %%DocumentFonts: Times-Roman %%BoundingBox: 50 50 410 226 %%EndComments /gnudict 40 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /vshift -120 def /dl {10 mul} def /hpt 31.5 def /vpt 31.5 def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /PL { stroke gnulinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /P { stroke [] 0 setdash currentlinewidth 2 div sub M 0 currentlinewidth V stroke } def /D { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke P } def /A { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /B { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke P } def /C { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /T { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke P } def /S { 2 copy A C} def /F { stroke [] 0 setdash exch hpt sub exch vpt add moveto 0 vpt2 neg rlineto hpt2 0 rlineto 0 vpt2 rlineto hpt2 neg 0 rlineto closepath fill } def /I { stroke [] 0 setdash vpt 1.12 mul add moveto hpt neg vpt -1.62 mul rlineto hpt 2 mul 0 rlineto hpt neg vpt 1.62 mul rlineto closepath fill } def end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray /Times-Roman findfont 360 scalefont setfont newpath LTa 1728 541 M 0 2806 V LTb 1728 541 M 63 0 V 4914 0 R -63 0 V stroke 1512 541 M 0 (110) stringwidth pop add neg -120.000000 rmoveto (110) show 1728 1128 M 63 0 V 4914 0 R -63 0 V stroke 1512 1128 M 0 (115) stringwidth pop add neg -120.000000 rmoveto (115) show 1728 1715 M 63 0 V 4914 0 R -63 0 V stroke 1512 1715 M 0 (120) stringwidth pop add neg -120.000000 rmoveto (120) show 1728 2302 M 63 0 V 4914 0 R -63 0 V stroke 1512 2302 M 0 (125) stringwidth pop add neg -120.000000 rmoveto (125) show 1728 2889 M 63 0 V 4914 0 R -63 0 V stroke 1512 2889 M 0 (130) stringwidth pop add neg -120.000000 rmoveto (130) show 1728 541 M 0 63 V 0 2743 R 0 -63 V stroke 1728 181 M 0 (0) stringwidth pop add 2 div neg -120.000000 rmoveto (0) show 2439 541 M 0 63 V 0 2743 R 0 -63 V stroke 2439 181 M 0 (20) stringwidth pop add 2 div neg -120.000000 rmoveto (20) show 3150 541 M 0 63 V 0 2743 R 0 -63 V stroke 3150 181 M 0 (40) stringwidth pop add 2 div neg -120.000000 rmoveto (40) show 3861 541 M 0 63 V 0 2743 R 0 -63 V stroke 3861 181 M 0 (60) stringwidth pop add 2 div neg -120.000000 rmoveto (60) show 4572 541 M 0 63 V 0 2743 R 0 -63 V stroke 4572 181 M 0 (80) stringwidth pop add 2 div neg -120.000000 rmoveto (80) show 5283 541 M 0 63 V 0 2743 R 0 -63 V stroke 5283 181 M 0 (100) stringwidth pop add 2 div neg -120.000000 rmoveto (100) show 5994 541 M 0 63 V 0 2743 R 0 -63 V stroke 5994 181 M 0 (120) stringwidth pop add 2 div neg -120.000000 rmoveto (120) show 6705 541 M 0 63 V 0 2743 R 0 -63 V stroke 6705 181 M 0 (140) stringwidth pop add 2 div neg -120.000000 rmoveto (140) show 1728 541 M 4977 0 V 0 2806 V -4977 0 V 0 -2806 V LT0 50 setlinewidth 1775 2100 M 46 0 V 46 0 V 47 0 V 46 0 V 46 1 V 46 0 V 47 1 V 46 1 V 46 0 V 47 1 V 46 0 V 46 1 V 46 0 V 47 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 46 -1 V 47 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 46 0 V 47 0 V 46 0 V 46 0 V 47 -1 V 46 -1 V 46 14 V 46 35 V 47 49 V 46 59 V 46 67 V 47 75 V 46 82 V 46 88 V 46 94 V 47 100 V 46 105 V 46 109 V 47 115 V 46 119 V stroke grestore end showpage %%Trailer ---------------9811071409760--