The paper is 40 pages, LaTeX version 2.09. It is submitted to Commun. Math. Phys. BODY \documentstyle[11pt]{article} \newtheorem{Th}{Theorem} \newtheorem{Prop}{Proposition} \newtheorem{Lem}{Lemma} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\lb}{\label} \newcommand{\lam}{\lambda} \newcommand{\eps}{\epsilon} \newcommand{\en}{\epsilon} \newcommand{\ba}{{\bf a}} \newcommand{\br}{{\bf r}} \newcommand{\hbr}{{\widehat{\bf r}}} \newcommand{\hr}{{\widehat{r}}} \newcommand{\bv}{{\bf v}} \newcommand{\bk}{{\bf k}} \newcommand{\bx}{{\bf x}} \newcommand{\by}{{\bf y}} \newcommand{\bM}{{\bf M}} \newcommand{\bD}{{\bf D}} \newcommand{\bR}{{\bf R}} \newcommand{\bP}{{\bf P}} \newcommand{\bG}{{\bf G}} \newcommand{\bV}{{\bf V}} \newcommand{\bz}{{\bf 0}} \newcommand{\bZ}{{\bf z}} \newcommand{\BZ}{{\bf Z}} \newcommand{\bxi}{{\mbox{\boldmath $\xi$}}} \newcommand{\Bxi}{{\mbox{\boldmath $\Xi$}}} \newcommand{\BQ}{{\bf Q}} \newcommand{\oH}{\widetilde{{\cal H}}} \newcommand{\cH}{{\cal H}} \newcommand{\cS}{{\cal S}} \newcommand{\of}{\overline{f}} \newcommand{\cL}{{\cal L}} \newcommand{\ch}{\widetilde{h}} \newcommand{\grad}{{\mbox{\boldmath $\nabla$}}} \newcommand{\bdot}{{\mbox{\boldmath $\cdot$}}} \newcommand{\btimes}{{\mbox{\boldmath $\times$}}} \textwidth6.25in \textheight8.5in \oddsidemargin.25in \topmargin0in \renewcommand{\baselinestretch}{1.7} \begin{document} \title{Existence and Uniqueness of $L^2$-Solutions at Zero-Diffusivity in the Kraichnan Model of a Passive Scalar} \author{Gregory Eyink and Jack Xin\\ {\em Department of Mathematics}\\ {\em University of Arizona}} \date{\today} \maketitle \begin{abstract} We study Kraichnan's model of a turbulent scalar, passively advected by a Gaussian random velocity field delta-correlated in time, for every space dimension $d\geq 2$ and eddy-diffusivity (Richardson) exponent $0<\zeta<2$. We prove that at zero molecular diffusivity, or $\kappa = 0$, there exist unique weak solutions in $L^2\left(\Omega^{\otimes N}\right)$ to the singular-elliptic, linear PDE's for the stationary $N$-point statistical correlation functions, when the scalar field is confined to a bounded domain $\Omega$ with Dirichlet b.c. Under those conditions we prove that the $N$-body elliptic operators in the $L^2$ spaces have purely discrete, positive spectrum and a minimum eigenvalue of order $L^{-\gamma}$, with $\gamma =2-\zeta$ and with $L$ the diameter of $\Omega$. We also prove that the weak $L^2$-limits of the stationary solutions for positive, $p$th-order hyperdiffusivities $\kappa_p>0$, $p\geq 1$, exist when $\kappa_p \rightarrow 0$ and coincide with the unique zero-diffusivity solutions. These results follow from a lower estimate on the minimum eigenvalue of the $N$-particle eddy-diffusivity matrix, which is conjectured for general $N$ and proved in detail for $N=2,3,4$. Some additional issues are discussed: (1) H\"{o}lder regularity of the solutions; (2) the reconstruction of an invariant probability measure on scalar fields from the set of $N$-point correlation functions, and (3) time-dependent weak solutions to the PDE's for $N$-point correlation functions with $L^2$ initial data. \end{abstract} \newpage \section{Introduction} We study the model problem of a scalar field $\theta(\br,t)$ satisfying an advection-diffusion equation \be (\partial_t+\bv\bdot\grad_\br)\theta=\kappa\bigtriangleup_\br\theta+f \lb{pseq} \ee in a bounded domain $\Omega$ of Euclidean $d$-dimensional space ${\bf R}^d$, with Dirichlet conditions on the boundary $\partial\Omega$. The scalar source $f(\br,t)$ is assumed a Gaussian random field, white-noise in time but regular in space. Precisely, we take $f$ with mean $\langle f(\br,t)\rangle =\of(\br)\in L^2(\Omega)$ and covariance \be \langle f(\br,t)f(\br',t')\rangle-\langle f(\br,t)\rangle\langle f(\br',t')\rangle =F(\br,\br')\delta(t-t') \lb{Fcov} \ee with $F\in L^2\left(\Omega\otimes\Omega\right)$. The velocity field is also assumed Gaussian, white-noise in time, zero-mean with covariance \be \langle v_i(\br,t)v_j(\br',t')\rangle =V_{ij}(\br-\br')\delta(t-t') \lb{Vcov} \ee The velocity to be considered is a divergence-free random field in ${\bf R}^d$ and, for convenience, statistically homogeneous. There is no reason to insist on Dirichlet b.c. for the velocity field. The spatial covariance matrix $\bV$ we consider is defined by the Fourier integral \be V_{ij}(\br)= D_0\int {{d^d\bk}\over{(2\pi)^d}}\,\, \left(k^2+m^2\right)^{-(d+\zeta)/2}P^\perp_{ij}(\bk) e^{i\bk\bdot\br}. \lb{Vspec} \ee where $0<\zeta<2$ and $P^\perp_{ij}(\bk)$ is the projection in ${\bf R}^d$ onto the subspace perpendicular to $\bk$. This automatically defines a suitable positive-definite, symmetric matrix-valued function, divergence-free in each index. The model originates in the 1968 work of R. H. Kraichnan \cite{Kr68} and has been the subject of recent analytical investigations \cite{SS,Maj,GK-L,GK,BGK,CFKL,CF}. It is not hard to show that \be V_{ij}(\br)\sim V_0\delta_{ij}-D_1\cdot r^\zeta\cdot\left[\delta_{ij} +{{\zeta}\over{d-1}}\left(\delta_{ij}-{{r_ir_j}\over{r^2}}\right)\right]+\cdots \lb{scaleq} \ee asymptotically for $mr\ll 1$, with $V_0$ and $D_1$ constants proportional to $D_0$, given below. See also Section 4.1 of \cite{GK-L}. The exponent $\zeta$ has the physical interpretation of an ``eddy-diffusivity exponent'' analogous to the Richardson exponent $4/3$ \cite{Rich}. The remarkable feature of Kraichnan's model, which makes it, in a certain sense, ``exactly soluble'' is that $N$-th order correlation functions $\Theta_N(\br_1,...,\br_N;t)=\langle\theta(\br_1,t) \cdots\theta(\br_N,t)\rangle$ satisfy {\em closed} equations of the form \begin{eqnarray} \, & & \partial_t\Theta_N= -\oH_N^{(\kappa)}\Theta_N+\sum_n \of(\br_n)\Theta_{N-1}(...\widehat{\br_n}...) \cr \, & & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\sum_{{\rm pairs}\,\,\,\,\{nm\}} F(\br_n,\br_m)\Theta_{N-2}(...\widehat{\br_n}...\widehat{\br_m}...). \lb{closeq} \end{eqnarray} In this equation for the $N$-correlator only itself and lower-order correlators appear \cite{Kr68,SS,Maj,GK-L}. Here $\oH_N^{(\kappa)}$ is an elliptic partial-differential operator in $\Omega^{\otimes N}$ defined as \be \oH_N^{(\kappa)}= -{{1}\over{2}}\sum_{i,j=1}^d\sum_{n,m=1}^N\,\,{{\partial}\over{\partial x_{in}}} \left[V_{ij}(\br_n-\br_m){{\partial}\over{\partial x_{jm}}}\cdot\right] -\kappa\sum_{n=1}^N\bigtriangleup_{\br_n}, \lb{singell} \ee with Dirichlet b.c., where $x_{in}$ are Cartesian coordinates in $\left({\bf R}^d\right)^{\otimes N}$. However, the operator $\oH_N$ obtained by taking $\kappa\rightarrow 0$ is degenerate, i.e. it is singular-elliptic. We refer to $\oH_N$ as the $N$-body {\em convective operator} because it accounts for the effects of the velocity advection alone in the equation (\ref{closeq}) for $N$-point correlations. Because of the degeneracy for $\kappa\rightarrow 0$, the solutions of the parabolic equation are expected in that limit to lie only in a H\"{o}lder class $C^{\gamma}\left(\Omega^{\otimes N} \right)$ with $\gamma=2-\zeta$. As the differential operator is of second-order, these solutions must then be taken in a suitable weak sense. Despite the degeneracy, the linear operator $\oH_N$ is formally self-adjoint and nonnegative in the $L^2$ inner product of functions on $\Omega^{\otimes N}$. This suggests that an $L^2$-theory of weak solutions to Eq.(\ref{closeq}) may be appropriate. We shall develop here such a theory in detail. The key to the analysis of the $\kappa\rightarrow 0$ limiting solutions is a proof of existence and uniqueness directly for $\kappa=0$. Let us state precisely the main theorems of this work. We shall actually consider a somewhat more general model than Eq.(\ref{pseq}), namely, \be (\partial_t+\bv\bdot\grad_\br)\theta= -\kappa_p(-\bigtriangleup_\br)^p\theta+f \lb{ppseq} \ee with $p\geq 1$, in which $\kappa_p$ is a so-called hyperdiffusivity of order $p$. This allows us to establish a universality result concerning the independence of limits on $p$. In this case, the closed correlation equations (\ref{closeq}) are still satisfied, with the operator (\ref{singell}) replaced by \be \oH_N^{(\kappa_p)}= -{{1}\over{2}}\sum_{i,j=1}^d\sum_{n,m=1}^N\,\, {{\partial}\over{\partial x_{in}}}\left[V_{ij}(\br_n-\br_m){{\partial}\over{\partial x_{jm}}}\cdot\right] +\kappa_p\sum_{n=1}^N(-\bigtriangleup_{\br_n})^p. \lb{psingell} \ee Note that this operator requires higher-order Dirichlet b.c., namely, elements in its domain must have zero trace on the boundary for the first $k=[\![p-(1/2)]\!]$ derivatives. However, our first main result is for the solution of that equation directly at $\kappa_p=0$: \begin{Th} Assume that $d\geq 2$ and $0<\zeta<2$. Then, for integers $N\geq 1$, the equation (\ref{closeq}) at $\kappa=0$ has a unique stationary weak solution $\Theta_N^*$ in $L^2\left(\Omega^{\otimes N}\right)$. Away from the codimension-$d$ set where pairs of points in $\bR=(\br_1,...,\br_N)$ coincide, the solution $\Theta_N^*(\bR)$ is in $H^1_0\left(\Omega^{\otimes N} \right)$. \end{Th} This ideal zero-diffusivity solution is, in fact, the physically relevant one in the limits $\kappa_p\rightarrow 0$, as shown by our second main result: \begin{Th} Assume that $d\geq 2$ and $0<\zeta<2$, and also $p\geq 1$. \noindent (i) For integers $N\geq 1$, the equation (\ref{closeq}), with $\oH^{(\kappa)}$ generalized to $\oH^{(\kappa_p)}$, has a unique stationary weak solution $\Theta_N^{(\kappa_p)*}$ in $L^2\left(\Omega^{\otimes N}\right)$, which, in fact, belongs to the Sobolev space $H^p_0\left(\Omega^{\otimes N}\right)$. \noindent (ii) The weak-$L^2$ limit exists as $\kappa_p\rightarrow 0$ and $w-\lim_{\kappa_p\rightarrow 0} \Theta_N^{(\kappa_p)*}=\Theta_N^*$. \end{Th} \noindent To prove these results requires a spectral analysis of the $N$-body convective operator $\oH_N$. In fact, we show that this operator has pure point spectrum, using a criterion borrowed from a work of R. T. Lewis \cite{Lew}. Discreteness of the spectrum was already shown by Majda \cite{Maj} in his simple version of the model. For our theorems above, we do not really require that $\oH_N$ have a compact inverse, but merely a bounded inverse. To prove this, we require an estimate from below on the quadratic form associated to $\oH_N$. This is proved in two steps. First, for each integer $N\geq 1$ we define the $(Nd)\times(Nd)$-dimensional matrix $[\bG_N(\bR)]_{in,jm}= \langle v_i(\br_n)v_j(\br_m)\rangle,\,\,\,i,j=1,...,d,n,m=1,...,N.$ Physically, this is interpreted as an {\em N-particle eddy-diffusivity matrix}. Mathematically, it is the nonnegative Gramian matrix of the $Nd$ elements $v_i(\br_n)$ in the $L^2$ inner-product space of the random velocity field. It is nonsingular if and only if these $Nd$ elements are linearly independent. We shall prove below (Proposition 2) that its minimum eigenvalue obeys $\lambda_N^{\min}(\bR)\geq C_N [\rho(\bR)]^\zeta$, where $\rho(\bR)=\min_{n\neq m}|\br_n-\br_m|$, when $N=2,3,4$. The second step of the proof uses only this property of $\bG_N(\bR)$, which is conjectured to hold for all $N\geq 1$. As a consequence of this estimate, we prove a lower bound on the operator quadratic form, reminiscent of the well-known {\em Hardy inequality} \cite{HLP} (Theorem 330). For the operator with Dirichlet b.c. we may adapt a convenient proof of the Hardy-type inequality due also to Lewis \cite{Lew}. Unfortunately, as explained below, this proof does not work with periodic b.c. although the inequality is likely to hold there as well (for zero-mean functions). Lewis' argument is also too restrictive to permit treatment of other models with more natural b.c. on the velocity field. In a real turbulent flow with velocity field governed by the Navier-Stokes equation, the realizations of the velocity field would satisfy also Dirichlet b.c. This behavior may be mimicked with the Gaussian random velocity fields by taking as their covariance \be V_{ij}^{(\Omega)}(\br,\br')=\Delta_\Omega(\br)V_{ij}(\br-\br') \Delta_\Omega(\br'), \lb{dampvel} \ee in which $\Delta_\Omega(\br)$ is a suitable ``wall-damping function''. It should be taken as some decreasing function of the distance to the boundary $\partial\Omega$, vanishing there as some power. Of course, with this choice of velocity covariance, a lower bound directly follows from our present work that $\lambda_N^{\min}(\bR)\geq C_N [\rho(\bR)]^\zeta[\Delta_\Omega(\bR)]^2$, where $\Delta_\Omega(\bR)=\min_{1\leq n\leq N}\Delta_\Omega(\br_n)$. While we expect the main results of this work to carry over to such models, it requires a different proof of the generalized Hardy inequality. We will return to this problem in a later work. Let us summarize the contents of this paper: In Section 2 we establish the required properties of the model velocity covariance and the resulting $N$-particle eddy-diffusivity matrix, in particular the lower bound on the minimum eigenvalue. In Section 3 we study the operator quadratic form, and prove its principal properties, such as the generalized Hardy inequality. Finally, in Section 4 we exploit these results to prove the main Theorems 1 and 2 above. In the conclusion Section 5 we briefly discuss three other problems: regularity of solutions, the reconstruction of an invariant measure from the stationary $N$-point correlation functions, and time-dependent solutions to the parabolic PDE's for the $N$-correlators. \newpage \section{Properties of the N-Particle Eddy-Diffusivity Matrix} \noindent {\em (2.1) The Velocity Covariance Matrix} \noindent We first state and prove the regularity properties of the velocity covariance matrix elements $(V_{ij}(\br))$ that we will need for later analysis. We have made the choice of Eq.(\ref{Vspec}) just for specificity. In fact, any velocity covariance with the following properties would suffice. \begin{Lem} The elements of velocity covariance matrix $V_{ij}(\br)$, $\br \in \bR^d$, are $C^{\infty}$ in $\br$ if $\br \neq 0$, and $C^{\zeta}$ near $\br =\bz$, with $\zeta \in (0,2)$. Moreover, there is a positive number $\rho_{0}$ such that if $r \in [0,\rho_{0}]$, we have the local expansion: \be V_{ij}(\br) = V_{0}\delta_{ij} - D_{1}\cdot r^\zeta\cdot\left[\delta_{ij} +{{\zeta}\over{d-1}}\left(\delta_{ij}- {{r_ir_j}\over{r^2}}\right)\right]+ O\left (m^2r^2\right ) \label{eq:A1} \ee \end{Lem} \noindent {\em Proof:} The matrix $V_{ij}(\br)$ can be written as \be V_{ij}(\br)= V(r)\delta_{ij}+\partial_i\partial_jW(r), \lb{Vrep} \ee where the function $V(r)$ is defined by the integral \be V(r)= D_0\int {{d^d\bk}\over{(2\pi)^d}}\,\,\left(k^2+m^2\right)^{-(d+\zeta)/2} e^{i\bk\bdot\br} \lb{Bespot} \ee and $W(r)$ is given by the (for $d=2$, principal part) integral \be W(r)= D_0\int {{d^d\bk}\over{(2\pi)^d}}\,\,\left(k^2+m^2\right)^{-(d+\zeta)/2}{{1}\over{k^2}} e^{i\bk\bdot\br}, \lb{Wftn} \ee so that $-\bigtriangleup W=V$. The scalar function $V(r)$ is essentially just the standard Bessel potential kernel \cite{ArS}, and may thus be expressed in terms of a modified Bessel function: \be V(r)=D_0{{2^{1-(\zeta/2)}m^{-\zeta}}\over{(4\pi)^{d/2} \Gamma\left({{d+\zeta}\over{2}}\right)}}\cdot (mr)^{\zeta/2}K_{\zeta/2}(mr). \lb{Besrep} \ee The Hessian $\partial_i\partial_jW(r)$ of the function $W$ of magnitude $r=|\br|$ alone is \be \partial_i\partial_jW(r)=\delta_{ij} A(r)+ \hr_i\hr_j\cdot r{{dA}\over{dr}}(r), \lb{Hess} \ee with $A(r)= W'(r)/r$ and $\widehat{\br}=\br/r$. However, because ${\rm Tr}\left(\grad\otimes\grad W\right)= -V$, a Cauchy-Euler equation follows for $A(r)$: \be r{{dA}\over{dr}}(r)+ d\cdot A(r)= -V(r). \lb{CEeq} \ee Due to the rapid decay of its Fourier transform, the function $A(r)$ is continuous. Thus, the relevant solution is found to be \be A(r) = -r^{-d}\int_0^r \rho^{d-1} V(\rho) d\rho. \lb{AinV} \ee in terms of $V(r)$. Using this expression for $A(r)$, along with Eq.(\ref{Hess}), we thus find \be V_{ij}(\br)= (V(r)+A(r))\delta_{ij}-(V(r)+d\cdot A(r))\hr_i\hr_j, \lb{Gexp} \ee for $V_{ij}$ as a linear functional of $V$. If $V$ has a power-law form, $V(r)=B r^\xi$, then it is easy to calculate that \be V_{ij}(\br)=Br^\xi{{d-1}\over{d+\xi}}\left[\delta_{ij} +{{\xi}\over{d-1}}\left(\delta_{ij}-\hr_i\hr_j\right)\right]. \lb{powfrm} \ee By means of the known Frobenius series expansions for the modified Bessel functions (e.g. \cite{AbS}, (9.6.2),(9.6.10)), it follows that \be z^\nu K_\nu(z)= {{\Gamma(\nu)}\over{2^{1-\nu}}}-{{\Gamma(1-\nu)}\over{\nu\cdot 2^{1+\nu}}}z^{2\nu} +O\left(z^2\right). \lb{Frobser} \ee {}From these terms for $K_\nu(z)$ we obtain, upon substituting Eq.(\ref{Besrep}) into Eq.(\ref{Gexp}), the claimed asymptotic expression for $V_{ij}(\br)$ in Eq.(\ref{eq:A1}), with \be V_0= D_0{{(d-1)\Gamma\left({{\zeta}\over{2}}\right)}\over{(4\pi)^{d/2}\cdot d\cdot \Gamma\left({{d+\zeta}\over{2}}\right)}}\cdot m^{-\zeta}, \lb{Vzero} \ee and \be D_1= D_0{{(d-1)\Gamma\left({{2-\zeta}\over{2}}\right)}\over{(4\pi)^{d/2}\cdot 2^\zeta\cdot\zeta\cdot \Gamma\left({{d+\zeta+2}\over{2}}\right)}}. \lb{Done} \ee Finally, the Bessel function $K_\nu(z)$ is analytic in the complex plane with a branch cut along the negative real axis. Thus, the stated smoothness properties of $V_{ij}$ follow. $\,\,\,\,\,\Box$ \vspace{.1in} \noindent We shall denote the second term on the right hand side of (\ref{eq:A1}) as $-r^{\zeta}Q_{ij}$. Obviously, $(Q_{ij})$ is positive definite uniformly in $r$. We will denote by $\br_{nm}= \br_{n} - \br_{m}$ the vector, and $r_{nm}=|\br_{n} -\br_{m}|$ the scalar distance from $\br_{n}$ to $\br_{m}$; $V_{ij}$ the matrix elements, and $\bV_{nm}$ the matrix evaluated at $\br_{nm}$. We show two more lemmas. \begin{Lem} Let $\br_{i}$, $r=1,2,3$, be any three points in $R^{d}$, and $r_{12} \leq r_{13}$, $r_{12} \leq r_{23}$. Then there is a constant $\bar{C}$ depending on $\rho_{0}$ and $\zeta$ in Lemma 1 but independent of $r_{12}$, $r_{13}$, and $r_{23}$ such that: \[ | \bV_{13} - \bV_{23} | \leq \bar{C}r_{12}\max(r_{13}^{\zeta -1}, r_{23}^{\zeta-1}). \] \end{Lem} {\em Proof}: If $\zeta \in (1,2)$, then $\nabla \bV \in C^{\zeta -1}$, and so by Lemma 1: \[ |\bV_{13} -\bV_{23}| = | \br_{12} \cdot \nabla_{\br_{1}} \bV |_{\br_{\theta}}|= |\br_{12}\cdot (\nabla_{\br_{1}}\bV|_{\br_{\theta}} - \nabla_{\br_{1}}\bV|_{\br = 0})|, \] \[ \leq \bar{c}r_{12} r_{\theta}^{\zeta -1} \leq \bar{c} r_{12} \max(r_{12}^{\zeta -1},r_{23}^{\zeta -1}), \] where $\br_{\theta} =\theta \br_{1} + (1-\theta)\br_{2}$, for some $\theta \in (0,1)$. The case $\zeta =1$ is obviously true by the mean value theorem. Now if $\zeta \in (0,1),\; \max (r_{13},r_{23}) \geq \rho_{0},$ then using $\bV \in C^1$ away from zero, we have: \[ | \bV_{13} -\bV_{23} | \leq \bar{c} r_{12} \leq \bar{c}r_{12}(m\max (r_{13}, r_{23}))^{\zeta -1} \] \[ \leq \bar{c}(\rho_{0},m)r_{12} \max (r_{13}^{\zeta -1},r_{23}^{\zeta -1}). \] If $\zeta \in (0,1)$, and $\max(r_{13},r_{23}) < \rho_{0}$, we employ local expansion to calculate for any $\bx \not = \by$: \[ | V_{ij}(\bx) -V_{ij}(\by) | \leq \bar{c} |(|\bx|^{\zeta} - |\by|^{\zeta})[\delta_{ij}+{\zeta \over (d-1)}(\delta_{ij} - { x_{i} x_{j} \over x^{2}})] | \] \[ + \bar{c}|y^{\zeta}({x_{i}x_{j}\over x^{2}} - {y_{i}y_{j}\over y^{2}})| \] \[ \leq \bar{c}\max(x^{\zeta -1},y^{\zeta -1})|\bx -\by| + \bar{c}y^{\zeta}\left |{x_{i}x_{j}y^{2} -y_{i}y_{j}x^{2} \over x^{2}y^{2}}\right |. \] The latter term is just: \[ \bar{c}y^{\zeta}|{(x_{i}x_{j}-y_{i}y_{j})y^{2} + y_{i}y_{j}(y^{2}-x^{2}) \over x^{2}y^{2}} | \] \[ = \bar{c}y^{\zeta}\left ({|\bx -\by|\over |x|} + {|\bx -\by|y \over x^{2}} + {|\bx -\by|(x+y)\over x^{2}}\right ). \] With no loss of generality, we assume that $y \leq x$; otherwise, we simply switch $\bx $ and $\by$. It follows that \[ |V_{ij}(\bx) -V_{ij}(\by)| \leq \bar{c}|\bx -\by|\max(x^{\zeta -1}, y^{\zeta -1}) + \bar{c}|\bx -\by|x^{\zeta -1} \leq \bar{c}|\bx -\by|\max(x^{\zeta -1},y^{\zeta -1}). \] We complete the proof with $\bx = \br_{13}$, and $\by =\br_{23}$. \begin{Lem} Assume that $r_{12} \leq r_{34}$; $r_{13} =O(r_{14}) =O(r_{23}) = O(r_{24})$; ${r_{34} \over r_{13}} \leq \eps \in (0,1)$. Then there exist $\eps_{0}$ and a positive constant $\bar{c}_{1}$ depending on $\rho_{0}$, $\zeta$, maximum and minimum ratios of $r_{13}$, $r_{14}$, $r_{23}$, and $r_{24}$, such that: \[ |\bV_{13} -\bV_{14} - (\bV_{23} -\bV_{24})| \leq \bar{c}_{1}r_{12} r_{34}r_{13}^{\zeta -2}, \] for all $\eps \in (0,\eps_{0})$. \end{Lem} {\em Proof}: Applying the mean value theorem to $F(\br_1) \equiv \bV_{13}- \bV_{14}$, we get for $\br_{\theta}=\theta \br_{1} + (1-\theta)\br_{2}$ that: \[ F(\br_{1}) -F(\br_{2}) = \br_{12} \cdot\nabla_{\br_{1}}F|_{\br_{\theta}}.\] If $\max (r_{13},r_{24}) \geq {\rho_{0}\over 2}$, then \[ \nabla_{\br_{1}}F|_{\br_{\theta}} =\nabla_{\br_{1}}\bV_{13} -\nabla_{\br_{1}}\bV_{14}|_{\br_{1} =\br_{\theta}}. \] By the smoothness of $\nabla_{\br_1}\bV_{1i}$ when the distance of $\br_1$ from $\br_{i},\,\,\,i=3,4$ is larger than ${\rho_{0} \over 4}$ (which is possible if $\eps$ is small enough), we obtain: \[ | \nabla_{\br_{1}}F|_{\br =\br_{\theta}} | \leq \bar{c}_{1}\rho_{0}r_{34}, \] from which it follows that: \[ |F(\br_{1}) -F(\br_{2})| \leq \bar{c}_{1}r_{12} r_{34} \leq \bar{c}_{1}r_{12}r_{34}r_{13}^{\zeta -2}. \] On the other hand, if $\max(r_{13},r_{24}) < {\rho_{0} \over 2}$, we use local expansion in Lemma 1 to get for each matrix element: \begin{eqnarray} (F(\br_{1}) - F(\br_{3}))_{ij} & = & (-D_{1})r_{13}^{\zeta}[ \delta_{ij} + {\zeta \over d-1}(\delta_{ij} -{\br_{13}^{(i)}\br_{13}^{(j)} \over r_{13}^{2}})] \nonumber \\ & + & D_{1} r_{14}^{\zeta}[\delta_{ij}+{\zeta \over d-1}( \delta_{ij} -{\br_{14}^{(i)}\br_{14}^{(j)} \over r_{14}^{2}})] \nonumber \\ & - & (1 \rightarrow 2) \nonumber \\ & = & (-D_{1})\br_{12} \cdot \nabla_{\br_{1}}( r_{13}^{\zeta}[ \delta_{ij} + {\zeta \over d-1}(\delta_{ij} -{\br_{13}^{(i)}\br_{13}^{(j)} \over r_{13}^{2}})] \nonumber \\ & - & r_{14}^{\zeta}[\delta_{ij}+{\zeta \over d-1}( \delta_{ij} -{\br_{14}^{(i)}\br_{14}^{(j)} \over r_{14}^{2}})]) (\br_{1} =\br_{\theta}), \label{eq:L1} \end{eqnarray} where the notation $(1 \rightarrow 2)$ means the same terms as before except that subscript $1$ is replaced by $2$. Let us calculate the $\br_{1}$ gradient in (\ref{eq:L1}) as ($k$ meaning the $k$th component of this gradient): \[ \zeta r_{13}^{\zeta -1}{\br_{1}^{(k)} -\br_{3}^{(k)} \over r_{13}} [\delta_{ij}+{\zeta \over d-1}( \delta_{ij} -{\br_{13}^{(i)}\br_{13}^{(j)} \over r_{13}^{2}})] + r_{13}^{\zeta}{-\zeta\over d-1}\cdot \nabla_{\br_{1}} {\br_{13}^{(i)}\br_{13}^{(j)}\over r_{13}^{2}} - (3\rightarrow 4) \] \begin{eqnarray} & =& \zeta (r_{13}^{\zeta -1}-r_{14}^{\zeta -1}) {\br_{1}^{(k)} -\br_{3}^{(k)} \over r_{13}} [\delta_{ij}+{\zeta \over d-1}( \delta_{ij} -{\br_{13}^{(i)}\br_{13}^{(j)} \over r_{13}^{2}})] \nonumber \\ & + & \zeta r_{14}^{\zeta -1}\left ( {\br_{1}^{(k)} -\br_{3}^{(k)} \over r_{13}} [\delta_{ij}+{\zeta \over d-1}( \delta_{ij} -{\br_{13}^{(i)}\br_{13}^{(j)} \over r_{13}^{2}})] - (3 \rightarrow 4) \right ) \nonumber \\ & + & {-\zeta \over d-1}\left ( r^{\zeta}_{13}{-2\br_{13}^{(i)}\br_{13}^{(j)}\br_{13}^{(k)}\over r_{13}^{4}} +r^{\zeta}_{13}{\delta_{ik}\br_{13}^{(j)}\over r_{13}^{2}} + r_{13}^{\zeta}{\br_{13}^{(i)}\delta_{jk}\over r_{13}^{2}} - (3 \rightarrow 4) \right ). \label{eq:L2} \end{eqnarray} Note that the first term of the right hand side of (\ref{eq:L2}) is bounded by: \[ C(\zeta,d)|r_{13}^{\zeta -1} -r_{14}^{\zeta -1}| \leq C(\zeta,d) r_{23}^{\zeta -2}r_{34}. \] We can think of \[ {\br_{1}^{(k)} -\br_{3}^{(k)} \over r_{13}} [\delta_{ij}+{\zeta \over d-1}( \delta_{ij} -{\br_{13}^{(i)}\br_{13}^{(j)} \over r_{13}^{2}})] \] as a bounded $C^{1}$ function of the unit vector $\hat{\br}_{13}$ along $\br_{13}$. Hence the second term of (\ref{eq:L2}) being the difference of two values of this function at two points $\hat{\br}_{13}$ and $\hat{\br}_{14}$ is of the order $O({r_{34}\over r_{13}})$. Thus the second term is bounded by \[ \bar{c}_{1}r_{14}^{\zeta -1}r_{34}r_{13}^{-1}\leq \bar{c}_{1}r_{13}^{\zeta -2}r_{34}.\] Similarly, the third term is bounded as such. Combining the above with (\ref{eq:L1}) we deduce that $|F(\br_{1})-F(\br_{3})| \leq \bar{c}_{1}r_{12}r_{34}r_{13}^{\zeta -2}$. The proof of the lemma is complete. \noindent {\em (2.2) The N-Point Eddy-Diffusivity (Gramian) Matrix} \noindent As in the Introduction, we define for each integer $N\geq 1$ the $(Nd)\times(Nd)$-dimensional Gramian matrix $[\bG_N(\bR)]_{in,jm}=\langle v_i(\br_n)v_j(\br_m)\rangle.$ For the moment we consider general velocity covariances, given by a Fourier integral \be V_{ij}(\br)= \int {{d^d\bk}\over{(2\pi)^d}}\,\,\widehat{V}_{ij}(\bk)e^{i\bk\bdot\br}, \lb{gVspec} \ee with $\widehat{\bV}(\bk)\geq \bz$ for each $\bk\in\bR^d$. The basic properties are contained in: \begin{Prop}For each $N\geq 2$ the matrix $\bG_N(\bR)$ has the following properties: (i) $\bG_N(\bR)\geq \bz$. (ii) Assume that for all $\bk\in\bR^d$ the velocity spectral matrix $\widehat{\bV}(\bk)>\bz$ on the subspace orthogonal to the vector $\bk$. In that case, $\bG_N(\bR)$ has a nontrivial null space if and only if $\br_n=\br_m$ for some pair of points $n\neq m$. (iii) For the same hypothesis as (ii), if $\{\br_1,...,\br_N\}$ has $K$ subsets of coinciding points, with $N_k$ points in the $k$th subset, $k=1,...,K,$ then the dimension of the null space of $\bG_N(\bR)$ is $\sum_{k=1}^K (N_k-1)d.$ The null space consists precisely of vectors $\Bxi=(\bxi_1,...,\bxi_N)$ with the property that \be \sum_{n_k=1}^{N_k} \bxi_{n_k}=\bz, \lb{kernel} \ee for each $k=1,...,K$, where the sum runs over the $N_k$ coinciding points in the $k$th subset. \end{Prop} {\em Proof}: {\em (i)} Obvious from the stochastic representation. {\em (ii)}\& {\em (iii)} Let us assume that the $Nd$-dimensional vector $\Bxi=(\bxi_1,...,\bxi_N)$ belongs to ${\rm Ker}\bG_N(\bR)$. Then, using the definition of $\bG_N(\bR)$ and the Fourier integral representation Eq.(\ref{gVspec}), it follows that \be 0=\langle\Bxi,\bG_N(\bR)\Bxi\rangle= \int {{d^d\bk}\over{(2\pi)^d}}\,\,\overline{\left(\sum_{n=1}^N\bxi_n e^{i\bk\bdot\br_n}\right)}\bdot \widehat{\bV}(\bk)\bdot\left(\sum_{n=1}^N\bxi_n e^{i\bk\bdot\br_n}\right). \lb{nullcon1} \ee This can only occur if the nonnegative integrand vanishes for a.e. $\bk\in\bR^d$. Because of our assumption on $\widehat{\bV}(\bk)$, this implies that \be \sum_{n=1}^N\bxi_n e^{i\bk\bdot\br_n}=\alpha(\bk)\cdot\bk, \lb{nullcon2} \ee for a.e. $\bk\in\bR^d$ with some complex coefficient $\alpha(\bk)$. Taking the vector cross product with respect to $\bk$ and then Fourier transforming, we obtain that \be \sum_{n=1}^N \bxi_n\btimes \grad\delta(\br-\br_n)=\bz, \lb{nullcon3} \ee in the sense of distributions. Therefore, for any smooth test function $\varphi$, \be \sum_{n=1}^N \bxi_n\btimes (\grad\varphi)(\br_n)=\bz. \lb{nullcon4} \ee Because the values of $\grad\varphi$ may be arbitrarily specified at any set of distinct points, it follows that \be \sum_{k=1}^K \left(\sum_{n_k=1}^{N_k}\bxi_{n_k}\right)\btimes \ba_k=\bz \lb{nullcon5} \ee with $\ba_k\in\bR^d$ arbitrary. This immediately implies that Eq.(\ref{kernel}) is both necessary and sufficient for $\Bxi$ to belong to ${\rm Ker}\bG_N(\bR)$. Furthermore, this subspace has dimension $\sum_{k=1}^K (N_k-1)d$, which completes the proof of {\em (iii)}. Finally, {\em (ii)} follows from {\em (iii)} by observing that ${\rm Ker}\bG_N(\bR) =0$ if and only if $K=N$ and $N_k=1$ for all $k=1,...,N$. $\,\,\,\,\,\Box$ For the particular choice of covariance function defined by Eq.(\ref{Vspec}) for $0<\zeta<2$, we need also the following crucial lower bound: \begin{Prop} For each $0<\zeta<2$ and $d\geq 2$, there exists for each $N\geq 2$ a constant $C_N=C_N(d,\zeta)>0$ so that the minimum eigenvalue $\lambda_N^{\min}(\bR)$ of $\bG_N(\bR)$ satisfies \be \lambda_N^{\min}(\bR)\geq C_N\cdot [\rho(\bR)]^\zeta, \lb{lowbd} \ee with $\rho(\bR)=\min_{n\neq m}r_{nm}$. \end{Prop} \noindent The above property will be proved in detail in this paper for $N=2,3,4$. While the proof in these cases strongly suggests the result is true for all $N\geq 2$, the argument becomes increasingly complicated for larger values of $N$. We shall leave the discussion of the general $N$ to a future publication, although we point out that many parts of the argument below apply for the general case. Note that we can view $\bG_N$ as a matrix parametrized by the $\zeta$ power of the minimum distance, $\eps \equiv \rho^{\zeta}$. Let $\lambda_N^{\min}=\lambda_N(\eps)$ be the minimum positive eigenvalue of $\bG_N$ with corresponding unit eigenvector $\Bxi_N^{\min}=\Bxi_N(\eps)$. Then, by the standard formulae of degenerate first-order perturbation theory (see Kato, \cite{Kato}): \be \lam_N(\eps) = \langle\Bxi_N(0),\bG_N(\eps)\Bxi_N(0)\rangle+ O(\eps^{2}). \label{eq:A3} \ee We have used the fact that $\lam_N(\eps)$ is at least twice differentiable in $\eps$ near zero: see \cite{Kato}, Theorems II.1.8 and II.6.8. Furthermore, $\Bxi_N(0)$ is in the null space of $\bG_N(0)$. Thus, by Proposition 1{\em (iii)}, $\Bxi_N(0)=(\bxi_{1},...,\bxi_{N})$ such that $\sum_{n=1}^N\bxi_{n} = 0$. By simply minimizing over this entire subspace of vectors $\Bxi$, we shall show that the righthand side quadratic form of (\ref{eq:A3}), denoted by $Q_N(\bxi_{1},\cdots, \bxi_{N-1})$, is bounded from below by a constant times $\eps$. Thus $\lam(\eps)$ obeys the same type of lower bound. \noindent {\bf Proposition 2, N=3 Case} {\bf \noindent Remark:} The following proof for Proposition 2, $N=3$, also implies the lower bound $C_2r_{12}^{\zeta}$ for the $N=2$ case. \noindent {\em Proof:} Let $\br_{n}$, $n=1,2,3$, be three distinct points in $\bR^{d}$, $d\geq 2$. Then we show that there is a positive constant $C_3= C_3(\rho_{0})$, where $\rho_{0}$ is the scale of local approximation (\ref{eq:A1}), such that the minimum eigenvalue of $\bG_3$ is bounded from below by $C_3 \rho^{\zeta}$. It suffices to treat the situation where $\rho \leq \rho_{0}$, otherwise, we conclude with Proposition 1. Let $C_{0}$ be a large but $O(1)$ constant to be determined, and let $r_{12}=\rho$ for definiteness. \noindent Case I: Suppose now that ${ r_{13} \over \rho}\leq C_{0}$, and ${r_{23} \over \rho} \leq C_{0}$. By further reducing the size of $\rho$, we can ensure that $\rho C_{0} \leq \rho_{0}$. Now write: \[ \left ( \begin{array}{r} \bxi_{1} \\ \bxi_{2} \\ -\bxi_{1} -\bxi_{2} \end{array} \right ) = \left ( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & -1 & 1 \end{array} \right ) \cdot \left ( \begin{array}{r} \bxi_{1} \\ \bxi_{2} \\ 0 \end{array} \right ), \] then: \[ Q_3 = \langle(\bxi_{1},\bxi_{2}); \left ( \begin{array}{rr} 2(\bV(0) -\bV_{13}) & \bV(0)+\bV_{12} -\bV_{13} -\bV_{23} \\ \bV(0) +\bV_{12} -\bV_{13} -\bV_{23} & 2(\bV(0) -\bV_{23}) \end{array} \right ) \cdot \left ( \begin{array}{rr} \bxi_{1} \\\bxi_{2} \end{array} \right ) \rangle. \] Since all the three distances are less than $\rho_{0}$, we apply lemma 1 to see that ${|\bV(0) -\bV_{ij}|\over r_{ij}^{\zeta}} \leq C_{0}$. Therefore we can factor out $\rho^{\zeta}$. The remaining entries are bounded by $C_{0}$, and we also know that they form a positive definite matrix. Hence by continuity of eigenvalues on the matrix entries, we get the bound: \be Q_3 \geq \mu_{1}(C_{0})\rho^{\zeta}, \label{eq:A4} \ee for some positive constant $\mu_{1}=\mu_{1}(C_{0})$. \noindent Case II: Suppose ${r_{13}\over \rho} > {C_{0} \over 2}$, ${r_{23} \over \rho} > { C_{0}\over 2}$. By geometric constraint, $\frac{r_{13}}{r_{23}} = 1 + O(C_{0}^{-1})$. To estimate $Q_3$ from below, we decompose the vectors $\{ (\bxi_1,\bxi_2,-(\bxi_1 + \bxi_2)) \}$ into the orthogonal sum of $\{ (\bar{\bxi}_{1},-\bar{\bxi}_{1},0)\}$ and $\{(\bxi'_1,\bxi'_1,-2\bxi'_1)\}$. Then $Q_3$ is expressed into the sum of three terms as: \[ Q_3(\bxi_1,\bxi_2) = \langle (\bxi_1,\bxi_2,-(\bxi_1 + \bxi_2)), \bG_3 (\bxi_1,\bxi_2,-(\bxi_1 +\bxi_2))^{T}\rangle \] \[ = \langle(\bar{\bxi}_{1},-\bar{\bxi}_{1},0), \bG_3(\bar{\bxi}_{1},-\bar{\bxi}_{1},0)^{T}\rangle \] \[ + \langle (\bxi'_{1},\bxi'_1,-2\bxi'_1),\bG_3(\bxi'_1,\bxi'_1,-2\bxi'_1)^{T}\rangle \] \be + 2\langle(\bar{\bxi}_1,-\bar{\bxi}_{1},0), \bG_3(\bxi'_1,\bxi'_1,-2\bxi'_1)^{T}\rangle. \label{eq:A5} \ee Write: \[ \left (\begin{array}{r} \bar{\bxi}_{1}\\ - \bar{\bxi}_{1}\\ 0 \end{array} \right ) = \left (\begin{array}{rrr} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right ) \left (\begin{array}{r} \bar{\bxi}_1\\ 0\\ 0 \end{array} \right ), \] then the bar term of (\ref{eq:A5}): \[ \langle(\bar{\bxi}_{1},0,0); \left (\begin{array}{rrr} 1 & -1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right )\left (\begin{array}{rrr} \bV(0)-\bV_{12} & \bV_{12} & \bV_{13}\\ \bV_{12} -\bV(0) & \bV(0) & \bV_{23} \\ \bV_{13}-\bV_{23} & \bV_{23} & \bV(0) \end{array} \right )\left ( \begin{array}{r} \bar{\bxi}_{1} \\0 \\0 \end{array} \right ) \rangle \] \be = 2 \langle\bar{\bxi}_1,(\bV(0)-\bV_{12})\bar{\bxi}_1\rangle \geq \bar{c}_1\rho^{\zeta} |\bar{\bxi}_1|^{2}, \label{eq:A6} \ee where $\bar{c}$ here and after will denote a positive constant depending only on $\rho_{0}$. Also $1$ is a shorthand for $d\times d$ identity matrix. Similarly, we express: \[ \left (\begin{array}{r} \bxi'_1 \\ \bxi'_{1}\\ -2\bxi'_{1} \end{array} \right ) = \left (\begin{array}{rrr} 1 & 0 & 0 \\ 1 & 1 & 0 \\ -2 & 0 & 1 \end{array} \right )\left (\begin{array}{r} \bxi'_1 \\ 0 \\ 0 \end{array} \right ) \] and write the prime term by Lemma 2 as: \[ \langle(\bxi'_{1},0,0), \left (\begin{array}{rrr} 1 & 1 & -2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right )\left (\begin{array}{rrr} \bV(0)+\bV_{12}-2\bV_{13} & \bV_{12} & \bV_{13}\\ \bV_{12} +\bV(0)-2\bV_{23} & \bV(0) & \bV_{23} \\ \bV_{13}+\bV_{23}-2\bV(0) & \bV_{23} & \bV(0) \end{array} \right )\left (\begin{array}{r} \bxi'_1 \\ 0 \\ 0 \end{array} \right )\rangle\] \[ = \langle \bxi'_1,(6\bV(0)+2\bV_{12} -4\bV_{13}-4\bV_{23})\bxi'_1\rangle \] \[ = \langle\bxi'_1,8(\bV(0)-\bV_{13})\bxi'_1\rangle + \langle\bxi'_1,\left(2(\bV_{12}-\bV(0)) + 4(\bV_{13}-\bV_{23})\right )\bxi'_{1}\rangle\] \[ \geq \bar{c}r_{13}^{\zeta}|\bxi'_{1}|^{2} - \bar{c}(\rho^{\zeta} + r_{12}r_{13}^{\zeta -1})|\bxi'_{1}|^{2} \] \be \geq \bar{c}r_{13}^{\zeta}|\bxi'_{1}|^{2}( 1-\bar{c}\left( (\rho r_{13}^{-1})^{\zeta} + (\rho r_{13}^{-1})\right ))\geq \bar{c}_1 r^{\zeta}_{13} |\bxi_1'|^{2}. \label{eq:A7} \ee The mixed term is equal to : \[ \langle (\bar{\bxi}_1,-\bar{\bxi}_{1},0), \left (\begin{array}{rrr} \bV(0)+\bV_{12}-2\bV_{13} & \bV_{12} & \bV_{13}\\ \bV_{12} +\bV(0)-2\bV_{23} & \bV(0) & \bV_{23} \\ \bV_{13}+\bV_{23}-2\bV(0) & \bV_{23} & \bV(0) \end{array} \right )\left (\begin{array}{r} \bxi'_1 \\ 0 \\ 0 \end{array} \right )\rangle\] \[ = \langle\bar{\bxi}_1,(\bV(0) +\bV_{12} -2\bV_{13})\bxi'_{1}\rangle - \langle\bar{\bxi}_{1}, (\bV_{12}+\bV(0)-2\bV_{23})\bxi'_{1}\rangle, \] \[ = \langle\bar{\bxi}_1,2(\bV_{23}-\bV_{13})\bxi'_{1}\rangle, \] and so is bounded by: \be | mixed\;\; term| \leq \bar{c}_2 |\bxi'_1|\cdot |\bar{\bxi}_{1}|\cdot r_{12} r_{13}^{\zeta -1}. \label{eq:A8} \ee Thus: \be Q_3 = Q_3(\bxi_1,\bxi_2) \geq \bar{c}_1\rho^{\zeta}|\bar{\bxi}_{1}|^{2} + \bar{c}_1 r_{13}^{\zeta}|\bxi'_{1}|^{2} - \bar{c}_2 |\bxi'_1|\cdot |\bar{\bxi}_{1}|r_{12}r_{13}^{\zeta -1} \label{eq:A9} \ee The mixed term may then be controlled by the positive terms through the following Young's inequality: \begin{eqnarray} |\bxi'_1|\cdot |\bar{\bxi}_{1}|\rho r_{13}^{\zeta -1} & = & \sqrt{\theta\rho^\zeta}|\bar{\bxi}_{1}|\cdot {{\rho^{1-{{\zeta}\over{2}}}r_{13}^{\zeta -1}}\over{\sqrt{\theta}}}|\bxi'_1| \cr \, & \leq & {{1}\over{2}}\theta\cdot \rho^\zeta|\bar{\bxi}_{1}|^2+ {{(\rho/r_{13})^{2-\zeta}}\over{2\theta}}\cdot r_{13}^{\zeta}|\bxi'_1|^2, \label{eq:A10} \end{eqnarray} with $\theta$ a small number in $(0,1)$. Then, since $\rho/r_{13}<2C_0^{-1}$, it follows that for any $\zeta<2$, $(\rho/r_{13})^{2-\zeta}<\theta^2$ for $C_0$ large enough. Thus, \be |\bxi'_1|\cdot |\bar{\bxi}_{1}|\rho r_{13}^{\zeta -1} \leq {{1}\over{2}}\theta\cdot \rho^\zeta|\bar{\bxi}_{1}|^2+ {{1}\over{2}}\theta\cdot r_{13}^{\zeta}|\bxi'_1|^2, \label{eq:A11} \ee which allows the mixed term to be absorbed into the positive bar and prime terms. Combining (\ref{eq:A9}-\ref{eq:A11}), we conclude that: \be Q_3(\bxi_1,\bxi_2) \geq \bar{c}\rho^{\zeta}|\bar{\bxi}_{1}|^{2} + \bar{c}r_{13}^{\zeta}|\bxi'_1|^{2}, \label{eq:A12} \ee which in the original $(\bxi_1,\bxi_2)$ variables reads: \be Q_3(\bxi_1,\bxi_2) \geq \bar{c}\rho^{\zeta}|\bxi_1 -\bxi_2|^{2} + \bar{c} r_{13}^{\zeta}|\bxi_1 + \bxi_2|^{2}. \label{eq:A13} \ee We finish the proof with inequality (\ref{eq:A13}) and (\ref{eq:A4}). $\,\,\,\,\,\Box$ \noindent {\bf Proposition 2, N=4 Case} \noindent We now turn to $N=4$, for which inequality (\ref{eq:A13}) is very helpful. Let $\br_{n}$, $n=1,2,3,4$, be four distinct points in $\bR^d$, $d\geq 2$, and assume that $r_{12}$ is the minimum length $\rho$. Then we show that there is a positive constant $\bar{c}$ depending only on $\rho_0$ so that the minimum eigenvalue of $\bG_4$ is bounded from below by $\bar{c}\rho^{\zeta}$. \noindent {\em Proof:} We order $r_3$ and $r_4$ according to the lengths of the three sides intersecting at them. The longest length at $r_4$ is larger than that at $r_3$. If they are equal, then the second longest length at $r_4$ is larger than its counterpart at $r_3$, and so on. Generically, we are able to order $r_3$ and $r_4$ this way. Now $r_i$, $i=1,2,3,4$, determine a tetrahedra in $R^d$. Due to geometric constraint, $r_{23}$ and $r_{13}$ are on the same order. So are $r_{14}$ and $r_{24}$. With no loss of generality, we can assume that $r_{13}=r_{23} =\alpha$, and $r_{14} =r_{24} =\beta$. Let $r_{34}$ be $\gamma$, which satisfies the inequalities: \be \gamma \leq \alpha +\beta, \; \beta \leq \alpha + \gamma; \alpha \leq \beta. \label{eq:A14} \ee We consider all the possibilities under (\ref{eq:A14}). \noindent Case I. Suppose $2 \geq {\gamma \over \beta} \geq C_{2}^{-1}$, where $C_2 >0$ is a large constant to be selected. We have four subcases: I 1.1: $ 1 \leq {\beta \over \alpha}\leq C_1$ and $1\leq {\alpha \over \rho}\leq C_0$; I 1.2: $1 \leq {\beta \over \alpha}\leq C_1$ and ${\alpha \over \rho} > C_0$; I 2.1: ${\beta \over \alpha} > C_1$ and $1\leq {\alpha \over \rho}\leq C'_0$; I 2.2: ${\beta \over \alpha} > C_1$ and $1\leq {\alpha \over \rho}> C'_0$. Case II: $ {\gamma \over \beta} < C_{2}^{-1}$, which implies with (\ref{eq:A14}) that $1\leq {\beta \over\alpha}\leq {C_2 \over C_{2} -1}$. We have two subcases: II 1.1: $1\leq {\alpha \over \rho}\leq C_0$ and II 1.2: $1\leq {\alpha \over \rho}>C_0$ \noindent As in the analysis for $N=3$, we assume that $\rho$ is smaller than $\rho_0$. The I1.1 is very similar to the first case of $N=3$, in that all lengths are comparable to each other. Writing \[ (\bxi_1,\bxi_2,\bxi_3,-(\bxi_1+\bxi_2+\bxi_3)) = \left(\begin{array}{rrrr} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 \\-1 & -1 & -1 & 0 \end{array} \right )(\bxi_1,\bxi_2,\bxi_3,0)^{T}, \] then: \[ Q_4(\bxi_1,\bxi_2,\bxi_3)= (\bxi_1,\bxi_2,\bxi_3) \] \[ \left ( \begin{array}{rrr} 2(\bV(0)-\bV_{14}) & \bV(0) + \bV_{12} -\bV_{14} -\bV_{24} & \bV(0) + \bV_{13} -\bV_{14} -\bV_{24} \\ \bV(0) + \bV_{12} -\bV_{14} -\bV_{24} & 2(\bV(0)-\bV_{24}) & \bV(0)+\bV_{23}-\bV_{24}-\bV_{34}\\ \bV(0) + \bV_{13} -\bV_{14} -\bV_{24} & \bV(0)+\bV_{23}-\bV_{24}-\bV_{34} & 2(\bV(0)-\bV_{34}) \end{array}\right ) \left( \begin{array}{r} \bxi_1\\\bxi_2 \\\bxi_3 \end{array}\right ). \] Using lemma 1 again, we can factor out $\rho^{\zeta}$ with remaining matrix being positive and bounded. We find that there is $\mu=\mu(C_0,C_1,C_2)$ such that: \be Q_4 \geq \mu \rho^{\zeta}. \label{eq:A15} \ee \noindent Now for I 1.2, we decompose $\{(\bxi_1,\bxi_2,\bxi_3,-(\bxi_1+\bxi_2+\bxi_3))\}$ into the orthogonal sum of \[ \{(\bar{\bxi}_1, -\bar{\bxi}_1, 0,0)\}\] and \[ \{(\bxi'_1,\bxi'_1,\bxi'_2,-2\bxi'_1-\bxi'_2)\}.\] Then: \[ Q_4(\bxi_1,\bxi_2,\bxi_3) = \langle(\bar{\bxi}_1,-\bar{\bxi}_1,0,0), \bG_4(\bar{\bxi}_1,-\bar{\bxi}_1,0,0)^{T}\rangle \] \[ +\langle(\bxi'_1,\bxi'_1,\bxi'_2,-2\bxi'_1-\bxi'_2), \bG_4(\bxi'_1,\bxi'_1,\bxi'_2,-2\bxi'_1-\bxi'_2)^{T}\rangle\] \be + 2 \langle(\bar{\bxi}_1,-\bar{\bxi}_1,0,0), \bG_4(\bxi'_1,\bxi'_1,\bxi'_2,-2\bxi'_1-\bxi'_2)^{T}\rangle. \label{eq:A16} \ee Writing: \[ \left ( \begin{array}{r} \bar{\bxi}_1 \\ -\bar{\bxi}_{1} \\ 0 \\ 0 \end{array} \right ) = \left ( \begin{array}{rrrr} 1 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0& 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right )\left ( \begin{array}{r} \bar{\bxi}_1 \\ 0 \\ 0 \\ 0 \end{array} \right ), \] we see that the bar term is equal to: \be \langle\bar{\bxi}_{1},2(\bV(0)-\bV_{12})\bar{\bxi}_{1}\rangle \geq \bar{c}\rho^{\zeta} |\bar{\bxi}_{1}|^{2}, \label{eq:A17} \ee Writing: \[ \left ( \begin{array}{r} \bxi'_1 \\ \bxi'_{1} \\ \bxi'_2 \\ -2\bxi'_1 -\bxi'_2 \end{array} \right ) = \left ( \begin{array}{rrrr} 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ -2 & -1 & 0 & 1 \end{array}\right )\left ( \begin{array}{r} \bxi_1' \\ \bxi_2'\\ 0 \\ 0 \end{array} \right ), \] the mixed term is equal to: \be \langle\bar{\bxi}_{1},-2(\bV_{14}-\bV_{24})\bxi'_{1}\rangle + \langle\bar{\bxi}_{1},(\bV_{13}-\bV_{23}+\bV_{24}-\bV_{14})\bxi'_{2}\rangle. \label{eq:A18} \ee Similarly the prime term is equal to: \[ \langle (\bxi'_1,\bxi'_2),\left (\begin{array}{rr} 8\bV(0)-8\bV_{24} & 2\bV_{23}-2\bV_{24}-2\bV_{34}+2\bV(0) \\ \bV_{23}-2\bV_{24}-2\bV_{34}+2\bV(0) & 2\bV(0)-2\bV_{34} \end{array} \right )(\bxi'_1,\bxi'_2)^{T}\rangle\] \be +\langle(\bxi'_1,\bxi'_2), \left (\begin{array}{rr} 2\bV_{12}-2\bV(0)+4(\bV_{24}-\bV_{14}) & \bV_{13}-\bV_{23}+\bV_{24}-\bV_{14}\\ \bV_{13}-\bV_{23}+\bV_{24}-\bV_{14} & 0 \end{array} \right )(\bxi'_1,\bxi'_2)^{T} \rangle. \label{eq:A19} \ee The first matrix of (\ref{eq:A19}) can be expressed as the product: \[\left ( \begin{array}{rrr} 2 & 0 & 0 \\0 & 1 & 0 \end{array}\right ) \left (\begin{array}{rrr} 1 & 0 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{array}\right ) \left ( \begin{array}{rrr} \bV(0) & \bV_{23} & \bV_{24} \\\bV_{23} & \bV(0) & \bV_{34} \\ \bV_{24} & \bV_{34} & \bV(0) \end{array}\right ) \left (\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & -1 & 1 \end{array}\right )\left (\begin{array}{rr} 2 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}\right ), \] hence is positive definite and bounded from below by a positive constant $\mu_{1}(C_1,C_2)$ times $r_{14}^{\zeta}|(\bxi'_1,\bxi'_2)|^{2}$. It follows that: \[ Q_4(\bxi_1,\bxi_2,\bxi_3) \geq \bar{c}\rho^{\zeta}|\bar{\bxi}_1|^{2} + \mu_{1}(C_1,C_2)(|\bxi'_{1}|^{2} + |\bxi'_2|^{2})r_{14}^{\zeta} -\bar{c}\rho r_{14}^{\zeta -1}|\bar{\bxi}_{1}|(|\bxi'_1| + |\bxi'_{2}|) \] \[ -\bar{c}(\rho^{\zeta} + \rho r_{14}^{\zeta -1})|\bxi'_1|^{2} -\bar{c}\rho (r_{14}^{\zeta-1} + r_{13}^{\zeta -1})(|\bxi'_1|\cdot |\bxi'_2|)\] \be \geq \bar{c}\rho^{\zeta}|\bar{\bxi}_1|^{2} + \mu_{1}(C_1,C_2)(|\bxi'_{1}|^{2} + |\bxi'_{2}|^{2})r_{14}^{\zeta} \geq \bar{c} \rho^{\zeta}, \label{eq:A20} \ee where the mixed term is handled as for $N=3$ with Young's inequality and $C_0$ is chosen large enough for given $C_1$ and $C_{2}$. \noindent We now consider I 2.1 and I 2.2. Decompose $\{(\bxi_1,\bxi_2,\bxi_3,-(\bxi_1+\bxi_2 +\bxi_3)\}$ into the orthogonal sum of $\{(\bar{\bxi}_1,\bar{\bxi}_2,-(\bar{\bxi}_1 +\bar{\bxi}_2), 0) \}$ and $\{(\bxi'_1,\bxi'_1,\bxi'_1,-3\bxi'_1)\}$. Then: \[ Q_4(\bxi_1,\bxi_2,\bxi_3) = \langle(\bar{\bxi}_1,\bar{\bxi}_2,-(\bar{\bxi}_1 +\bar{\bxi}_2), 0), \bG_4(\bar{\bxi}_1,\bar{\bxi}_2,-(\bar{\bxi}_1 +\bar{\bxi}_2), 0)^{T}\rangle \] \[ + \langle((\bxi'_1,\bxi'_1,\bxi'_1,-3\bxi'_1),\bG_4 (\bxi'_1,\bxi'_1,\bxi'_1,-3\bxi'_1)^{T}\rangle\] \be + 2 \langle(\bar{\bxi}_1,\bar{\bxi}_2,-(\bar{\bxi}_1 +\bar{\bxi}_2), 0), \bG_4(\bxi'_1,\bxi'_1,\bxi'_1,-3\bxi'_1)^{T}\rangle. \label{eq:A21} \ee Write: \[ \left (\begin{array}{r} \bar{\bxi}_1 \\\bar{\bxi}_2 \\ -(\bar{\bxi}_1 + \bar{\bxi}_2)\\ 0 \end{array}\right ) = \left ( \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\-1 & -1 & 1 & 0 \\0 & 0 & 0 & 1 \end{array}\right )\left ( \begin{array}{r} \bar{\bxi}_1 \\ \bar{\bxi}_2 \\ 0\\ 0 \end{array}\right ). \] Then the bar term is equal to: \[ \langle (\bar{\bxi}_{1},\bar{\bxi}_{2}), \left ( \begin{array}{rr} 2(\bV(0) -\bV_{13}) & \bV(0) + \bV_{12}-\bV_{13} -\bV_{23}\\ \bV(0) + \bV_{12}-\bV_{13} -\bV_{23} & 2(\bV(0)-\bV_{23}) \end{array}\right ) (\bar{\bxi}_{1},\bar{\bxi}_{2})^{T}\rangle,\] which is larger than: \be \bar{c}(\rho^{\zeta} |\bar{\bxi}_{1} -\bar{\bxi}_{2}|^{2} + r_{13}^{\zeta}|\bar{\bxi}_{1} + \bar{\bxi}_{2}|^{2}), \label{eq:A22} \ee by applying (\ref{eq:A13}) and the $N=3$ result. We express: \[ (\bxi'_1,\bxi'_1,\bxi'_1,-3\bxi'_1)^{T} = \left (\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\1 & 0 & 1 & 0 \\-3 & 0 & 0 & 1 \end{array}\right ) \left (\begin{array}{r} \bxi'_{1} \\0 \\ 0 \\ 0 \end{array}\right ), \] and so: \[ \bG_4\left (\begin{array}{r} \bxi'_{1} \\\bxi'_{1} \\\bxi'_{1} \\ -3\bxi'_{1} \end{array}\right ) = \left (\begin{array}{rrrr} \bV(0)+\bV_{12}+\bV_{13}-3\bV_{14} & \bV_{12} & \bV_{13} & \bV_{14} \\ \bV_{12}+\bV(0)+\bV_{23}-3\bV_{24} & \bV(0) & \bV_{23} & \bV_{24} \\ \bV_{13} + \bV_{23} + \bV(0) -3\bV_{34} & \bV_{23} & \bV(0) & \bV_{34} \\ \bV_{14} + \bV_{24} + \bV_{34} -3\bV(0) & \bV_{24} & \bV_{34} & \bV(0) \end{array}\right ) \left (\begin{array}{r} \bxi'_{1} \\0 \\ 0 \\ 0 \end{array}\right ). \] The mixed term is equal to: \[ 2(\bar{\bxi}_{1},\bar{\bxi}_{2},0,0) \left (\begin{array}{rrrr} 1 & 0 & -1 & 0 \\ 0 & 1 & -1 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \end{array}\right )\left (\begin{array}{rrrr} \bV(0)+\bV_{12}+\bV_{13}-3\bV_{14} & \bV_{12} & \bV_{13} & \bV_{14} \\ \bV_{12}+\bV(0)+\bV_{23}-3\bV_{24} & \bV(0) & \bV_{23} & \bV_{24} \\ \bV_{13} + \bV_{23} + \bV(0) -3\bV_{34} & \bV_{23} & \bV(0) & \bV_{34} \\ \bV_{14} + \bV_{24} + \bV_{34} -3\bV(0) & \bV_{24} & \bV_{34} & \bV(0) \end{array}\right ) \left (\begin{array}{r} \bxi'_{1} \\0 \\ 0 \\ 0 \end{array}\right ). \] \[ = 2(\bar{\bxi}_{1},\bar{\bxi}_{2})\left ( \begin{array}{rr} \bV_{12} -\bV_{23} - 3(\bV_{14} -\bV_{34}) & \bV_{12} -\bV_{23} \\ \bV_{12} -\bV_{13} - 3(\bV_{24}-\bV_{34}) & \bV(0) - \bV_{23} \end{array}\right ) \left (\begin{array}{r} \bxi'_{1}\\0 \end{array}\right ) \] \be = 2 \langle\bar{\bxi}_{1},(\bV_{12} -\bV_{23} -3(\bV_{14}-\bV_{34}))\bxi'_{1}\rangle + 2 \langle\bar{\bxi}_{2},(\bV_{12}-\bV_{13}-3(\bV_{24}-\bV_{34}))\bxi'_{1}\rangle, \label{eq:A23} \ee which can be written as: \be = 2 \langle \bar{\bxi}_{1} +\bar{\bxi}_{2}, (\bV_{12}-\bV_{23}-3(\bV_{14}-\bV_{34}))\bxi'_{1}\rangle + 2 \langle \bar{\bxi}_{2}, ((\bV_{23}-\bV_{13})-3(\bV_{24}-\bV_{14}))\bxi'_{1}\rangle. \label{eq:A24} \ee It follows that the mixed term is bounded by: \[ \bar{c}|\bar{\bxi}_{1} +\bar{\bxi}_{2}|\cdot |\bxi'_{1}|r_{13}( \max(r_{12}^{\zeta -1},r_{13}^{\zeta -1}) +r_{14}^{\zeta -1} \mu(C_{2})) \] \[ + \bar{c}|\bar{\bxi}_{2}|r_{12}r_{13}^{\zeta -1}|\bxi'_{1}|+ \bar{c}|\bxi'_{1}|\cdot |\bar{\bxi}_{2}|r_{12}r_{14}^{\zeta -1}. \] \noindent The prime term is equal to: \begin{eqnarray} & & \langle\bxi'_{1}, (12 \bV(0) + 2\bV_{12}+2\bV_{13}+2\bV_{23}-6\bV_{14}-6\bV_{24}-6\bV_{34})\bxi'_{1}\rangle \nonumber \\ & = & \langle\bxi'_{1},18(\bV(0)-\bV_{14})\bxi'_{1}\rangle + \langle\bxi'_{1},(2(\bV_{12}-\bV(0))+ 2(\bV_{13}-\bV(0)) \nonumber \\ & + & 2(\bV_{23}-\bV(0))+6(\bV_{24}-\bV_{34}) -12 (\bV_{24}-\bV_{14}) )\bxi'_{1}\rangle\nonumber \\ & \geq & \bar{c}r_{14}^{\zeta}|\bxi'_{1}|^{2} - \bar{c} (r_{12}^{\zeta} + r_{13}^{\zeta}+r_{23}^{\zeta})|\bxi'_{1}|^{2} -\mu(C_{2})(r_{23}r_{24}^{\zeta -1} + r_{12}r_{24}^{\zeta -1})|\bxi'_{1}|^{2} \nonumber \\ & = & \bar{c}r_{14}^{\zeta}|\bxi'_{1}|^{2}(1 - \mu(C_{2})C_{1}^{-\zeta} -\mu(C_{2})C_{1}^{-1}) \geq \bar{c}r_{14}^{\zeta}|\bxi'_{1}|^{2}, \nonumber \end{eqnarray} if $C_{1}$ is chosen large enough for given $C_{2}$. In case of I 2.1, the mixed terms involving $r_{14}^{\zeta -1}$ can be controlled by a Young's inequality as in $N=3$, using $C_{1}$ sufficiently large. The terms $r_{12}r_{13}^{\zeta -1}|\bxi'_{1}|\cdot |\bar{\bxi}_{2}|$ and $r_{13}\max\{r_{12}^{\zeta -1},r_{13}^{\zeta -1}\}|\bxi'_{1}|\cdot |\bar{\bxi}_{1} + \bar{\bxi}_{2}|$ can be estimated by $(C'_{0})^p r_{12}^{\zeta} = r_{12}^{\zeta/2}\cdot (C'_{0})^p r_{12}^{\zeta/2}$, $\left(p=\max\{\zeta,1\}\right)$, times the $\bxi$ bar or prime factors, then using again Young's inequality, thanks to the relatively large coefficient $r_{14}^{\zeta}$ in front of $|\bxi'_{1}|^{2}$. In other words, we use $C_{1}$ being much larger than any chosen $C'_{0}$. Observe that $|\bar{\bxi}_2|^2\leq {{1}\over{2}}|\bar{\bxi}_2-\bar{\bxi}_1|^2 +{{1}\over{2}}|\bar{\bxi}_2+\bar{\bxi}_1|^2$, so that the mixed terms are again controlled by the prime and bar terms. In case of I 2.2, we make $C'_{0}$ itself large to control the term $r_{12}r_{13}^{\zeta -1}|\bxi'_{1}|\cdot |\bar{\bxi}_{2}|$. The other terms involving $r_{14}$ are standard and controlled by large $C_{1}$. Note that if $\zeta \in (0,1]$ \begin{eqnarray} r_{13}\max\{r_{12}^{\zeta -1}, r_{13}^{\zeta -1}\} & = & r_{13}^{\zeta/2}r_{12}^{\zeta/2} \left({{r_{13}}\over{r_{12}}}\right)^{1-{{\zeta}\over{2}}} \cr \, & \leq & (C_0')^{1-{{\zeta}\over{2}}}r_{13}^{\zeta/2}r_{12}^{\zeta/2} \nonumber \end{eqnarray} Thus when multiplied to $|\bar{\bxi}_{1} +\bar{\bxi}_{2}|\cdot |\bxi'_{1}|$ it is bounded by \[ {{\theta}\over{2}}r_{13}^{\zeta}|\bar{\bxi}_{1}+\bar{\bxi}_{2}|^{2} +{{(C_0')^{2-\zeta}}\over{2\theta}}r_{12}^{\zeta}|\bxi'_{1}|^{2} \leq {{\theta}\over{2}}r_{13}^{\zeta}|\bar{\bxi}_{1} +\bar{\bxi}_{2}|^{2} +{{\theta}\over{2}}r_{14}^{\zeta}|\bxi'_{1}|^{2}, \] with $C_1$ much larger than chosen $C_0'$. If $\zeta \in (1,2)$, $r_{13} \max\{r_{12}^{\zeta -1}, r_{13}^{\zeta -1}\}=r_{13}^{\zeta}$, and its product with $|\bar{\bxi}_{1} +\bar{\bxi}_{2}|\cdot |\bxi'_{1}|$ is bounded by ${{\theta}\over{2}}r_{13}^{\zeta}|\bar{\bxi}_{1} +\bar{\bxi}_{2}|^{2} + {{r_{13}^\zeta}\over{2\theta}}|\bxi'_{1}|^{2}\leq {{\theta}\over{2}}r_{13}^{\zeta}|\bar{\bxi}_{1} +\bar{\bxi}_{2}|^{2} + {{\theta}\over{2}}r_{14}^{\zeta}|\bxi'_{1}|^{2}$, since $C_1^{-\zeta}<\theta^2$ for large $C_1$. Summarizing the above, we conclude that: \be Q_4(\bxi_1,\bxi_2,\bxi_3) \geq \bar{c}r_{14}^{\zeta}|\bxi'_{1}|^{2} + \bar{c}\rho^{\zeta}|\bar{\bxi}_{1}-\bar{\bxi}_{2}|^{2} + \bar{c}r_{13}^{\zeta} |\bar{\bxi}_{1} + \bar{\bxi}_{2}|^{2}, \label{eq:A25} \ee which, in $(\bxi_1,\bxi_2,\bxi_3)$ variables, is: \be Q_4(\bxi_1,\bxi_2,\bxi_3) \geq \bar{c}\left( \rho^{\zeta}|\bxi_1 -\bxi_2|^{2} +\alpha^{\zeta}|\bxi_{1}+\bxi_{2}-2\bxi_{3}|^{2} + \beta^{\zeta}|\bxi_{1}+\bxi_{2}+\bxi_{3}|^{2}\right ). \label{eq:A26} \ee \noindent Finally we consider II. The case II 1.1 is no different from I 1.1. Notice that for II 1.2, we have essentially two separate scales $\beta >> \gamma$, thanks to $\alpha $ and $\beta $ being on the same scale. Decompose $\{(\bxi_1,\bxi_{2},\bxi_{3},-(\bxi_{1} + \bxi_{2} + \bxi_{3}))\}$ into the orthogonal sum of $\{(\bar{\bxi}_{1},-\bar{\bxi}_{1},\bar{\bxi}_{3},-\bar{\bxi}_{3})\}$ and $\{(\bxi'_{1},\bxi'_{1},-\bxi'_{1},-\bxi'_{1})\}$. The bar term is: \[ \langle(\bar{\bxi}_{1},-\bar{\bxi}_{1},\bar{\bxi}_{3},-\bar{\bxi}_{3}), \bG_4 (\bar{\bxi}_{1},-\bar{\bxi}_{1},\bar{\bxi}_{3},-\bar{\bxi}_{3})^{T}\rangle. \] By writing: \[ \left (\begin{array}{r} \bar{\bxi}_{1} \\ -\bar{\bxi}_{1} \\ \bar{\bxi}_{3} \\ -\bar{\bxi}_{3} \end{array}\right ) = \left (\begin{array}{rrrr} 1 & 0 & 0 & 0 \\-1 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\ 0 & 0 & -1 & 1 \end{array} \right ) \left (\begin{array}{r} \bar{\bxi}_{1} \\ 0 \\ \bar{\bxi}_{3} \\ 0 \end{array} \right ), \] we simplify the bar term into: \be \langle (\bar{\bxi}_{1},\bar{\bxi}_{3}), \left ( \begin{array}{rr} 2(\bV(0)-\bV_{12}) & \bV_{13} + \bV_{24} -\bV_{14}-\bV_{23} \\ \bV_{13} + \bV_{24} -\bV_{14}-\bV_{23} & 2 (\bV(0)-\bV_{34}) \end{array}\right )\left (\begin{array}{r} \bar{\bxi}_{1} \\ \bar{\bxi}_{3} \end{array} \right ) \rangle. \label{eq:A27} \ee Then the bar term is bounded as: \[ = 2 \langle\bar{\bxi}_{1},(\bV(0)-\bV_{12})\bar{\bxi}_{1}\rangle + 2\langle\bar{\bxi}_{3},(\bV(0)-\bV_{34})\bar{\bxi}_{3 }\rangle+ 2\langle\bar{\bxi}_{1},(\bV_{13}-\bV_{23}+ \bV_{24}-\bV_{14})\bar{\bxi}_{3}\rangle\] \[ \geq \bar{c}\rho^{\zeta}|\bar{\bxi}_{1}|^{2} + \bar{c}r_{34}^{\zeta}|\bar{\bxi}_{3}|^{2} - \bar{c}|\bar{\bxi}_{1}|\cdot |\bar{\bxi}_{3}\left|\bV_{13}-\bV_{14}-(\bV_{23} -\bV_{24})\right|\] \be \geq {\bar{c}\over 2}\rho^{\zeta}|\bar{\bxi}_{1}|^{2} + {\bar{c}\over 2}r_{34}^{\zeta}|\bar{\bxi}_{3}|^{2}. \label{eq:A28} \ee To obtain the last inequality we used lemma 3: \[ \left|\bV_{13}-\bV_{14}-(\bV_{23} -\bV_{24})\right| \leq \bar{c} r_{12}r_{34}r_{13}^{\zeta -2} = \bar{c}r_{12}^{\zeta/2}r_{34}^{\zeta/2}\frac{r_{12}^{1-\zeta/2} r_{34}^{1-\zeta/2}}{r_{13}^{1-\zeta/2}r_{13}^{1-\zeta/2}}\] \[\leq\bar{c}r_{12}^{\zeta/2}r_{34}^{\zeta/2} C_1^{-(2-\zeta)/2}(C_2-1)^{-(2-\zeta)/2}. \] The last term is small for large $C_1,C_2$ when $\zeta<2$. Applying Young's inequality yields the same bound as (\ref{eq:A28}). \noindent Next the prime term is simplified by using: \[ \left (\begin{array}{r} \bxi'_{1} \\ \bxi'_{1} \\ -\bxi'_{1} \\ -\bxi'_{1} \end{array}\right ) = \left (\begin{array}{rrrr} 1 & 0 & 0 & 0 \\1 & 1 & 0 & 0 \\ -1 & 0 & 1 & 0 \\ -1 & 0 & 0 & 1 \end{array} \right ) \left (\begin{array}{r} \bxi'_{1} \\ 0 \\ 0 \\ 0 \end{array} \right ). \] The prime term becomes: \[ \langle\bxi'_{1},\left( 2\bV(0) +2\bV_{12} -2 \bV_{13}-2\bV_{14}-2\bV_{24}-2\bV_{23} + 2\bV(0) + 2\bV_{34}\right )\bxi'_{1} \rangle \] \[ = \langle\bxi'_{1},\left (8\bV(0)-2\bV_{13}-2\bV_{14}-2\bV_{23}-2\bV_{24}\right )\bxi'_{1}\rangle \] \be -\langle\bxi'_{1},\left ( 4\bV(0)-2\bV_{12}-2\bV_{34}\right )\bxi'_{1}\rangle \geq \bar{c}r_{13}^{\zeta} |\bxi'_{1}|^{2} \label{eq:A29} \ee \noindent The mixed bar-prime term is: \[\left ( \begin{array}{r} \bar{\bxi}_{1} \\ 0 \\ \bar{\bxi}_{3} \\0 \end{array} \right ) \left (\begin{array}{rrrr} 1 & -1 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 1 \end{array} \right ) \left (\begin{array}{rrrr} \bV(0)+\bV_{12}-\bV_{13}-\bV_{14} & \bV_{12} & \bV_{13} & \bV_{14} \\ \bV_{12}+\bV(0)-\bV_{23}-\bV_{24} & \bV(0) & \bV_{23} & \bV_{24} \\ \bV_{13} + \bV_{23} - \bV(0) -\bV_{34} & \bV_{23} & \bV(0) & \bV_{34} \\ \bV_{14} + \bV_{24} -\bV_{34} -\bV(0) & \bV_{24} & \bV_{34} & \bV(0) \end{array}\right ) \left ( \begin{array}{r} \bxi'_{1} \\ 0 \\ 0 \\ 0 \end{array}\right ) \] \[ = \langle\bar{\bxi}_{1},(\bV_{23}-\bV_{13}+\bV_{24}-\bV_{14})\bxi'_{1}\rangle + \langle\bar{\bxi}_{3},(\bV_{13}+\bV_{23}-\bV_{14}-\bV_{24})\bxi'_{1}\rangle. \] Hence the mixed term is bounded by: \be \bar{c}(r_{12}r_{13}^{\zeta -1} + r_{12} r_{14}^{\zeta -1})|\bxi'_{1}|\cdot |\bar{\bxi}_{1}| + \bar{c}(r_{34}r_{14}^{\zeta -1} +r_{34}r_{24}^{\zeta -1}) |\bxi'_{1}|\cdot |\bar{\bxi}_{3}|. \label{eq:A30} \ee All the terms in (\ref{eq:A30}) can be estimated as before with Young's inequality, and we have: \[ Q_4(\bxi_{1},\bxi_{2},\bxi_{3}) \geq \bar{c} \rho^{\zeta}|\bar{\bxi}_{1}|^{2} + \bar{c}r_{34}^{\zeta}|\bar{\bxi}_{3}|^{2}, \] which is: \be Q_4(\bxi_{1},\bxi_{2},\bxi_{3}) \geq \bar{c}\rho^{\zeta}|\bxi_1 -\bxi_2|^{2} + \gamma^{\zeta}|2\bxi_{3}-\bxi_{1}-\bxi_{2}|^{2} + \beta^{\zeta}|\bxi_{1}+\bxi_{2}|^{2}. \label{eq:A31} \ee \noindent Summarizing all the cases, we finish the proof of the proposition. $\,\,\,\,\,\Box$ \newpage \section{Properties of the N-Body Convective Operator} We now define a sesquilinear form $h_N[\Psi_N,\Phi_N]$ for $\Psi_N,\Phi_N\in L^2\left(\Omega^{\otimes N}\right)$, by the expression \be h_N[\Psi_N,\Phi_N]=\int_{\Omega^{\otimes N}} d\bR\,\,\overline{\grad_\bR\Psi_N(\bR)}\bdot\bG_N(\bR) \bdot\grad_\bR\Phi_N(\bR). \lb{sesq} \ee and a quadratic form $h_N[\Psi_N]=h_N[\Psi_N,\Psi_N]$. We take as the form domain \begin{eqnarray} \, & & \bD(h_N)=\{\Psi_N\in L^2\left(\Omega^{\otimes N}\right): \Psi_N\in C^\infty\left(\Omega^{\otimes N}\right), {\rm supp}\Psi_N\subseteq\overline{{\Omega^{\otimes N}}_k}\,\,\,\,\,{\rm for}\,\,\,\,\,{\rm some}\,\,\,\,\,k, \cr \, & & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm and}\,\,\,\,\,\Psi_N(\bR)=0\,\,\,\,\, {\rm for}\,\,\,\,\,\bR\in\partial\Omega^{\otimes N}\}. \lb{formdom} \end{eqnarray} Here we made use of an increasing sequence of open subsets of $\Omega^{\otimes N}$ defined as \be {\Omega^{\otimes N}}_k=\{\bR\in \Omega^{\otimes N}: \rho(\bR)>{{1}\over{k}}\}. \lb{incrseq} \ee Clearly, this form can be expressed as $h_N[\Psi_N,\Phi_N]=\langle\Psi_N,\cH_N\Phi_N\rangle$ where $\cH_N$ is the positive, symmetric differential operator \be \cH_N= -{{1}\over{2}}\sum_{i,j=1}^d\sum_{n,m=1}^N\,\, {{\partial}\over{\partial x_{in}}}\left[V_{ij}(\br_n-\br_m){{\partial}\over{\partial x_{jm}}}\cdot\right] \lb{symop}\ee with $\bD(\cH_N)=\bD(h_N)$. Our basic object of interest is the self-adjoint (Friedrichs) extension $\oH_N$ of $\cH_N$, which corresponds to the operator with Dirichlet b.c. on $\partial\Omega^{\otimes N}$. Note that it will follow from our discussion below that the same extension $\oH_N$ also arises if one chooses $\bD(\cH_N)=C_0^\infty \left(\Omega^{\otimes N}\right)$, rather than as above. The main properties of $\oH_N$ follow from those of the form $h_N$ which we now consider. The basic properties of the form are contained in: \begin{Prop} The sesquilinear form $h_N[\Psi_N,\Phi_N]$ enjoys the following: (i) $h_N$ is a nonnegative, closable form. (ii) For all $\Psi_N\in\bD(h_N)$ and for the same constant $C_N$ in Proposition 2, \be h_N[\Psi_N]\geq C_N\int_{\Omega^{\otimes N}} d\bR\,\,[\rho(\bR)]^\zeta|\grad_\bR\Psi_N(\bR)|^2. \lb{1stineq} \ee (iii)For all $\Psi_N\in\bD(h_N)$ and for the same constant $C_N$ in Proposition 2, \be h_N[\Psi_N]\geq C_N\cdot{{(d-\gamma)^2}\over{2}}\int_{\Omega^{\otimes N}} d\bR\,\, [\rho(\bR)]^{-\gamma}|\Psi_N(\bR)|^2. \lb{2ndineq} \ee \end{Prop} \noindent {\em Proof of Proposition 3}. {\em Ad (i)}: non-negativity is obvious from the definition Eq.(\ref{sesq}) and Proposition 1{\em (i)}. That $h_N$ is closable follows from \cite{Kato}, Theorem VI.1.27 and its Corollary VI.1.28. {\em Ad (ii)}: This follows directly from the definition Eq.(\ref{sesq}) and the variational formula for the minimum eigenvalue of $\bG_N(\bR)$. {\em Ad (iii)}: For the proof of this inequality, we use the Lemma 2 of Lewis \cite{Lew}. That lemma states that, given an open domain $\Lambda$ with smooth boundary, then for any function $g\in H^2(\Lambda)$ such that $\bigtriangleup_{\bR}g(\bR)>0$ for all $\bR\in\Lambda$ and for any function $\varphi\in C_0^\infty(\Lambda)$ (i.e. $=0$ on $\partial\Lambda$), the inequality holds that \begin{eqnarray} \, & & \int_\Lambda d\bR\,\,|\bigtriangleup_\bR g(\bR)||\varphi(\bR)|^2 \cr \, & & \,\,\,\,\,\,\, \leq 4\int_\Lambda d\bR\,\,|\bigtriangleup_\bR g(\bR)|^{-1}|\grad_\bR g(\bR)|^2|\grad_\bR\varphi(\bR)|^2, \lb{Hardeq} \end{eqnarray} This is proved by applying Green's first formula and the Cauchy-Schwartz inequality (see \cite{Lew}). Let us take for each integer $k\geq 1$ the domain $\Lambda_k={\Omega^{\otimes N}}_k$ defined as in Eq.(\ref{incrseq}). If we define $g(\bR)=[\rho(\bR)]^\zeta$, then $g\in H^2(\Lambda_k)$ for each $k$ (and, in fact, $g\in C^\infty(\Lambda_k)$). Furthermore, \be \bigtriangleup_\bR g(\bR)=2\zeta(d-\gamma)[\rho(\bR)]^{-\gamma}>0, \lb{laplace} \ee for $d>\gamma$ (which certainly holds if $\zeta>0$ and $d\geq 2$) and also \be |\grad_\bR g(\bR)|^2=2\zeta^2[\rho(\bR)]^{2\zeta-2}. \lb{sqrgrad} \ee If $\Psi_N\in\bD(h_N)$, then for some $k$ sufficiently large $\Psi_N\in C_0^\infty(\Lambda_k)$, and all the conditions for the inequality (\ref{Hardeq}) are satisfied. Hence, we find by substitution that \begin{eqnarray} \, & & \int_{\Omega^{\otimes N}}d\bR\,\,[\rho(\bR)]^\zeta|\grad_\bR\Psi_N(\bR)|^2 \cr \, & & \,\,\,\,\,\,\, \geq {{(d-\gamma)^2}\over{2}}\int_{\Omega^{\otimes N}}d\bR\,\,[\rho(\bR)]^{-\gamma}|\Psi_N(\bR)|^2, \lb{mainineq} \end{eqnarray} whenever $\Psi_N\in\bD(h_N)$, for $\zeta>0$ and $d\geq 2$. If we now use together {\em (ii)} and inequality (\ref{mainineq}), we obtain {\em (iii)}. $\,\,\,\,\,\Box$ \noindent Because of item {\em (i)} we may now pass to the closed form $\ch_N$ (see \cite{Kato}, VI.1.4). Its properties are given in the following Proposition 4: \newpage \begin{Prop}The sesquilinear form $\ch_N[\Psi_N,\Phi_N]$ enjoys the following: (i) $\ch_N$ is a nonnegative, closed form. (ii) The domain $\bD(\ch_N)$ consists of the Hilbert space $H_{h_N}\left(\Omega^{\otimes N}\right)$ obtained by completion of $C_0^\infty\left(\Omega^{\otimes N}\right)$ in the inner product \be\langle\Psi_N,\Phi_N\rangle_{h_N}=\langle\Psi_N, \Phi_N\rangle+h_N[\Psi_N,\Phi_N]. \lb{innprod} \ee In particular, $H_0^1\left(\Omega^{\otimes N}\right)\subset\bD(\ch_N)$. Alternatively, $\Psi_N\in\bD(\ch_N)$ iff $\Psi_N\in L^2\left(\Omega^{\otimes N}\right)$, its 1st distributional derivative satisfies $h_N[\Psi_N]<\infty$, and $\gamma_k\left(\left.\Psi_N\right|_{{\Omega^{\otimes N}}_k}\right)=0$ for all integer $k\geq 1$, where $\gamma_k$ is the trace operator from $H^1\left({\Omega^{\otimes N}}_k\right)$ into $L^2\left(\partial\Omega^{\otimes N} \bigcap{\Omega^{\otimes N}}_k\right)$. (iii) Both the items (ii) and (iii) of Proposition 3 hold for $\ch_N[\Psi_N]$ and for all $\Psi_N\in\bD(\ch_N)$. Furthermore, \be h_N[\Psi_N]\geq C_N L^{-\gamma}\cdot{{(d-\gamma)^2}\over{2}}\|\Psi_N\|^2_{L^2} \lb{3rdineq} \ee also for all $\Psi_N\in\bD(\ch_N)$. In particular, $\ch_N$ is strictly positive. \end{Prop} \noindent {\em Proof of Proposition 4:} {\em (i)} is immediate. \noindent {\em (ii)} We first prove the statement that $H_0^1\left(\Omega^{\otimes N}\right)\subset\bD(\ch_N)$. To see this, we remark that $\bD(h_N)$ is dense in $H^1_0\left(\Omega^{\otimes N}\right)$ for $d\geq 2$. In fact, it is well-known that in a bounded domain $\Lambda$ the set of functions $C^\infty_0(\Lambda-\Gamma)$, i.e. functions vanishing on $\Gamma\subset\Lambda$ in addition to $\Lambda^c$, is dense in $H^l_0(\Lambda)$ if $\Gamma$ is a finite union of submanifolds with codimension $k\geq 2l$. This follows from standard density theorems for Sobolev spaces: see Ch.III of Adams \cite{Adams} or Ch.9 of Maz'ja \cite{Maz'ja}. The Theorem 3.23 of \cite{Adams} states that $C^\infty_0(\Lambda -\Gamma)$ is dense in $H^l_0(\Lambda)$ iff $\Gamma$ is a $(2,l)$-polar set, when $\Lambda=\bR^D$. However, the same result is true for any open domain $\Lambda$. In fact, repeating Adams' argument, if $C^\infty_0(\Lambda-\Gamma)$ is not dense in $H^l_0(\Lambda)$, then there must be a $u\in H^l_0(\Lambda)$ and an element $T\in H^{-l}_0(\Lambda)$, the Banach dual, so that $T(u)=1$ but $T(f)=0$ for all $f\in C^\infty_0(\Lambda-\Gamma)$. However, by \cite{Adams}, Theorem 3.10, this $T$ can be identified with an element of ${\cal D}'(\Lambda)$ supported on $\Gamma$. Since this can further be canonically identified with an element of ${\cal D}'(\bR^D)$ supported on $\Gamma$, the set $\Gamma$ cannot be $(2,l)$-polar. The other direction is even simpler. These arguments go back to \cite{HL}. On the other hand, by Theorem 9.2.2 of \cite{Maz'ja} the set $\Gamma$ is $(2,l)$-polar iff its lower $H^l$-capacity vanishes, $\underline{{\rm Cap}}\left(\Gamma,H^l\right)=0$. A convenient sufficient condition for zero $H^l$-capacity is that the Hausdorff $(D-2l)$-dimensional measure of $\Gamma$ be finite, ${\cal H}^{D-2l}\left(\Gamma\right)<\infty$. See Proposition 7.2.3/3 and Theorem 9.4.2 of \cite{Maz'ja}. (This is essentially just the converse of the Frostman theorem, due originally to Erd\"{o}s \& Gillis \cite{EG}.) In the case considered, the set $\Gamma$ is of Hausdorff dimension $D-k$, so that ${\cal H}^{D-2l}\left(\Gamma\right)<\infty$ for $k\geq 2l$ ($=0$ for $k>2l$). Thus, the set $\Gamma$ has zero $H^l$-capacity as required. Clearly, $\bD(h_N)$ defined in the statement of the Proposition 3 above coincides with $C^\infty_0\left(\Omega^{\otimes N}-\Gamma\right)$, where the set $\Gamma=\{\bR\in\Omega^{\otimes N}:\br_n=\br_m,n\neq m\}$ has codimension $=d\geq 2$. Therefore, taking $D=Nd$, $l=1,\,\,k=d$ and $\Lambda=\Omega^{\otimes N}$ we obtain the density of $\bD(h_N)$ in $H^1_0\left(\Omega^{\otimes N}\right)$, as claimed. As a consequence, for any $\Psi_N\in H^1_0\left(\Omega^{\otimes N}\right)$ there exists a sequence of elements $\Psi_N^{(m)}\in \bD(h_N)$ converging in $H^1$-norm to $\Psi_N$. Next, we observe that \be h_N[\Psi_N]\leq B_N\|\Psi_N\|_{H^1}^2 \lb{upper} \ee for some coefficient $B_N>0$. This may be proved by using the variational principle for the maximum eigenvalue $\lambda_N^{\max}(\bR)$ of $\bG_N(\bR)$ and then the continuity in $\bR$ of $\lambda_N^{\max}(\bR)$ over the compact set $\overline{\Omega^{\otimes N}}$ to infer $\lambda_N^{\max}(\bR)\leq B_N$. This inequality states that the $H^1$-norm is stronger than the $h_N$-seminorm. Thus, convergence in $H^1$ norm of $\Psi_N^{(m)}\in \bD(h_N)$ to $\Psi_N\in H^1_0\left( \Omega^{\otimes N}\right)$ implies both that $\Psi_N^{(m)}\rightarrow\Psi_N$ in $L^2$ and also that $h_N[\Psi_N^{(m)} -\Psi_N^{(n)}]\rightarrow 0$ as $m,n\rightarrow\infty$. Comparing with \cite{Kato},Section VI.1.3 we see that this means precisely that $\Psi_N\in\bD(\ch_N)$. Therefore, $H^1_0\left(\Omega^{\otimes N}\right)\subset \bD(\ch_N)$. This is the first statement of {\em (ii)}. Next, we recall from \cite{Kato}, Section VI.1.3 that $\bD(\ch_N)$ is characterized as the Hilbert space obtained by completion of $\bD(h_N)$ in the inner-product (\ref{innprod}). Since $\bD(h_N)\subset C_0^\infty\left(\Omega^{\otimes N} \right)$, this is certainly contained in the Hilbert space defined in {\em (ii)} above. However, since we have shown that $H^1_0\left(\Omega^{\otimes N}\right)\subset \bD(\ch_N)$, the completions of $\bD(h_N)$ and $C_0^\infty \left(\Omega^{\otimes N}\right)$ are the same. Finally, we prove the alternative characterization of $\bD(\ch_N)$ in {\em (ii)}. We note by Proposition 3{\em (ii)} that for each $\Psi_N\in \bD(h_N)$ and for each $k$ \be \|\Psi_N\|_{H^1\left({\Omega^{\otimes N}}_k\right)}\leq k^\zeta C_N^{-1}\cdot\|\Psi_N\|_{h_N}. \lb{lower} \ee Thus, the $H_{h_N}$-norm is stronger than the $H^1\left({\Omega^{\otimes N}}_k\right)$-norm on $\left.\bD(h_N) \right|_{{\Omega^{\otimes N}}_k}$. By definition, for each $\Psi_N\in\bD(\ch_N)$ there is a sequence $\Psi_N^{(m)}\in \bD(h_N)$ converging to $\Psi_N$ in $H_{h_N}$-norm. This sequence must also then converge to $\left.\Psi_N \right|_{{\Omega^{\otimes N}}_k}$ in $H^1\left({\Omega^{\otimes N}}_k\right)$-norm. Passing to the limit in (\ref{lower}), one then obtains its validity for all $\bD(\ch_N)$. This implies that $\left.\bD(\ch_N)\right|_{{\Omega^{\otimes N}}_k} \subset H^1\left({\Omega^{\otimes N}}_k\right)$ for each integer $k$. Furthermore, the trace $\gamma_k$ onto the codimension-1 set $(\partial\Omega^{\otimes N})\bigcap {\Omega^{\otimes N}}_k$ is continuous from $H^1\left( {\Omega^{\otimes N}}_k\right)$ into $H^{1/2}\left((\partial\Omega^{\otimes N})\bigcap {\Omega^{\otimes N}}_k\right)$. Since $\Psi_N^{(m)}\in\bD(h_N)$, we see that $\gamma_k\left(\left.\Psi_N^{(m)}\right|_{{\Omega^{\otimes N}}_k}\right)=0$ and, passing to the limit, $\gamma_k\left(\left.\Psi_N\right|_{{\Omega^{\otimes N}}_k}\right)=0$ as an element of $H^{1/2}\left((\partial\Omega^{\otimes N})\bigcap {\Omega^{\otimes N}}_k\right)$. That is the ``only if'' part of the characterization. The ``if'' part is very standard. For each $\Psi_N$ obeying the alternative set of conditions and $k\geq 1$, we may define $\widetilde{\Psi}_N^{(k)}$ by extending the restriction $\left.\Psi_N\right|_{{\Omega^{\otimes N}}_k}$ again to the whole of $\Omega^{\otimes N}$, defining it to be $0$ outside of ${\Omega^{\otimes N}}_k$. Because of the conditions on $\Psi_N$, the new function $\widetilde{\Psi}_N^{(k)}\in H^1_0\left(\Omega^{\otimes N}\right)$ for each $k\geq 1$. See Theorems 3.16 and 7.55 of \cite{Adams}. Thus, $\widetilde{\Psi}_N^{(k)}\in \bD(\ch_N)$ for all $k\geq 1$. However, \be \|\Psi_N-\widetilde{\Psi}_N^{(k)}\|_{h_N}=\left\|\left(1-\chi_{{\Omega^{\otimes N}}_k}\right)\Psi_N\right\|_{h_N} \lb{dens} \ee where $\chi_{{\Omega^{\otimes N}}_k}$ is the characteristic function of ${\Omega^{\otimes N}}_k$. Because $\|\Psi_N\|_{h_N}<\infty$ by assumption, the righthand side goes to zero by dominated convergence as $k\rightarrow\infty$. Thus, we conclude that $\lim_{k\rightarrow\infty}\|\Psi_N-\widetilde{\Psi}_N^{(k)}\|_{h_N}=0$, which implies that $\Psi_N\in\bD(\ch_N)$. For {\em (iii)}: We note that the righthand side of inequalities (\ref{1stineq}) and (\ref{2ndineq}) in Proposition 3 {\em (ii)} \& {\em (iii)} are just certain weighted $H^1$-norms and $L^2$-norms, respectively, and both of these are bounded by the $h_N$-norm on $\bD(h_N)$. Thus, the argument used to extend inequality (\ref{lower}) from $\bD(h_N)$ to $\bD(\ch_N)$ applies also to extending (\ref{1stineq})-(\ref{2ndineq}). Noting that $\rho(\bR)\leq {\rm diam}\,\Omega=L$ for all $\bR\in\Omega^{\otimes N}$, we derive inequality (\ref{3rdineq}) from (\ref{2ndineq}). $\,\,\,\,\Box$ \vspace{.1 in} \noindent {\bf Remark:} The proof does not work for $\Omega={\bf T}^d$, the $d$-dimensional torus. In that case, inequality (\ref{1stineq}) of Proposition 3{\em (ii)} is still valid, where $\rho(\bR)=\min_{n\neq m,\bk\in\BZ^d} |\br_n-\br_m+L\cdot\bk|$ has now period $L$ in each direction as required. Unfortunately, the function $g(\bR)= [\rho(\bR)]^\zeta$ does not belong to $H^2\left(({\bf T}^d)^{\otimes N}\right)$ away from the set $\Gamma$ where $\rho(\bR)=0$. It has singularities also on the codimension-1 set $\Gamma'$ of points where $|\br_n-\br_m|= |\br_n^*-\br_m|$, with $\br_n^*$ a periodic image of $\br_n$. Unless the domain of $\bD(h_N)$ is chosen to be $=0$ on $\Gamma'$, these singularities would contribute a surface term in the Green's formula, invalidating (\ref{2ndineq}). However, if that condition on $\bD(h_N)$ is imposed, then the resulting closed form $\ch_N$ has Dirichlet b.c. on $\Gamma'$, which is unphysical. On the other hand, we expect that these are really just problems with the proof and that the inequality (\ref{2ndineq}) still holds with periodic b.c. Methods used to derive general Hardy-Sobolev inequalities (\cite{Maz'ja}, Ch.2) should apply. We now exploit the previous results to study the Friedrichs extension $\oH_N$ of $\cH_N$. Its existence is provided by the First Representation Theorem of forms (\cite{Kato}, Theorem VI.2.1) which states that there is a unique self-adjoint operator $\oH_N$ whose domain $\bD(\oH_N)$ is a core for $\ch_N$ and for which $\ch_N[\Psi_N,\Phi_N]=\langle \Psi_N,\oH_N\Phi_N\rangle$ for every $\Psi_N\in\bD(\ch_N)$ and $\Phi_N\in\bD(\oH_N)$. We now discuss the essential properties of this operator that we will need later: \begin{Prop} The Friedrichs extension $\oH_N$ enjoys the following: {\em (i)} $\oH_N$ is strictly positive, with lower bound $\geq C_N L^{-\gamma}\cdot {{(d-\gamma)^2}\over{2}}$. {\em (ii)} The spectrum of $\oH_N$ is pure point. \end{Prop} \noindent {\em Proof of Proposition 5:} {\em Ad (i):} (\ref{3rdineq}) and \cite{Kato}, Theorem VI.2.6. {\em Ad (ii)}: We use the Corollary to Lemma 1 of Lewis \cite{Lew}. His hypothesis ${\cal H}1$ is satisfied by the increasing sequence ${\Omega^{\otimes N}}_k$ for integer $k\geq 1$. His hypothesis ${\cal H}2$ is true with $H^m=H^1$ and $c_k=C_N\cdot k^{-\zeta}$ as a consequence of (\ref{lower}). Finally, his third hypothesis holds, with the role of his function $p(x)$ played by $C_N{{(d-\gamma)^2}\over{2}}\cdot[\rho(\bR)]^{-\gamma}$ and $\varepsilon_k=C_N{{(d-\gamma)^2}\over{2}} \cdot k^{-\gamma}$, by (\ref{2ndineq}). Lewis' proof exploits the Rellich lemma for the domain ${\Omega^{\otimes N}}_k$ to show that the identity injection $I: H_{h_N}\left(\Omega^{\otimes N}\right)\rightarrow L^2\left(\Omega^{\otimes N} \right)$ is compact, by approximating it in norm with compact operators $I_k(\Psi_N)=\widetilde{\Psi}^{(k)}_N$, defined above. The segment property holds for ${\Omega^{\otimes N}}_k$, since its boundary is $C^\infty$ except for a finite number of corners where the two parts of its boundary, $\{\bR\in \Omega^{\otimes N}:\rho(\bR)=k\}$ and $\partial\Omega^{\otimes N}$, intersect. $\,\,\,\,\,\Box$ \newpage \section{Proofs of the Main Theorems} We now prove the main results of the paper, using the properties of $\oH_N$ proved in the preceding section. We start with: \noindent {\em Proof of Theorem 1:} By a stationary weak solution of (\ref{closeq}) at $\kappa=0$, we mean a sequence of $\Theta_N^*\in L^2\left(\Omega^{\otimes N}\right)$ indexed by $N\geq 1$, such that, for each $N\geq 1$ and for all $\Phi_N\in \bD(\oH_N)$, \be \langle\oH_N\Phi_N,\Theta_N^*\rangle=\langle \Phi_N,G_N^*\rangle \lb{weakeq*} \ee where for $N\geq 2$ \be G_N^*(\bR)=\sum_n \of(\br_n)\Theta_{N-1}^*(...\widehat{\br_n}...)+ \sum_{{\rm pairs}\,\,\,\,\{nm\}}F(\br_n,\br_m) \Theta_{N-2}^*(...\widehat{\br_n}...\widehat{\br_m}...) \lb{inhom} \ee is the inhomogeneous term of Eq.(\ref{closeq}) and $G_1^*(\br_1)=\overline{f}(\br_1)$. Because this quantity for $N>1$ involves the correlations of lower order, our construction will proceed inductively. We may assume that $G_N^*\in L^2\left(\Omega^{\otimes N}\right)$ (in fact, $G_N^*\in H^1_0\left(\Omega^{\otimes N}\right)$ away from the set $\Gamma$). This statement is true for $N=1$ and, for $N\geq 2$, may be assumed to be true for all $M0$. Second, because $\oH_N$ is closed and $-{{1}\over{\en}}$ is in its resolvent set, it follows from the first equality of (\ref{resop}) that $\cS^\en_N:L^2\left(\Omega^{\otimes N}\right)\rightarrow \bD(\oH_N)$. This exhibits the ``smoothing'' property of the $\cS^\en_N$. Third, $\cS^\en_N$ for each $\en>0$ commutes with $\oH_N$, or, more correctly, $\cS^\en_N\oH_N\subset\oH_N\cS^\en_N$. Finally, because $\lim_{\en\rightarrow 0}\en\oH_N=0$ in the strong resolvent sense, it follows from the second equality of (\ref{resop}) that \be \lim_{\en\rightarrow 0}\|\cS_N^\en\Psi_N-\Psi_N\|_{L^2}=0 \lb{converg} \ee for all $\Psi_N\in L^2\left(\Omega^{\otimes N}\right)$. We now observe that, if $\Theta_N$ satisfies (\ref{weakeq}) for any $G_N$ in $L^2$, then for any $\Phi_N\in \bD(\oH_N)$, \begin{eqnarray} \ch_N[\Phi_N,\cS^\en_N\Theta_N] & = & \langle\oH_N\Phi_N,\cS^\en_N\Theta_N\rangle \cr & = & \langle\cS^\en_N\oH_N\Phi_N,\Theta_N\rangle \cr & = & \langle\oH_N\cS^\en_N\Phi_N,\Theta_N\rangle \cr & = & \langle\cS^\en_N\Phi_N,G_N\rangle=\langle\Phi_N,\cS^\en_NG_N\rangle. \lb{smeq} \end{eqnarray} In particular, if we apply this to $\Phi_N=\cS^\en_N\Theta_N$, then we find for the quadratic form $\ch_N[\cS^\en_N\Theta_N] =\langle\cS^\en_N\Theta_N,\cS^\en_NG_N\rangle$ and, thus, \be \ch_N[\cS^\en_N\Theta_N]\leq \|\Theta_N\|_{L^2}\cdot\|G_N\|_{L^2} \lb{unbd} \ee uniformly in $\en>0$. Since, in addition, the form $\ch_N$ is closed and $s-\lim_{\en\rightarrow 0}\cS^\en_N\Theta_N =\Theta_N$ by Eq.(\ref{converg}), it follows from Theorem VI.1.16 of \cite{Kato} that $\Theta_N\in\bD(\ch_N)$. In that case, for any $\Phi_N\in \bD(\oH_N)$, the equation (\ref{weakeq}) may be rewritten \be \ch_N[\Phi_N,\Theta_N]=\langle \Phi_N,G_N\rangle. \lb{2ndweakeq} \ee Furthermore, $\bD(\oH_N)$ is a core for $\bD(\ch_N)$ by the First Representation Theorem for forms: see \cite{Kato}, Theorem VI.2.1,item {\em (ii)}. By the same Theorem VI.2.1, item {\em (iii)}, it follows from (\ref{2ndweakeq}) that $\Theta_N\in \bD(\oH_N)$ and that \be \oH_N\Theta_N=G_N \lb{opeq} \ee with equality as elements of $L^2\left(\Omega^{\otimes N}\right)$. We observe, since $\oH_N^{-1}$ is bounded, that the equation (\ref{opeq}) is equivalent to \be \Theta_N=\oH_N^{-1}G_N \lb{invopeq} \ee However, for any $G_N\in L^2\left(\Omega^{\otimes N}\right)$ the righthand side of (\ref{invopeq}) exists, again by boundedness of $\oH_N^{-1}$, and it defines an element $\Theta_N=\oH_N^{-1}G_N\in \bD(\oH_N)$. Thus, the weak solution exists and is unique. $\,\,\,\,\,\Box$ \noindent {\em Proof of Theorem 2 (i)}: The proof of existence and uniqueness here very closely parallels the previous one, but is even easier. For this reason, we will discuss only a few details. As in the previous case, we may begin by introducing a symmetric sesquilinear form, \be h_N^{(\kappa_p)}[\Psi_N,\Phi_N]=h_N[\Psi_N,\Phi_N] +\sum_{n=1}^N\int_{\Omega^{\otimes N}} d\bR\,\, \overline{(-\bigtriangleup_{\br_n})^{p/2}\Psi_N(\bR)} \cdot(-\bigtriangleup_{\br_n})^{p/2}\Phi_N(\bR). \lb{psesq} \ee densely defined on either the same domain as before, $\bD\left(h_N^{(\kappa_p)}\right)=\bD(h_N)$, or, with identical results, $\bD\left(h_N^{(\kappa_p)}\right)=C^\infty_0\left(\Omega^{\otimes N}\right)$. Clearly, this is the same as $h_N^{(\kappa_p)}[\Psi_N,\Phi_N] =\langle\Psi_N,\cH_N^{(\kappa_p)}\Phi_N\rangle$, where $\cH_N^{(\kappa_p)}$ is the differential operator in Eq.(\ref{psingell}) with $\bD\left(\cH_N^{(\kappa_p)}\right)= \bD\left(h_N^{(\kappa_p)}\right)$. We may now consider the self-adjoint (Friedrichs) extensions of these operators, denoted $\oH_N^{(\kappa_p)}$, just as before. We may observe that there is a basic inequality, \be \ch_N^{(\kappa_p)}[\Psi_N]\geq {{\kappa_p A_N}\over{L^{2p}}}\|\Psi_N\|_{L^2}^2 \lb{poinc} \ee with some constant $A_N>0$, for all $\Psi_N\in\bD\left(\ch_N^{(\kappa_p)}\right)$. This plays the same role in the present proof as inequality (\ref{3rdineq}) of Proposition 4 {\em (iii)} did in the previous one. It is proved first for $h_N^{(\kappa_p)}[\Psi_N]$ with $\Psi_N\in\bD\left(h_N^{(\kappa_p)}\right)$, by expanding the elements of $\bD\left(h_N^{(\kappa_p)}\right)$ in a series of eigenfunctions of the Dirichlet Laplacian $(-\bigtriangleup_\bR)_D$, which are complete in $H^p_0\left(\Omega^{\otimes N}\right)$. Then, the result is extended to $\ch_N^{(\kappa_p)}[\Psi_N]$ by taking limits. Note that the inequality (\ref{poinc}), in particular, implies that the operator $\oH_N^{(\kappa_p)}$ is strictly positive, with lower bound $\geq \kappa_p A_N/L^{2p}$. Therefore, the inverse operator $\left[\oH_N^{(\kappa_p)} \right]^{-1}$ is bounded, as before, and unique weak solutions $\Theta_N^{(\kappa_p)*}$ of the stationary equations are easily constructed with its aid. A last point which requires some explanation is the regularity $\Theta_N^{(\kappa_p)*}\in H^p_0\left(\Omega^{\otimes N} \right)$ of solutions. In fact, it follows as before that $\Theta_N^{(\kappa_p)*}\in \bD\left(\ch_N^{(\kappa_p)}\right)$. It therefore suffices to show that $\bD\left(\ch_N^{(\kappa_p)}\right)\subset H^p_0\left(\Omega^{\otimes N}\right)$. We may identify $\bD\left(\ch_N^{(\kappa_p)}\right)$ as the completion of the pre-Hilbert space $C^\infty_0\left( \Omega^{\otimes N}\right)$ with the inner product \be \langle\Psi_N,\Phi_N\rangle_{h_N^{(\kappa_p)}} =\langle\Psi_N,\Phi_N\rangle+h_N^{(\kappa_p)}[\Psi_N,\Phi_N]. \lb{pinnprod} \ee See \cite{Kato}, Section VI.1.3. However, we have the elementary inequality \be \left[\sum_{n=1}^N \,k_n^2\right]^{p/2}\leq C_{N,p}\left[\sum_{n=1}^N \,(k_n^2)^{p/2}\right], \lb{elemineq} \ee with $C_{N,p}=N^{(p-2)/2}$ for $p\geq 2$ and $=1$ for $1\leq p\leq 2$. Using then the Parseval's equality for Fourier integrals, it follows that the norm $\|\Psi_N\|_{h_N^{(\kappa_p)}}$ is stronger on $C^\infty_0\left( \Omega^{\otimes N}\right)$ than the Sobolev norm \be \|\Psi_N\|_{H^p}^2\equiv \|\Psi_N\|^2_{L^2}+\|(-\bigtriangleup_\bR)^{p/2}\Psi_N\|^2_{L^2}. \lb{sobnorm} \ee Since $H^p_0\left(\Omega^{\otimes N}\right)$ is defined to be the completion of $C^\infty_0\left(\Omega^{\otimes N}\right)$ in the norm $\|\cdot\|_{H^p}$, it follows that $\bD\left(\ch_N^{(\kappa_p)}\right)\subset H^p_0\left(\Omega^{\otimes N} \right)$, as required. $\,\,\,\,\,\Box$ \noindent {\em Proof of Theorem 2 (ii):} To construct the weak-$L^2$ limits of $\Theta_N^{(\kappa_p)*}$ for $\kappa_p\rightarrow 0$, the main thing that is required are a priori estimates on the $L^2$-norms uniform in $\kappa_p>0$. These are provided as follows. First, we note that $\Theta_N^{(\kappa_p)*}\in \bD(\ch_N)$ because $\Theta_N^{(\kappa_p)*}\in \bD\left(\oH_N^{(\kappa_p)}\right)$ and $\bD\left(\oH_N^{(\kappa_p)}\right) \subset H^p\left(\Omega^{\otimes N}\right)\subset \bD(\ch_N)$ for $p\geq 1$. Thus, we may apply Proposition 4 {\em (iii)}, inequality (\ref{3rdineq}), to calculate that \begin{eqnarray} \|\Theta_N^{(\kappa_p)*}\|_{L^2}^2 & \leq & C_N' L^\gamma \ch_N\left[\Theta_N^{(\kappa_p)*}\right] \cr \, & \leq & C_N' L^\gamma \ch_N^{(\kappa_p)}\left[\Theta_N^{(\kappa_p)*}\right] \cr \, & = & C_N' L^\gamma \langle \Theta_N^{(\kappa_p)*},G^{(\kappa_p)*}_N\rangle \cr \, & \leq & C_N' L^\gamma \|\Theta_N^{(\kappa_p)*}\|_{L^2} \|G^{(\kappa_p)*}_N\|_{L^2}. \lb{preL2} \end{eqnarray} with $C_N'=[C_N(d-\gamma)^2/2]^{-1}$. In other words, \be \|\Theta_N^{(\kappa_p)*}\|_{L^2}\leq C_N' L^\gamma \|G^{(\kappa_p)*}_N\|_{L^2}. \lb{1stL2ineq} \ee Using the expression (\ref{inhom}) for $G^{(\kappa_p)*}_N$ in terms of the lower-order $\Theta_{M}^{(\kappa_p)*}$, for $M0$. Passing to the limit along subsequence $\kappa_p^{(n')}$, we then obtain \be \langle\oH_N\Phi_N,\Theta_N^{(0)*}\rangle=\langle \Phi_N,G_N^{(0)*}\rangle, \lb{0weakeq} \ee for all $\Phi_N\in C_0^\infty\left(\Omega^{\otimes N}\right)$. This is not quite the statement that $\Theta_N^{(0)*}$ is a weak solution of the zero-diffusivity equation, with our definitions. For that to be true it is required that (\ref{0weakeq}) hold for all $\Phi_N\in\bD(\oH_N)$. By the same argument as above, $C_0^\infty\left(\Omega^{ \otimes N}\right)$ is a dense subset of $\bD(\ch_N)$ in the Hilbert space $H_{h_N}$. Thus, we would like to take the limit in $H_{h_N}$ to obtain (\ref{0weakeq}) for all $\Phi_N\in\bD(\oH_N)$, as required. To do so, however, requires that $\Theta_N^{(0)*}\in \bD(\ch_N)$, so that we may write \be \ch_N\left[\Phi_N,\Theta_N^{(0)*}\right]=\langle \Phi_N,G_N^{(0)*}\rangle, \lb{0weakeq'} \ee In this form, the limit may be taken to obtain (\ref{0weakeq}) for all $\Phi_N\in\bD(\oH_N)$. Thus, to complete the proof, it is enough to show that $\Theta_N^{(0)*}\in \bD(\ch_N)$. To demonstrate the latter regularity of $\Theta_N^{(0)*}$, we shall use the second characterization of $\bD(\ch_N)$ in Proposition 4{\em (ii)}. We already have the estimate (\ref{0hNest}). All that is required in addition is to show that \be \gamma_k\left(\left.\Theta_N^{(0)*}\right|_{{\Omega^{\otimes N}}_k}\right)=0 \lb{zerotr} \ee for all $k\geq 1$. To obtain this, we remark that for each $k$ the identity injection $\iota_k:H_{h_N}\left({\Omega^{ \otimes N}}_k\right)\rightarrow H^s\left({\Omega^{\otimes N}}_k\right)$ is compact for any $s<1$, because the identity injection from $H_{h_N}\left({\Omega^{\otimes N}}_k\right)$ to $H^1\left({\Omega^{\otimes N}}_k\right)$ is bounded by (\ref{lower}) and the identity injection $H^1\left({\Omega^{\otimes N}}_k\right)$ into $H^s\left({\Omega^{\otimes N}}_k\right)$ is compact, by the Rellich lemma. We may use the above compact embedding for each fixed $k$ to extract by a diagonal argument a further subsequence $\kappa_p^{(n''')}$ such that \be \lim_{n'''\rightarrow\infty}\left\|\Theta_N^{(\kappa_p^{(n''')})*} -\Theta_N^{(0)*}\right\|_{H^s\left({\Omega^{\otimes N}}_k\right)}=0 \lb{stHslim} \ee for {\em all} $k\geq 1$. However, for each $k$, the trace $\gamma_k$ is continuous as a map from $H^s\left({\Omega^{ \otimes N}}_k\right)$ into $L^2\left(\partial\Omega^{\otimes N}\bigcap{\Omega^{\otimes N}}_k\right)$ when $s>1/2$. Furthermore, \be \gamma_k\left(\left.\Theta_N^{(\kappa_p^{(n''')})*}\right|_{{\Omega^{\otimes N}}_k}\right)=0 \lb{pzerotr} \ee for all $n'''$. Thus, passing to the limit, we obtain (\ref{zerotr}). $\,\,\,\,\,\Box$ \section{Concluding Remarks} We make here just a few remarks on some further results of our analysis and some outstanding problems for future work. \noindent {\em (i) Regularity of the Solutions} \noindent The construction above produces solutions $\Theta_N^*\in L^2\left(\Omega^{\otimes N}\right)$ and $\in H^1_0\left(\Omega^{\otimes N}\right)$ away from the singular set $\Gamma$. In fact, as was mentioned in the Introduction, it is expected that $\Theta_N^*$ are H\"{o}lder regular, $\Theta_N^*\in C^\gamma \left(\Omega^{\otimes N}\right)$. Such additional regularity of the solutions of the singular-elliptic equations may follow from Harnack inequalities \cite{Mos,Trud}. \noindent {\em (ii) $N$-Dependence of Spectral Gap and Invariant Measure on Scalar Fields} \noindent The Proposition 2 has only been fully proved here for $N\leq 4$. Assuming that it holds for general $N$, the question of the $N$-dependence of the constant $C_N$ appearing in its statement has also some importance. As we have seen, the solutions $\Theta_N^*$ constructed for $\kappa=0$ obey an $L^2$-bound \be \|\Theta_N^*\|_{L^2\left(\Omega^{\otimes N}\right)}\leq B^N\cdot N! \lb{BmainL2} \ee in which $B$ is proportional to the inverse of $\min_{N\geq 1}C_N$. If $C_N$ is bounded from below uniformly in $N$, then the above constant $B<\infty$. 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