Content-Type: multipart/mixed; boundary="-------------9810140415368" This is a multi-part message in MIME format. ---------------9810140415368 Content-Type: text/plain; name="98-643.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-643.keywords" effective potential, many electron system, BCS theory ---------------9810140415368 Content-Type: application/x-tex; name="gm3.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="gm3.tex" \magnification=\magstep1 \font\gross=cmbx12 scaled \magstep0 \font\mittel=cmbx10 scaled \magstep0 \font\Gross=cmr12 scaled \magstep2 \font\Mittel=cmr12 scaled \magstep0 \overfullrule=0pt %\input amssym \def\Bbb#1{{\bf #1}} \def\blacksquare{\sqcup\!\!\!\!\sqcap} \baselineskip=12pt %\nopagenumbers \def\up{\uparrow} \def\down{\downarrow} \def\k{{\bf k}} \def\n{{\bf n}} \def\m{{\bf m}} \def\p{{\bf p}} \def\q{{\bf q}} \def\x{{\bf x}} \def\y{{\bf y}} \def\I{{\rm I}} \def\II{{\rm II}} \def\III{{\rm III}} \def\IV{{\rm IV}} \def\ts{\textstyle} \def\ds{\displaystyle} \def\tr{\Delta} \def\la{\langle} \def\ra{\rangle} \def\pro{\mathop\Pi} \def\1cm{\hskip 1cm} \def\vp{\varphi} \def\U{{\cal U}} \def\V{{\cal V}} \def\u{\underline} \def\ep{\epsilon} \def\vep{\varepsilon} \def\ka{\kappa} \def\wk{\sqrt{\kappa}} { \nopagenumbers \baselineskip=14pt $$ $$ \vskip 3cm \centerline{\Gross The Global Minimum of the Effective Potential } \smallskip \centerline{\Gross of the Many-Electron System with Delta-Interaction} \bigskip \bigskip \centerline{by} \bigskip \bigskip \centerline{{\Mittel Detlef Lehmann}\footnote{$\phantom{}^*$}{ e-mail: lehmann@math.tu-berlin.de}} \centerline{\Mittel Technische Universit\"at Berlin} \centerline{\Mittel Fachbereich Mathematik Ma 7-2} \centerline{\Mittel Sta{\ss}e des 17. Juni 136} \centerline{\Mittel D-10623 Berlin, GERMANY} \vskip 4cm \noindent{\bf Abstract:} We prove that the global minimum of the real part of the full effective potential of the many electron system with attractive delta interaction is in fact given by the BCS mean field configuration. This is a consequence of a simple bound which is obtained by applying Hadamard's inequality to the functional determinant. The second order Taylor expansion around the minimum is computed. \bigskip \bigskip %\centerline{\it (preliminary version)} \vfill \eject } \baselineskip=14pt \noindent{\gross I. Introduction and Results} \bigskip In this article we consider the effective potential $V$ of the nonrelativistic many electron system with attractive delta function interaction. In momentum space, it reads $$\eqalignno{ V(\{\phi_q\})&=\sum_q |\phi_q|^2 -\log{ \det\left[\matrix{ Id & {ig\over \sqrt{\beta L^d} }\, C \phi^* \cr {ig\over\sqrt{\beta L^d} }\, \bar C\phi & Id \cr}\right] } & (1) \cr & \cr &=\sum_q |\phi_q|^2-\ts\log{\det\left[Id+{g^2\over\beta L^d}\,\bar C\phi C\phi^* \right]} \cr}$$ where $g^2=\lambda>0$ is an attractive coupling, $C=(\delta_{k,p}C_k)_{k,p}$ is a diagonal matrix with entries $C_k={1\over ik_0-\vep_\k+\mu}$, $\vep_\k=\vep_{-\k}$ being the single particle energy momentum relation and $\mu$ denoting the chemical potential. In the following we set $e_\k=\vep_\k-\mu$ such that the Fermi surface is given by $e_\k=0$. Furthermore $\phi$ is a short notation for the matrix $(\phi_{k-p})_{k,p}$. The momenta $k=(k_0,\k),p=(p_0,\p)$ range over some finite subset of ${\pi\over\beta}(2\Bbb Z+1)\times \left({2\pi\over L}\Bbb Z\right)^d$ if the system is kept in finite volume $[0,L]^d$ and at some small but positive temperature $T={1\over \beta}>0$. To be specific, choose $$\ts k,p\in M_\nu:=\left\{ (k_0,\k)\in {\pi\over\beta}(2\Bbb Z+1)\times \left({2\pi\over L}\Bbb Z\right)^d\>\bigr|\> |e_\k| \le 1,\; |k_0|\le \nu\right\} \eqno (2)$$ where $1<\!\!<\nu<\infty$ is some cuttoff. The momenta $q=(q_0,\q)$ are given by $q\in\{k-p\>|\>k,p\in M_\nu\}$. \par It is widely believed that the global minimum of the real part of $V$ (in general, the determinant in (1) is complex) is given by the mean field configuration $\phi_x=const$ in coordinate space or $\phi_q\sim\delta_{q,0}$ in momentum space [1,2,3]. Although this is suggested by diagrammatic arguments [4], there was, to the authors knowledge, no rigorous proof of that. In Theorem 1 below we show that this result can be obtained by applying Hadamard's inequality (the absolute value of the determinant of $n$ column vectors is less than the volume of the cube spanned by the $n$ vectors) to the determinant in (1). \par More precisely, all global minima of the real part $${\rm Re}V(\{\phi_q\})=\sum_q |\phi_q|^2-\log{\left| \det\left[\matrix{ Id & {ig\over \sqrt{\beta L^d} }\, C \phi^* \cr {ig\over\sqrt{\beta L^d} }\, \bar C\phi & Id \cr}\right]\right| } \eqno(3)$$ of $V$ are given by $$\phi_q=\delta_{q,0}\sqrt{\beta L^d}\, r_0\, e^{i\theta} \eqno (4)$$ where $\theta\in[0,2\pi]$ is an arbitrary phase and $\Delta^2=\lambda r_0^2$ is a solution of the BCS equation $${\ts {\lambda\over \beta L^d}}\sum_{k\in M_\nu} \ts{1\over k_0^2+e_\k^2+\lambda r_0^2}=1 \eqno (5)$$ Equation (4) describes the BCS mean field configuration. \par In Theorem 2 we expand $V$ up to second order in $\xi=(\xi_{k-p})_{k,p}$ where $$\eqalignno{ \xi_{q}&=\phi_{q}-\delta_{q,0}\sqrt{\beta L^d}\,r_0\, e^{i\theta_0} =\cases{ (\rho_0-\sqrt{\beta L^d}\,r_0)e^{i\theta_0} &for $q=0$ \cr \rho_{q} e^{i\theta_{q}} &for $q\ne 0$. \cr} &(6) \cr}$$ We remark that a priori there is no need to introduce a small external field to fix the phase $\theta_0$ and then to expand with respect to radial and tangential components as it is usually done like in [2] or [3]. The expansion around $\xi$ gives $$\eqalignno{ V(\{\phi_q\})&=V_{\rm min}+2\beta_0 \, (\rho_0-\sqrt{\beta L^d}\,r_0)^2 + \sum_{q\ne 0} (\alpha_q+i\gamma_q)\, \rho_q^2 \cr &\phantom{m} + {\ts{1\over2}}\sum_{q\ne 0} \beta_q \,|e^{-i\theta_0} \phi_q +e^{i\theta_0} \bar\phi_{-q}|^2 +O(\xi^3) & (7) \cr} $$ where $V_{\rm min}\sim \beta L^d\Delta\log(1/\Delta)$ is identical the global minimum of the BCS effectiv potential $$V_{\rm BCS}(\rho)=\beta L^d\left(\rho^2-\ts {1\over\beta L^d}\sum_{k} \log\left[ 1+ {\lambda \rho^2\over k_0^2+e_\k^2}\right] \right) \eqno (8)$$ and the coefficients $\alpha_q, \;\beta_q$ are real and positive and $\gamma_q$ is real. They are given in Theorem 2 below. In particular, $\alpha_q\sim q^2$ for small $q$. \par The error term in (7) is not uniform in the coupling $\lambda$. Since $\beta_q\to 0$ and $\alpha_q+i\gamma_q\to 1$ for $\lambda\to 0$ (see (54)), equation (7) becomes in the limit $\lambda\to 0$ $$ \sum_q|\phi_q|^2=\sum_{q\ne 0}|\phi_q|^2+O(\xi^3) \eqno (9)$$ or $|\phi_0|^2=O(\xi^3)$. The reason for this can already be seen if one expands $V_{\rm BCS}$ above around its minimum $r_0$. One obtains, abbreviating $E_k^2:=k_0^2+e_\k^2+\Delta^2$, $$\eqalignno{ {\ts {1\over \beta L^d}}\bigl(V_{\rm BCS}&(\rho) -V_{\rm min}\bigr) =\rho^2-r_0^2-\ts {1\over\beta L^d}\sum_k \log\left[ 1+{\lambda(\rho^2-r_0^2)\over k_0^2+e_\k^2+\Delta^2} \right] \cr &=\left\{1-\ts {\lambda\over \beta L^d}\sum_k{\ts {1\over E_k^2}} \right\} (\rho^2-r_0^2)+ \ts {1\over2} {1\over \beta L^d}\sum_k \ts {\lambda^2\over E_k^4}(\rho^2-r_0^2)^2 +O\bigl( (\rho-r_0)^3\bigr) \cr &=\ts {1\over2} {1\over \beta L^d}\sum_k \ts {\lambda^2\over E_k^4}\left[ 4r_0^2(\rho-r_0)^2+ 4r_0(\rho-r_0)^3+(\rho-r_0)^4\right] +O\bigl( (\rho-r_0)^3\bigr)\;\;&(10)\cr}$$ where in the last line the BCS equation (5) has been used. Since ${1\over \beta L^d}\sum_k{1\over E_k^4} \sim{1\over\Delta^2} ={1\over\lambda r_0^2}$, the quadratic term on the right hand side of (10) goes to 0, whereas the third and fourth order terms diverge. \bigskip Thus it is not clear to what extent one may draw conclusions from the quadratic approximation. Nevertheless, we write down the results for the partition function and the expectation value of the energy obtained by using (7). To this end we shortly recall the relation between the Hamiltonian and the functional integral representation of the model. One may look in [5] to see more explicit computations. \bigskip The Hamiltonian for the many-electron system with delta interaction in finite volume at some small but positive temperature $T={1\over \beta}>0$ described in the grand canonical ensemble is given by $$H=H_0-\lambda H_{\rm int}={\ts {1\over L^d}}\sum_{\k\sigma}(\vep_\k-\mu)a_{\k\sigma}^+ a_{\k\sigma}-{\ts {\lambda\over L^{3d}}} \sum_{\k\p\q} a_{\k\up}^+ a_{\q-\k\down}^+ a_{\q-\p\down} a_{\p\up} \eqno (11)$$ We are interested in the grand canonical partition function $Tr\,e^{-\beta H}$, in the correlation function $$\tilde \Lambda(\q)={\ts {1\over L^{3d}}}\sum_{\k,\p}Tr[ a_{\k\up}^+ a_{\q-\k\down}^+ a_{\q-\p\down} a_{\p\up} e^{-\beta H}] \bigr/ Tr\, e^{-\beta H} \eqno (12)$$ and in the expectation value $E_{\rm int}=\la H_{\rm int}\ra=\sum_\q \tilde \Lambda(\q)$. \medskip In terms of Grassmann integrals, the perturbation series for the normalized partition function $Z=Tr e^{-\beta(H_0-\lambda H_{\rm int})}/Tr e^{-\beta H_0}$ is given by $$Z=\int e^{ {\lambda\over(\beta L^d)^3}\sum_{k,p,q}\bar\psi_{k\up} \bar\psi_{q-k\down}\psi_{q-p\down}\psi_{p\up} } d\mu_C\eqno (13)$$ and $\tilde \Lambda(\q)$ is given by $$\tilde \Lambda(\q)={\ts {1\over \beta}}\sum_{q_0\in {2\pi\over\beta}\Bbb Z} \Lambda(q_0,\q) \eqno (14)$$ where $$\eqalignno{ \Lambda(q_0,\q)& ={\ts {1\over (\beta L^d)^3}}\sum_{k,p}\la\bar\psi_{k\up} \bar\psi_{q-k\down}\psi_{q-p\down}\psi_{p\up}\ra &(15) \cr &={\ts {1\over (\beta L^d)^3}}\sum_{k,p} {\ts {1\over Z}} \int \bar\psi_{k\up} \bar\psi_{q-k\down}\psi_{q-p\down}\psi_{p\up}\> e^{ {\lambda\over(\beta L^d)^3}\sum_{k,p,q}\bar\psi_{k\up} \bar\psi_{q-k\down}\psi_{q-p\down}\psi_{p\up} } d\mu_C \cr &={\ts {d\over d\lambda_q }}_{|_{\lambda_q=\lambda}}\log \int e^{ {1\over(\beta L^d)^3}\sum_{k,p,q}\lambda_q \bar\psi_{k\up} \bar\psi_{q-k\down}\psi_{q-p\down}\psi_{p\up} } d\mu_C&(16) \cr}$$ By making a Hubbard Stratonovich transformation, that is, by applying the formula ($\phi_q=u_q+iv_q$, $\bar\phi_q=u_q-iv_q$, $d\phi_qd\bar\phi_q:= du_qdv_q$) $$ e^{\sum_q a_q b_q}=\int e^{\sum_q a_q\phi_q+b_q\bar\phi_q} e^{-\sum_q|\phi_q|^2}\pro_q{\ts {d\phi_qd\bar\phi_q\over \pi}}$$ with $a_q=ig_q(\beta L^d)^{-{3\over2}} \sum_k\bar\psi_{k\up}\bar\psi_{q-k\down}$ and $b_q=ig_q(\beta L^d)^{-{3\over2}} \sum_p \psi_{p\up}\bar\psi_{q-p\down}$, $g_q=\sqrt\lambda_q$, the exponent becomes quadratic in the Grassmann variables and the fermionic integral can be performed. The result is $$Z(\{\lambda_q\})=\int e^{-V(\{\phi_q\})} \pro_q {\ts {d\phi_qd\bar\phi_q\over \pi}} \eqno (17)$$ where, if the $\lambda_q$'s are not all equal, instead of (1) $V$ is given by $$\eqalignno{ V(\{\phi_q\})&=\sum_q |\phi_q|^2 \ts -\log{ \det\left[\matrix{ Id +{1\over\beta L^d} C (g\phi)^* \bar C(g\phi)}\right] } & (18) \cr}$$ where $(g\phi)$ stands for the matrix $(g_{k-p}\phi_{k-p})_{k,p}$. To perform the $\lambda_q$-derivative, one may change variables to obtain $$\eqalignno{ \Lambda(q)&={\ts {1\over Z}{d\over d\lambda_q}} \int e^{-\sum_q {|\phi_q|^2\over\lambda_q}}\ts \det\left[\matrix{ Id+{1\over\beta L^d} \bar C\phi C \phi^*}\right] \pro_q {\ts {d\phi_qd\bar\phi_q\over\lambda_q \pi}} \cr &={\ts {1\over Z}} \int\left( {\ts {|\phi_q|^2\over \lambda_q^2} -{1\over\lambda_q} }\right) e^{-\sum_q {|\phi_q|^2\over\lambda}} \det\left[\matrix{ Id+{1\over\beta L^d} \bar C\phi C \phi^*}\right] \pro_q {\ts {d\phi_qd\bar\phi_q\over\lambda\pi}} \cr &={\ts {1\over Z}} \int {\ts {|\phi_q|^2-1\over \lambda}} e^{-\sum_q {|\phi_q|^2}} \det\left[\matrix{ Id+{\lambda\over\beta L^d} \bar C\phi C \phi^* }\right] \pro_q {\ts {d\phi_qd\bar\phi_q\over\pi}} \cr &= {\ts {1\over Z}} \int {\ts {|\phi_q|^2-1\over \lambda}} e^{-V(\{\phi_q\})} \pro_q {\ts {d\phi_qd\bar\phi_q\over\pi}}&(19) \cr}$$ For the ideal Fermi gas $\lambda=0$ the above formula gives ${0\over0}$. In that case one may use (15) to obtain $$\eqalignno{ \Lambda(q)& ={\ts{1\over (\beta L^d)^3}}\sum_{k,p} \la\bar \psi_{k\up}\psi_{p\up}\ra \la \bar\psi_{q-k\down}\psi_{q-p\down}\ra ={\ts{1\over (\beta L^d)^3}}\sum_{k,p} \beta L^d \delta_{k,p} C_k \>\beta L^d \delta_{q-k,q-p} C_{q-k} \cr &={\ts{1\over \beta L^d}}\sum_k C_k C_{q-k} &(20) \cr}$$ which is the particle particle bubble. \bigskip If one substitutes $V$ by the second order approximation $$V_2(\{\phi_q\}):=V_{\rm min}+2\beta_0 \, (\rho_0-\sqrt{\kappa}\,r_0)^2 + \sum_{q\ne 0}( \alpha_q+i\gamma_q) \rho_q^2 + {\ts{1\over2}}\sum_{q\ne 0} \beta_q \,|e^{-i\theta_0} \phi_q +e^{i\theta_0} \bar\phi_{-q}|^2 \eqno (21)$$ the integrals in (17,19) become Gaussian and can be performed. The results are (the index ``2" in the following means that $V_2$ instead of $V$ has been used) $$Z_2=e^{-V_{\rm min}} \, \int_0^\infty e^{-2\beta_0^2(\rho_0-\sqrt\kappa r_0)^2 } 2\rho_0d\rho_0 \pro_{q\atop q_0>0} \ts {1\over \alpha_q^2+\gamma_g^2+2\alpha_q\beta_q} \>, \eqno (22)$$ $$\Lambda_2(q)=\ts {1\over\lambda} \left( {\alpha_q+i\gamma_q+\beta_q \over \alpha_q^2+\gamma_q^2+2\alpha_q\beta_q} -1 \right)\eqno (23) $$ and, since $\gamma_{-q}=-\gamma_q$, $$\eqalignno{ \varepsilon_{\rm int,2}&:={\ts {1\over L^d}} \la H_{\rm int}\ra_2 ={\ts {1\over \beta L^d }} \sum_q \Lambda_2(q) \cr &={\ts {1\over \lambda}{1\over \beta L^d}} \sum_q {\ts \left( {\alpha_q+\beta_q \over \alpha_q^2+\gamma_q^2+2\alpha_q\beta_q} -1 \right)}\ge {\ts {1\over \lambda} {1\over\beta L^d}} \sum_q \ts \left( {1\over \alpha_q}-1\right) &(24) \cr} $$ In particular, since $\alpha_q\le const {q^2\over \Delta^2}$ for small $q$, $$\varepsilon_{\rm int,2}\ge {\ts {1\over \lambda} {1\over\beta L^d}} \sum_q \ts \left( const {\Delta^2\over q^2}-1\right) \eqno (25) $$ which, for $L\to\infty$, is infrared singular for $d=1$ and, since $q_0=0$ is an allowed value for positive temperature, also logarithmically divergent for small $q$ in $d=2$. A similar observation is made in [3]. % % % THEOREM 1 % % \bigskip \bigskip \noindent{\gross II. Proof of Theorems} \bigskip \noindent{\bf Theorem 1:} {\it Let {\rm Re}$V$ be the real part of the effective potential for the many electron system with attractive delta interaction given by (3). Let $e_\k=\vep_\k-\mu$ satisfy $e_\k=e_{-\k}$ and: $\forall\k\;e_\k=e_{\k+\q}\Rightarrow \q=0$. Then all global minima of Re$V$ are given by (4), $$\phi_q=\delta_{q,0}\sqrt{\beta L^d} \, r_0\, e^{i\theta},\1cm \theta\in[0,2\pi]\;\;\;{\rm arbitrary} \eqno (4)$$ where $r_0$ is a solution of the BCS equation (5) or, equivalently, the global minimum of the function (8) $$\eqalignno{ V_{\rm BCS}(\rho)&=V(\{\delta_{q,0}\sqrt{\beta L^d}\rho\,e^{i\theta} \}) \cr &= \beta L^d\rho^2-\sum_k {\ts\log\left[ 1+{\lambda \rho^2\over k_0^2+e_\k^2}\right]} =\beta L^d \rho^2- \sum_\k {\ts\log\left[{ \cosh({\beta\over2} \sqrt{ e_\k^2+\lambda\rho^2})\over \cosh{\beta\over2}e_\k }\right]} & (8)\cr}$$ More specifically, there is the bound $$\eqalignno{ {\rm Re}V(\{\phi_q\})\ge \ts V_{\rm BCS}( \|\phi\|)& -\min_{k\in M_\nu}\ts \Biggl\{ \log\left[{\ds \pro_{q\ne 0} } \biggl(1-{|{\lambda\over \beta L^d}\sum_{p}\phi_p \bar \phi_{p+q}|^2\over (|a_k|^2+\lambda\|\phi\|^2) (|a_{k+q}|^2+\lambda\|\phi\|^2) } \biggr)^{1\over2} \right] &(26) \cr &\phantom{mmmm}+\ts\log\left[{\ds \pro_{q\ne 0} } \biggl(1-{ {\lambda\over \beta L^d}|\phi_q|^2|a_{k+q}-a_k|^2 \over (|a_k|^2+\lambda\|\phi\|^2) (|a_{k+q}|^2+\lambda\|\phi\|^2) } \biggr)^{1\over2} \right]\Biggr\} \cr} $$ where $\|\phi\|^2:={1\over\beta L^d}\sum_q|\phi_q|^2$ and $|a_k|^2:= k_0^2+e_\k^2$. In particular, $${\rm Re}V(\{\phi_q\})\ge V_{\rm BCS}( \|\phi\|)\eqno (27)$$ since the products in (26) are less or equal 1. } \bigskip \noindent{\bf Proof:} Suppose first that (26) holds. For each $q$, the round brackets in (26) are between 0 and 1 which means that $-\log(\pro_q\cdots)$ is positive. Thus $${\rm Re}V(\{\phi_q\})\ge \ts V_{\rm BCS}( \|\phi\|)\ge V_{\rm BCS}(r_0)\eqno (28)$$ which proves that $\phi_q=\delta_{q,0}\sqrt{\beta L^d}r_0\,e^{i\theta} $ are indeed global minima of Re$V$. On the other hand, if a configuration $\{\phi_q\}$ is a global minimum, then the logarithms in (26) must be zero for all $k\in M_\nu$ which in particular means that for all $q\ne 0$ $$\sum_{p}\phi_p \bar \phi_{p+q}=0 \eqno(29)$$ and for all $k\in M_\nu$ and $q\ne 0$ $$ |\phi_q|^2|a_{k+q}-a_k|^2 =|\phi_q|^2[q_0^2+(e_{\k+\q}-e_\k)^2]=0\eqno(30)$$ which implies $\phi_q=0$ for all $q\ne 0$. It remains to prove (26). \par To this end, we write (recall that $C_k={1\over a_k}={1\over ik_0-e_\k}$) $$\det\left[\matrix{ Id & {ig\over \sqrt{\beta L^d} }\, \bar C \phi^* \cr {ig\over\sqrt{\beta L^d} }\, C\phi & Id \cr}\right]=\det\left[ \matrix{&|& &|& \cr &\vec b_k& &\vec b'_{k'}& \cr &|& &| & \cr}\right] \eqno (31)$$ where $$\vec b_k=\pmatrix{ \cr 1\cr \cr {\ts {ig\over\sqrt{\beta L^d}}} {\phi_{k-p}\over a_k} \cr \cr}\1cm \vec b'_{k'}= \pmatrix{ \cr {\ts {ig\over\sqrt{\beta L^d}}}{\bar\phi_{p-k'}\over \bar a_{k'}}\cr \cr 1 \cr \cr} \eqno (32) $$ If $|\vec b_k|$ denotes the euclidean norm of $\vec b_k$, then we have $$\ts |\vec b_k|^2=1+{\lambda\over\beta L^d}\sum_p{|\phi_{k-p}|^2\over |a_k|^2}=1+ \lambda{\|\phi\|^2\over |a_k|^2}=|\vec b'_{k}|^2 \eqno (33)$$ Therefore one obtains, if $\vec e_k={\vec b_k\over |\vec b_k|}$, ${\vec e}^{\,\prime}_k={\vec b'_k\over |\vec b'_k|}$ $$\det\left[ \matrix{&|& &|& \cr &\vec b_k& &\vec b'_{k'}& \cr &|& &| & \cr}\right]= \pro_k\left\{ 1+ \lambda{\|\phi\|^2\over |a_k|^2} \right\} \det\left[ \matrix{&|& &|& \cr &\vec e_k& &{\vec e}^{\,\prime}_{k'} & \cr &|& &| & \cr}\right] \eqno (34)$$ From this the inequality (27) already follows since the determinant on the right hand side of (34) is less or equal 1. To obtain (26), we choose a fixed but arbitrary momentum $t\in M_\nu$ and orthogonalize all vectors $\vec e_k$, ${\vec e}^{\,\prime}_{k'}$ in the determinant with respect to $e_t$. That is, we write $$ \det\left[ \matrix{&|& &|& \cr &\vec e_k& &{\vec e}^{\,\prime}_{k'}& \cr &|& &| & \cr}\right]= \det\left[ \matrix{&|& &|& \cr &\vec e_k-(\vec e_k,\vec e_t)\vec e_t& &{\vec e}^{\,\prime}_{k'}-({\vec e}^{\,\prime}_{k'},\vec e_t)\vec e_t & \cr &|& &| & \cr}\right] \eqno(35)$$ Finally we apply Hadamard's inequality, $ |\det F|\le \pro_{j=1}^n |\vec f_j| =\Bigl\{ \pro_{j=1}^n \sum_{i=1}^n |f_{ij}|^2 \Bigr\}^{1\over2} $ if $F=(f_{ij})_{1\le i,j\le n}$ is a complex matrix, to the determinant on the right hand side of (35). Since $$|\vec e_k-(\vec e_k,\vec e_t)\vec e_t|^2=1-|(\vec e_k,\vec e_t)|^2$$ one obtains $$\left|\det\left[ \matrix{&|& &|& \cr &\vec e_k-(\vec e_k,\vec e_t)\vec e_t& &{\vec e}^{\,\prime}_{k'}-({\vec e}^{\,\prime}_{k'},\vec e_t)\vec e_t & \cr &|& &| & \cr}\right]\right| $$ $$ \le |\vec e_t|\pro_{k\in M_\nu\atop k\ne t} (1-|(\vec e_k,\vec e_t)|^2)^{1\over2} \pro_{k\in M_\nu} (1-|({\vec e}^{\,\prime}_k,\vec e_t)|^2)^{1\over2} \eqno(36)$$ or with (31) and (34) $$\eqalignno{ {\rm Re}V(\{\phi_q\})&=\sum_q|\phi_q|^2 -\log \left|\det\left[\matrix{ Id & {ig\over \sqrt{\beta L^d} }\, \phi^*\bar C \cr {ig\over\sqrt{\beta L^d} }\, \phi C & Id \cr}\right]\right|\cr &=\beta L^d\|\phi\|^2 -\sum_k\log \left\{ 1+ \lambda{\|\phi\|^2\over |a_k|^2} \right\} -\log\left| \det\left[ \matrix{&|& &|& \cr &\vec e_k& &{\vec e}^{\,\prime}_{k'}& \cr &|& &| & \cr}\right] \right|\cr &=V_{\rm BCS}(\|\phi\|) -\log\left| \det\left[ \matrix{&|& &|& \cr &\vec e_k& &{\vec e}^{\,\prime}_{k'}& \cr &|& &| & \cr}\right] \right|\cr &\ge V_{\rm BCS}(\|\phi\|)-\log \Bigl\{ \pro_{k\in M_\nu\atop k\ne t} (1-|(\vec e_k,\vec e_t)|^2)^{1\over2} \pro_{k\in M_\nu} (1-|({\vec e}^{\,\prime}_k,\vec e_t)|^2)^{1\over2}\Bigr\}&(37) \cr}$$ Finally one has $$(\vec b_k,\vec b_t)=\sum_p {\ts {ig\over\sqrt{\beta L^d}}} {\phi_{k-p}\over a_k}\> \overline{ {\ts {ig\over\sqrt{\beta L^d}}} {\phi_{t-p}\over a_t} }$$ which gives $$|(\vec e_k,\vec e_t)|^2=\ts {\left| {\lambda\over\beta L^d}\sum_p\phi_{k-p} \bar\phi_{t-p}\right|^2\over (|a_k|^2+\lambda\|\phi\|^2)\phantom{^{1\over2}} \!\!\!(|a_t|^2+\lambda\|\phi\|^2) } \eqno (38)$$ and $$\eqalignno{ (\vec b'_k,\vec b_t)& ={\ts {ig\over\sqrt{\beta L^d}}}{\bar\phi_{t-k}\over \bar a_{k}}+ \overline{ {\ts {ig\over\sqrt{\beta L^d}}} {\phi_{t-k}\over a_t} } = {\ts {ig\over\sqrt{\beta L^d}}}\bar\phi_{t-k}\ts \left({1\over \bar a_{k}}- {1\over \bar a_{t}} \right) \cr}$$ which gives $$|({\vec e}^{\,\prime}_k,\vec e_t)|^2=\ts { {\lambda\over\beta L^d}|\phi_{t-k}|^2 |a_t-a_k|^2 \over (|a_k|^2+\lambda\|\phi\|^2)\phantom{^{1\over2}} \!\!\!(|a_t|^2+\lambda\|\phi\|^2) } \eqno (39)$$ Substituting (38,39) in (37) proves Theorem 1 $\blacksquare$ \bigskip % % % THEOREM 2 % % \bigskip \noindent{\bf Theorem 2:} {\it Let $V$ be the effective potential (1), let $\kappa=\beta L^d$ and let $\xi=(\xi_{k-p})_{k,p}$ be the matrix with entries $$\eqalignno{ \xi_{q}&=\phi_{q}-\delta_{q,0}\sqrt{\kappa}\,r_0\, e^{i\theta_0} =\cases{ (\rho_0-\sqrt{\kappa}\,r_0)e^{i\theta_0} &for $q=0$ \cr \rho_{q} e^{i\theta_{q}} &for $q\ne 0$. \cr} &(40) \cr}$$ Then $$ V(\{\phi_q\})=V_{\rm min}+2\beta_0 \, (\rho_0-\sqrt{\kappa}\,r_0)^2 + \sum_{q\ne 0}( \alpha_q+i\gamma_q) \rho_q^2 + {\ts{1\over2}}\sum_{q\ne 0} \beta_q \,|e^{-i\theta_0} \phi_q +e^{i\theta_0} \bar\phi_{-q}|^2 +O(\xi^3) \eqno(41) $$ where, if $E_k^2=k_0^2+e_\k^2+\lambda r_0^2$, $$\alpha_q= {\ts {1\over2}}{\ts{\lambda\over\kappa}}\sum_k {\ts { q_0^2+(e_\k-e_{\k-\q})^2 \over E_k^2 E_{k-q}^2}} >0\,,\;\;\; \beta_q= {\ts{\lambda\over\kappa}}\sum_k \ts {\lambda r_0^2 \over E_k^2 E_{k-q}^2} >0\,,\;\;$$ $$\gamma_q =-{\ts {\lambda\over \kappa }} \sum_k \ts {k_0e_{\k-\q}-(k_0-q_0)e_\k\over E_k^2 E_{k-q}^2} \in \Bbb R \eqno (42)$$ and $$\ts V_{\rm min}=\kappa\left( r_0^2- {1\over\kappa} \sum_\k {\ts\log\left[{ \cosh({\beta\over2} \sqrt{ e_\k^2+\lambda r_0^2})\over \cosh{\beta\over2}e_\k }\right]}\right). \eqno (43)$$ } \bigskip \noindent{\bf Proof:} We abbreviate $\kappa=\beta L^d$ and write $$V(\{\phi_q\})=\sum_q |\phi_q|^2-\log{ \det\left[\matrix{ A& {ig\over\wk} \phi^* \cr {ig\over\wk} \phi & \bar A \cr}\right] \over \det\left[\matrix{ A& 0 \cr 0 & \bar A \cr}\right] } \eqno (44)$$ where $A=C^{-1}=(\delta_{k,p}a_k)_{k,p\in M_\kappa}$ and $a_k:=1/C_k= ik_0-e_\k$. Then $$ \eqalignno{ V(\{\phi_q\})- V(\{\sqrt\kappa \, \delta_{q,0}\, r_0 e^{i\theta_0} \}) &=\sum_q \rho_q^2 -\kappa r_0^2 -\log{ \det\left[\matrix{ A& {ig\over\wk} \phi^* \cr {ig\over\wk} \phi & \bar A \cr}\right] \over \det\left[\matrix{ A& igr_0\, e^{-i\theta_0} \cr igr_0\, e^{i\theta_0} & \bar A \cr}\right] } &(45) \cr}$$ where $ igr_0\, e^{i\theta_0}\equiv igr_0\, e^{i\theta_0}\,Id$ in the determinant above. Since $$\eqalignno{ \left[\matrix{ A& igr_0\, e^{-i\theta_0} \cr igr_0\, e^{i\theta_0} & \bar A \cr}\right]^{-1}& = \left[\matrix{ {\bar a_k\delta_{k,p}\over |a_k|^2+\lambda r_0^2} & -{ igr_0\, e^{-i\theta_0}\delta_{k,p}\over |a_k|^2+\lambda r_0^2} \cr - { igr_0\, e^{i\theta_0} \delta_{k,p}\over |a_k|^2+\lambda r_0^2} & {a_k \delta_{k,p}\over |a_k|^2+\lambda r_0^2} \cr}\right] \cr &\equiv {\ts {1\over|a|^2+\lambda r_0^2}} \left[\matrix{ \bar A& -igr_0\, e^{-i\theta_0} \cr -igr_0\, e^{i\theta_0} & A \cr}\right] &(46) \cr}$$ and because of $$\eqalignno{ \left[\matrix{ A& {ig\over\wk} \phi^* \cr {ig\over\wk} \phi & \bar A \cr}\right]&= \left[\matrix{ A& igr_0\, e^{-i\theta_0} \cr igr_0\, e^{i\theta_0} & \bar A \cr}\right]\>+\> \left[\matrix{ 0& {ig\over\wk} \phi^*-igr_0\, e^{-i\theta_0} \cr {ig\over\wk} \phi -igr_0\, e^{i\theta_0}& 0\cr}\right] \cr &= \left[\matrix{ A& ig \bar\gamma \cr ig\gamma & \bar A \cr}\right]\>+\> \left[\matrix{ 0& ig\,\xi^* \cr ig\,\xi & 0\cr}\right] &(47)\cr}$$ where $\gamma=r_0\, e^{i\theta_0}$ and $\xi=(\xi_{k,p})$ is given by (40), the quotient of determinants in (45) is given by $$\eqalignno{ \det&\left[ Id+ {\ts {1\over|a|^2+\lambda r_0^2}} \left(\matrix{ \bar A& -ig\bar\gamma \cr -ig\gamma & A \cr}\right) \left(\matrix{ 0& ig\,\xi^* \cr ig\,\xi & 0\cr}\right)\right] \cr &\phantom{mmmmmmm}= \det\left[ Id+ \left(\matrix{ \lambda {\bar\gamma \over|a|^2+\lambda r_0^2}\,\xi & ig {\bar A \over|a|^2+\lambda r_0^2}\,\xi^* \cr ig {A \over|a|^2+\lambda r_0^2}\,\xi & \lambda {\gamma \over|a|^2+\lambda r_0^2}\,\xi^* \cr}\right)\right] & (48)\cr}$$ Since $$\eqalignno{ \log\det[Id+B]&=Tr\log[Id+B] =\sum_{n=1}^\infty {(-1)^{n+1}\over n} Tr B^n \cr &=Tr\,B-{\ts{1\over2}}Tr\, B^2+{\ts{1\over3}} Tr\, B^3-+\cdots &(49) \cr}$$ one obtains to second order in $\xi$: $$\eqalignno{ \log&\det\left[ Id+ \left(\matrix{ \lambda {\bar\gamma \over|a|^2+\lambda r_0^2}\,\xi & ig {\bar A \over|a|^2+\lambda r_0^2}\,\xi^* \cr ig {A \over|a|^2+\lambda r_0^2}\,\xi & \lambda {\gamma \over|a|^2+\lambda r_0^2}\,\xi^* \cr} \right)\right] \cr &=Tr\left(\matrix{ \lambda {\bar\gamma \over|a|^2+\lambda r_0^2}\,\xi & ig {\bar A \over|a|^2+\lambda r_0^2}\,\xi^* \cr ig {A \over|a|^2+\lambda r_0^2}\,\xi & \lambda {\gamma \over|a|^2+\lambda r_0^2}\,\xi^* \cr}\right)\>-\> {\ts{1\over2}}Tr\biggl\{ \left(\matrix{ \lambda {\bar\gamma \over|a|^2+\lambda r_0^2}\,\xi & ig {\bar A \over|a|^2+\lambda r_0^2}\,\xi^* \cr ig {A \over|a|^2+\lambda r_0^2}\,\xi & \lambda {\gamma \over|a|^2+\lambda r_0^2}\,\xi^* \cr}\right)^2 \biggr\} +O(\xi^3) \cr &=\ts Tr {\lambda\bar\gamma \over|a|^2+\lambda r_0^2}\,\xi +Tr {\lambda\gamma \over|a|^2+\lambda r_0^2}\,\xi^* -{\ts{1\over2}}\biggl\{ Tr {\lambda\bar\gamma \over|a|^2+\lambda r_0^2}\,\xi {\lambda\bar\gamma \over|a|^2+\lambda r_0^2}\,\xi +Tr {ig\bar A \over|a|^2+\lambda r_0^2}\,\xi^* {ig A \over|a|^2+\lambda r_0^2}\,\xi \cr &\phantom{=}\ts + Tr {\lambda\gamma \over|a|^2+\lambda r_0^2}\,\xi^* {\lambda\gamma \over|a|^2+\lambda r_0^2}\,\xi^* +Tr { ig A \over|a|^2+\lambda r_0^2}\,\xi {ig \bar A \over|a|^2+\lambda r_0^2}\,\xi^* \biggr\}+O(\xi^3)&(50) \cr}$$ One has $$\ts \left({\lambda\bar\gamma \over|a|^2 +\lambda r_0^2}\,\xi\right)_{k,p} ={\lambda \bar\gamma \over|a_k|^2 +\lambda r_0^2}\,\xi_{k,p}\>, \;\;\;\; \left({\lambda\gamma \over|a|^2+\lambda r_0^2}\,\xi^*\right)_{k,p} ={\lambda\gamma \over|a_k|^2+\lambda r_0^2}\,\bar\xi_{p,k}$$ $$\eqalignno{ \ts \left({\lambda\bar\gamma \over|a|^2 +\lambda r_0^2}\,\xi{\lambda\bar\gamma \over|a|^2 +\lambda r_0^2}\,\xi\right)_{k,k}&=\sum_p \ts {\lambda \bar\gamma \over|a_k|^2+\lambda r_0^2}\,\xi_{k,p} {\lambda \bar\gamma \over|a_p|^2+\lambda r_0^2}\,\xi_{p,k} \cr &= {\ts \left({\lambda \bar\gamma \,\xi_{k,k} \over|a_k|^2+\lambda r_0^2}\right)^2 } +\sum_{p\atop p\ne k} \ts {\lambda \bar\gamma \over|a_k|^2+\lambda r_0^2} {\lambda \bar\gamma \over|a_p|^2+\lambda r_0^2}\,\xi_{k,p} \,\xi_{p,k} \cr &= {\ts {1\over\kappa} \left({\lambda r_0 (\rho_0-\sqrt\kappa r_0) \over|a_k|^2+\lambda r_0^2}\right)^2 } +{\ts {1\over\kappa}}\sum_{q\ne 0} \ts {\lambda \bar\gamma \over|a_k|^2+\lambda r_0^2} {\lambda \bar\gamma \over|a_{k-q}|^2+\lambda r_0^2}\,\phi_q \,\phi_{-q} \cr}$$ $$\eqalignno{ \ts \left({\lambda\gamma \over|a|^2 +\lambda r_0^2}\,\xi^*{\lambda\gamma \over|a|^2 +\lambda r_0^2}\,\xi^*\right)_{k,k}&=\sum_p \ts {\lambda \gamma \over|a_k|^2+\lambda r_0^2}\,\bar\xi_{p,k} {\lambda \gamma \over|a_p|^2+\lambda r_0^2}\,\bar\xi_{k,p} \cr &= {\ts \left({\lambda \gamma \,\bar\xi_{k,k} \over|a_k|^2+\lambda r_0^2}\right)^2 } +\sum_{p\atop p\ne k} \ts {\lambda \gamma \over|a_k|^2+\lambda r_0^2} {\lambda \gamma \over|a_p|^2+\lambda r_0^2}\,\bar\xi_{k,p} \,\bar\xi_{p,k} \cr &= {\ts {1\over\kappa} \left({\lambda r_0 (\rho_0-\sqrt\kappa r_0) \over|a_k|^2+\lambda r_0^2}\right)^2 } +{\ts {1\over\kappa}}\sum_{q\ne 0} \ts {\lambda \gamma \over|a_k|^2+\lambda r_0^2} {\lambda \gamma \over|a_{k-q}|^2+\lambda r_0^2}\,\bar\phi_q \,\bar\phi_{-q} \cr}$$ and $$\ts \left({ ig A \over|a|^2+\lambda r_0^2}\,\xi\right)_{k,p}= { ig a_k \over|a_k|^2+\lambda r_0^2}\,\xi_{k,p}\>,\;\;\;\; \ts \left({ ig \bar A \over|a|^2+\lambda r_0^2}\,\xi^*\right)_{k,p}= { ig \bar a_k \over|a_k|^2+\lambda r_0^2}\,\bar\xi_{p,k}$$ $$\eqalignno{\ts \left({ ig A \over|a|^2+\lambda r_0^2}\,\xi { ig \bar A \over|a|^2+\lambda r_0^2}\,\xi^* \right)_{k,k}&= \sum_p \ts { ig a_k \over|a_k|^2+\lambda r_0^2}\,\xi_{k,p} { ig \bar a_p \over|a_p|^2+\lambda r_0^2}\,\bar\xi_{k,p}\cr &= -\lambda{\ts { |a_k|^2 \over(|a_k|^2+\lambda r_0^2)^2} }|\xi_{k,k}|^2 -\lambda \sum_{p\atop p\ne k} \ts { a_k \over|a_k|^2+\lambda r_0^2}\,\xi_{k,p} { \bar a_p \over|a_p|^2+\lambda r_0^2}\,\bar\xi_{k,p}\cr &={\ts -{\lambda\over\kappa} { |a_k|^2 \over(|a_k|^2+\lambda r_0^2)^2} }(\rho_0-\sqrt\kappa r_0)^2 -{\ts {\lambda\over\kappa}} \sum_{q\ne 0} \ts { a_k \over|a_k|^2+\lambda r_0^2}\, { \bar a_{k-q} \over|a_{k-q}|^2+\lambda r_0^2}\, \rho_q^2\cr}$$ $$\eqalignno{\ts \left({ ig \bar A \over|a|^2+\lambda r_0^2}\,\xi^* { ig A \over|a|^2+\lambda r_0^2}\,\xi \right)_{k,k}&= \sum_p \ts { ig \bar a_k \over|a_k|^2+\lambda r_0^2}\,\bar\xi_{p,k} { ig a_p \over|a_p|^2+\lambda r_0^2}\,\xi_{p,k}\cr &= -\lambda{\ts { |a_k|^2 \over(|a_k|^2+\lambda r_0^2)^2} }|\xi_{k,k}|^2 -\lambda \sum_{p\atop p\ne k} \ts { \bar a_k \over|a_k|^2+\lambda r_0^2}\,\bar\xi_{p,k} { a_p \over|a_p|^2+\lambda r_0^2}\,\xi_{p,k}\cr &={\ts -{\lambda\over\kappa} { |a_k|^2 \over(|a_k|^2+\lambda r_0^2)^2} }(\rho_0-\sqrt\kappa r_0)^2 -{\ts {\lambda\over\kappa}} \sum_{q\ne 0} \ts { \bar a_k \over|a_k|^2+\lambda r_0^2}\, { a_{k+q} \over|a_{k+q}|^2+\lambda r_0^2}\, \rho_q^2\cr}$$ Therefore (50) becomes $$\eqalignno{ \log&\det\left[ Id+ \left(\matrix{ \lambda {\bar\gamma \over|a|^2+\lambda r_0^2}\,\xi & ig {\bar A \over|a|^2+\lambda r_0^2}\,\xi^* \cr ig {A \over|a|^2+\lambda r_0^2}\,\xi & \lambda {\gamma \over|a|^2+\lambda r_0^2}\,\xi^* \cr}\right)\right] \cr &=\ts Tr {\lambda\bar\gamma \over|a|^2+\lambda r_0^2}\,\xi +Tr {\lambda\gamma \over|a|^2+\lambda r_0^2}\,\xi^* -{\ts{1\over2}}\biggl\{ Tr {\lambda\bar\gamma \over|a|^2+\lambda r_0^2}\,\xi {\lambda\bar\gamma \over|a|^2+\lambda r_0^2}\,\xi +Tr {ig\bar A \over|a|^2+\lambda r_0^2}\,\xi^* {ig A \over|a|^2+\lambda r_0^2}\,\xi \cr &\phantom{=}\ts + Tr {\lambda\gamma \over|a|^2+\lambda r_0^2}\,\xi^* {\lambda\gamma \over|a|^2+\lambda r_0^2}\,\xi^* +Tr { ig A \over|a|^2+\lambda r_0^2}\,\xi {ig \bar A \over|a|^2+\lambda r_0^2}\,\xi^* \biggr\}+O(\xi^3) \cr &=2{\ts{\lambda\over\kappa}}\sum_k{\ts {\sqrt\kappa\,r_0(\rho_0- \sqrt\kappa\,r_0)\over |a_k|^2+\lambda r_0^2}} \cr &\phantom{=} -{\ts {1\over2}} \biggl\{ {\ts {1\over\kappa}} \sum_k {\ts \left({\lambda r_0 (\rho_0-\sqrt\kappa r_0) \over|a_k|^2+\lambda r_0^2}\right)^2 } +\sum_{q\ne 0}{\ts {1\over\kappa}}\sum_k \ts {\lambda \bar\gamma \over|a_k|^2+\lambda r_0^2} {\lambda \bar\gamma \over|a_{k-q}|^2+\lambda r_0^2}\,\phi_q \,\phi_{-q} \cr &\phantom{=}+ {\ts {1\over\kappa}} \sum_k {\ts \left({\lambda r_0 (\rho_0-\sqrt\kappa r_0) \over|a_k|^2+\lambda r_0^2}\right)^2 } +\sum_{q\ne 0}{\ts {1\over\kappa}}\sum_k \ts {\lambda \gamma \over|a_k|^2+\lambda r_0^2} {\lambda \gamma \over|a_{k-q}|^2+\lambda r_0^2}\,\bar\phi_q \,\bar\phi_{-q} &(51) \cr &\phantom{=} {\ts -{\lambda\over\kappa}}\sum_k{\ts { |a_k|^2 \over(|a_k|^2+\lambda r_0^2)^2} }(\rho_0-\sqrt\kappa r_0)^2 - \sum_{q\ne 0}{\ts {\lambda\over\kappa}}\sum_k \ts { \bar a_k \over|a_k|^2+\lambda r_0^2}\, { a_{k+q} \over|a_{k+q}|^2+\lambda r_0^2}\, \rho_q^2 \cr &\phantom{=} {\ts -{\lambda\over\kappa}}\sum_k {\ts { |a_k|^2 \over(|a_k|^2+\lambda r_0^2)^2} }(\rho_0-\sqrt\kappa r_0)^2 - \sum_{q\ne 0}{\ts {\lambda\over\kappa}}\sum_k \ts { a_k \over|a_k|^2+\lambda r_0^2}\, { \bar a_{k-q} \over|a_{k-q}|^2+\lambda r_0^2}\, \rho_q^2 \biggr\} +O(\xi^3) \cr}$$ Using the BCS equation (5), ${\lambda\over\kappa}\sum_k{1\over |a_k|^2+ \lambda r_0^2}=1$ and abbreviating $$E_k^2:=|a_k|^2+\lambda r_0^2=k_0^2+e_\k^2+\lambda r_0^2$$ this becomes $$\eqalignno{2\sqrt\kappa & r_0(\rho_0-\sqrt\kappa r_0)\cdot 1 -(\rho_0-\sqrt\kappa r_0)^2 {\ts{\lambda\over\kappa}} \sum_k {\ts{\lambda r_0^2-|a_k|^2\over E_k^4}}+ \sum_{q\ne 0} \rho_q^2 \biggl\{ {\ts{\lambda\over\kappa}} \sum_k \ts { \bar a_k a_{k-q} \over E_k^2 E_{k-q}^2} \biggr\} \cr &\phantom{=}-\sum_{q\ne 0}{\rm Re} \left( e^{-2i\theta_0}\phi_q\phi_{-q}\right) \biggl\{ {\ts{\lambda\over\kappa}}\sum_k \ts {\lambda r_0^2 \over E_k^2 E_{k-q}^2} \biggr\} \cr &=2\sqrt\kappa r_0(\rho_0-\sqrt\kappa r_0)\cdot 1 +(\rho_0-\sqrt\kappa r_0)^2-(\rho_0-\sqrt\kappa r_0)^2 2 {\ts{\lambda\over\kappa}} \sum_k {\ts{\lambda r_0^2 \over E_k^4}} \cr &\phantom{=} + \sum_{q\ne 0} \rho_q^2 \biggl\{ {\ts{\lambda\over\kappa}} \sum_k {\ts { \bar a_k a_{k-q} \over E_k^2 E_{k-q}^2} \biggr\} } -\sum_{q\ne 0}{\rm Re}\left( e^{-2i\theta_0}\phi_q\phi_{-q}\right) \biggl\{ {\ts{\lambda\over\kappa}}\sum_k \ts {\lambda r_0^2 \over E_k^2 E_{k-q}^2} \biggr\} \cr &=\rho_0^2 -\kappa r_0^2-(\rho_0-\sqrt\kappa r_0)^2 2 {\ts{\lambda\over\kappa}} \sum_k {\ts{\lambda r_0^2 \over E_k^4}} \cr &\phantom{=} + \sum_{q\ne 0} \rho_q^2 \biggl\{ {\ts{\lambda\over\kappa}} \sum_k {\ts { \bar a_k a_{k-q} \over E_k^2 E_{k-q}^2} \biggr\} } -\sum_{q\ne 0}{\rm Re}\left( e^{-2i\theta_0}\phi_q\phi_{-q}\right) \biggl\{ {\ts{\lambda\over\kappa}}\sum_k \ts {\lambda r_0^2 \over E_k^2 E_{k-q}^2} \biggr\} &(52) \cr}$$ Therefore one obtains, recalling that $\xi_{k,p}={1\over\sqrt\kappa}\,\phi_{k-p}-\gamma\,\delta_{k,p}$, $$ \eqalignno{ V(\{\phi_q\})&- V(\{\sqrt\kappa \, \delta_{q,0}\, r_0 e^{i\theta_0} \}) =\sum_q \rho_q^2 -\kappa\, r_0^2\>-\> \log\det\left[ Id+ \left(\matrix{ \lambda {\bar\gamma \over|a|^2+\lambda r_0^2}\,\xi & ig {\bar A \over|a|^2+\lambda r_0^2}\,\xi^* \cr ig {A \over|a|^2+\lambda r_0^2}\,\xi & \lambda {\gamma \over|a|^2+\lambda r_0^2}\,\xi^* \cr} \right)\right] \cr &=\sum_{q\ne 0} \rho_q^2 +(\rho_0-\sqrt\kappa r_0)^2 2 {\ts{\lambda\over\kappa}} \sum_k {\ts{\lambda r_0^2 \over E_k^4}} - \sum_{q\ne 0} \rho_q^2 \biggl\{ {\ts{\lambda\over\kappa}} \sum_k {\ts { \bar a_k a_{k-q}\over E_k^2 E_{k-q}^2} \biggr\} }\cr &\phantom{=} +\sum_{q\ne 0}{\rm Re}\left( e^{-2i\theta_0}\phi_q\phi_{-q}\right) \biggl\{ {\ts{\lambda\over\kappa}}\sum_k \ts {\lambda r_0^2 \over E_k^2 E_{k-q}^2} \biggr\} &(53) \cr}$$ % % % Coeffizient von rho^2 % % % Consider the coefficient of $\sum_{q\ne 0}\rho_q^2$. It is given by $$\eqalignno{ 1&-{\ts{\lambda\over\kappa}} \sum_k {\ts { \bar a_k a_{k-q} \over E_k^2 E_{k-q}^2}}= {\ts {1\over2}} (1+1)-{\ts {1\over2}}{\ts{\lambda\over\kappa}} \sum_k {\ts { 2\bar a_k a_{k-q} \over E_k^2 E_{k-q}^2}} \cr &= {\ts {1\over2}}\biggl({\ts{\lambda\over\kappa}} \sum_k{\ts {|a_k|^2+\lambda r_0^2\over E_k^2 E_{k-q}^2}}+ {\ts{\lambda\over\kappa}} \sum_k{\ts {|a_{k-q}|^2+\lambda r_0^2\over E_k^2 E_{k-q}^2}}\biggr) -{\ts {1\over2}}{\ts{\lambda\over\kappa}} \sum_k {\ts { 2\bar a_k a_{k-q} \over E_k^2 E_{k-q}^2}} \cr &= {\ts {1\over2}}{\ts{\lambda\over\kappa}}\sum_k {\ts {a_k\bar a_k-a_k\bar a_{k-q}-\bar a_k a_{k-q}+a_{k-q} \bar a_{k-q}\over E_k^2 E_{k-q}^2}}+ {\ts{\lambda\over\kappa}} \sum_k{\ts { \lambda r_0^2\over E_k^2 E_{k-q}^2}} -{\ts {1\over2}{\lambda\over\kappa} } \sum_k\ts { \bar a_k a_{k-q} -a_k\bar a_{k-q} \over E_k^2 E_{k-q}^2} \cr &= {\ts {1\over2}}{\ts{\lambda\over\kappa}}\sum_k {\ts {(a_k-a_{k-q})(\bar a_k-\bar a_{k-q})\over E_k^2 E_{k-q}^2}}+ {\ts{\lambda\over\kappa}} \sum_k{\ts { \lambda r_0^2\over E_k^2 E_{k-q}^2}}-i{\ts {\lambda\over\kappa} } \sum_k\ts { {\rm Im}( \bar a_k a_{k-q}) \over E_k^2 E_{k-q}^2} \cr &= {\ts {1\over2}}{\ts{\lambda\over\kappa}}\sum_k {\ts { q_0^2+(e_\k-e_{\k-\q})^2 \over E_k^2 E_{k-q}^2}}+ {\ts{\lambda\over\kappa}} \sum_k{\ts { \lambda r_0^2\over E_k^2 E_{k-q}^2}} -i{\ts {\lambda\over\kappa} } \sum_k\ts { k_0e_{\k-\q}-(k_0-q_0)e_\k \over E_k^2 E_{k-q}^2} \cr &= \alpha_q+i\gamma_q+\beta_q &(54) \cr}$$ % % % Inserting (54) in (53), one gets $$ \eqalignno{ V(\{\phi_q\})&- V(\{\sqrt\kappa \, \delta_{q,0}\, r_0 e^{i\theta_0} \})= (\rho_0-\sqrt\kappa r_0)^2 2 \beta_0 + \sum_{q\ne 0} \rho_q^2 \biggl\{1- {\ts{\lambda\over\kappa}} \sum_k {\ts { 2\bar a_k a_{k-q}\over E_k^2 E_{k-q}^2} \biggr\} }\cr &\phantom{=} +\sum_{q\ne 0}{\rm Re}\left( e^{-2i\theta_0}\phi_q\phi_{-q}\right) \biggl\{ {\ts{\lambda\over\kappa}}\sum_k \ts {\lambda r_0^2 \over E_k^2 E_{k-q}^2} \biggr\} \cr &= (\rho_0-\sqrt\kappa r_0)^2 2 \beta_0 + \sum_{q\ne 0} \rho_q^2 ( \alpha_q+i\gamma_q) \cr &\phantom{=}+\sum_{q\ne 0} \rho_q^2 \beta_q +\sum_{q\ne 0}{\ts { e^{-2i\theta_0}\phi_q\phi_{-q}+ e^{2i\theta_0}\bar\phi_q\bar\phi_{-q}\over2}} \beta_q &(55) \cr}$$ Since $\beta_q=\beta_{-q}$, the last two q-sums in (55) may be combined to give $$\eqalignno{ \sum_{q\ne 0}& {\ts {\rho_q^2+\rho_{-q}^2\over 2}} \beta_q +\sum_{q\ne 0}{\ts { e^{-2i\theta_0}\phi_q\phi_{-q}+ e^{2i\theta_0}\bar\phi_q\bar\phi_{-q}\over2}} \beta_q \cr &={\ts{1\over2}}\sum_{q\ne 0}( \phi_q\bar\phi_q +\phi_{-q}\bar\phi_{-q}+e^{-2i\theta_0}\phi_q\phi_{-q}+ e^{2i\theta_0}\bar\phi_q\bar\phi_{-q}) \beta_q \cr &={\ts{1\over2}}\sum_{q\ne 0}(e^{-i\theta_0} \phi_q +e^{i\theta_0} \bar\phi_{-q}) (e^{i\theta_0} \bar\phi_q+e^{-i\theta_0}\phi_{-q}) \beta_q\cr &={\ts{1\over2}}\sum_{q\ne 0}|e^{-i\theta_0} \phi_q +e^{i\theta_0} \bar\phi_{-q}|^2 \beta_q &(56) \cr}$$ This proves Theorem 2 $\blacksquare$ \bigskip \bigskip \bigskip \noindent{\gross References} \bigskip \item{[1]} J. Feldman, J. Magnen, V. Rivasseau, E. Trubowitz, {\it Fermionic Many Body Models}, in: CRM Proceedings and Lecture Notes Vol. 7, {\it Mathematical Quantum Theory I: Field Theory and Many Body Theory}, eds. J. Feldman, R. Froese, L. M. Rosen, 1994. \item{[2]} J. Feldman, J. Magnen, V. Rivasseau, E. Trubowitz, {\it Ward Identities and a Perturbative Analysis of a U(1) Goldstone Boson in a Many Fermion System}, Helvetia Physica Acta 66, 1993, 498-550. \item{[3]} T. Chen, J. Fr\"ohlich, M. Seifert, {\it Renormalization Group Methods: Landau-Fermi Liquid and BCS Superconductor}, Proceedings of the Les Houches session {\it Fluctuating Geometries in Statistical Mechanics and Field Theory}, eds. F. David, P. Ginsparg, J. Zinn-Justin, 1994. \item{[4]} J. Feldman, E. Trubowitz, {\it Perturbation Theory for Many Fermion Systems}, Helv. Phys. Acta 63, 1990, 156-260; {\it The Flow of an Electron Phonon System to the Superconducting State}, Helv. Phys. Acta 64, 1991, 214-357. \item{[5]} D. Lehmann, {\it The Many-Electron System in the Forward, Exchange and BCS Approximation}, Communications in Mathematical Physics, 1998, to appear. \end \noindent{\bf Corollary:}{\it Let $V_2$ be the quadratic approximation to the effective potential given in Theorem 2 above, $$\eqalignno{ V_2(\{\phi_q\})=V_{\rm min}&+2\beta^2_0 \, (\rho_0-\sqrt{\kappa}\,r_0)^2 + \sum_{q\ne 0} \alpha_q^2\, \rho_q^2 + {\ts{1\over2}}\sum_{q\ne 0} \beta_q^2 \,|e^{-i\theta_0} \phi_q +e^{i\theta_0} \bar\phi_{-q}|^2 &(47) \cr}$$ \item{\bf a)} The partition function becomes $$ Z=\int e^{-V_2(\{\phi_q\})} \pro_q{\ts {d\phi_qd\bar\phi_q\over \pi} } =e^{-V_{\rm min}} \, \int_0^\infty e^{-2\beta_0^2(\rho_0-\sqrt\kappa r_0)^2 } 2\rho_0d\rho_0 \pro_{q\atop q_0>0} \ts {1\over \alpha_q^2(\alpha_q^2 +2\beta_q^2)} \eqno (48)$$ and one has $\lim_{q_0\to\infty}\alpha_q^2=1$, $\lim_{q_0\to\infty}q_0^2\beta_q^2=\lambda r_0^2$ such that the product over $q$ is finite if the cuttoff $|q_0|\le \nu$ is removed. \item{\bf b)} For all $q\ne 0$ $$\eqalignno{\Lambda(q)&=\int {\ts {|\phi_q|^2-1\over\lambda}} e^{-V_2(\{\phi_q\})} \pro_q{\ts {d\phi_qd\bar\phi_q\over \pi} }\Bigr/ \int e^{-V_2(\{\phi_q\})} \pro_q{\ts {d\phi_qd\bar\phi_q\over \pi} } \cr &=\ts {1\over\lambda} \left( {\alpha_q^2+\beta_q^2\over \alpha_q^2(\alpha_q^2 +2\beta_q^2)}-1 \right) &(49) \cr} $$ \item{\bf c)} Let $e_\k=\k^2/2m-\mu$. Then there is a constant $K=K(m,\mu)$ such that for all $q$ with $|\q|\le 1$ $$\ts \alpha_q^2\le K{q^2\over \Delta^2}\;\;\;{\rm and}\;\;\; \Lambda(q)\ge \ts {\Delta^2\over 2\lambda K q^2}-{1\over\lambda}\>. \eqno (50) $$ In particular, $$\lim_{L\to\infty} \vep_{\rm int}(\beta,L)={\ts{1\over\beta}}\sum_{q_0\in {2\pi\over \beta}\Bbb Z} \int_{|\q|\le 1}\ts {d^d\q\over (2\pi)^d}\>\Lambda(q)\>=\>\infty\;\;\;\; {\rm if}\; d=1\;{\rm or}\;d=2. \eqno (51)$$ } \bigskip \noindent{\bf Proof:} {\bf a)} We have to show that $$\eqalignno{ \pro_{q\atop q_0>0} \ts {1\over \alpha_q^2(\alpha_q^2 +2\beta_q^2)}&= \int e^{ -\sum_{q\ne 0} \alpha_q^2\, \rho_q^2 - {\ts{1\over2}}\sum_{q\ne 0} \beta_q^2 \,|e^{-i\theta_0} \phi_q +e^{i\theta_0} \bar\phi_{-q}|^2} \pro_{q\ne 0} {\ts {d\phi_qd\bar\phi_q\over \pi} }{\ts {d\theta_0\over 2\pi}} \cr &=\int e^{ -\sum_{q\ne 0} \alpha_q^2\, \rho_q^2 - {\ts{1\over2}}\sum_{q\ne 0} \beta_q^2 \,|e^{i(\theta_q-\theta_0)} \rho_q +e^{-i(\theta_{-q}-\theta_0)} \rho_{-q}|^2} \pro_{q\ne 0} {\ts {\rho_q\,d\rho_qd\theta_q\over \pi} }{\ts {d\theta_0\over 2\pi}} \cr &=\int e^{ -\sum_{q\ne 0} \alpha_q^2\, \rho_q^2 - {\ts{1\over2}}\sum_{q\ne 0} \beta_q^2 \,|e^{i\theta_q} \rho_q +e^{-i\theta_{-q}} \rho_{-q}|^2} \pro_{q\ne 0} {\ts {\rho_q\,d\rho_qd\theta_q\over \pi} } &(52) \cr} $$ or, since $\alpha_q=\alpha_{-q}$, $\beta_q=\beta_{-q}$ and $| \phi_q+ \bar\phi_{-q}|^2=| \phi_{-q} +\bar\phi_{q}|^2$ $$\int e^{ -\alpha_q^2 (\rho_q^2 +\rho_{-q}^2) - \beta_q^2 \,| \phi_q +\bar\phi_{-q}|^2} {\ts {d\phi_qd\bar\phi_q\over \pi}{d\phi_{-q}d\bar\phi_{-q}\over \pi} }= \ts {1\over \alpha_q^2(\alpha_q^2 +2\beta_q^2)} \eqno (53)$$ which is the same as $${\ts {1\over \pi^2}} \int_{\Bbb R^2}d^2x\int_{\Bbb R^2}d^2y \> e^{-\alpha_q^2(x^2+y^2)-\beta_q(x+y)^2} = \ts {1\over \alpha_q^2(\alpha_q^2 +2\beta_q^2)} \eqno (54)$$ Since $${\ts {1\over\pi}} \int_{\Bbb R^2} d^2y\> e^{-\alpha_q^2 y^2-\beta_q^2 (y+x)^2}=\ts {1\over \alpha_q^2+\beta_q^2} \,e^{-{\alpha_q^2\beta_q^2 \over \alpha_q^2+\beta_q^2} x^2} \eqno (55)$$ the formula follows. The $q_0$-limits may be seen as follows: We had (33), if $E_\k^2=e_\k^2+\lambda r_0^2$, $$\alpha_q^2={\ts {\lambda\over \beta L^d}}\sum_k{\ts {q_0^2+(e_\k-e_{\k-\q})^2 \over (k_0^2+E_\k^2)[(k_0-q_0)^2+E_{\k-\q}^2]}}\buildrel q_0\to\infty \over \to {\ts {\lambda\over \beta L^d}}\sum_k\ts {1 \over k_0^2+E_\k^2}=1$$ where the last equality is just the BCS equation (5). The $\beta_q$-limit is obtained in the same way. \par \noindent{\bf b)} Since only the $\phi_q$ and $\phi_{-q}$ modes are coupled and because the $\theta_0$ variable can be transformed away as in (52) by substituting $\theta_q'=\theta_q+\theta_0$, one has $$\eqalignno{ \Lambda(q)& ={ \int {|\phi_q|^2-1\over\lambda} e^{ -\alpha_q^2 (|\phi_q|^2 +|\phi_{-q}|^2) - \beta_q^2 \,| \phi_q +\bar\phi_{-q}|^2} {\ts {d\phi_qd\bar\phi_q}{d\phi_{-q}d\bar\phi_{-q}} } \over \int e^{ -\alpha_q^2 (|\phi_q|^2 +|\phi_{-q}|^2) - \beta_q^2 \,| \phi_q +\bar\phi_{-q}|^2} {\ts {d\phi_qd\bar\phi_q }{d\phi_{-q}d\bar\phi_{-q}} } } \cr &={ \int_{\Bbb R^4} {x^2-1\over\lambda} e^{ -\alpha_q^2 (x^2 +y^2) - \beta_q^2 ( x +y)^2} d^2xd^2y \over \int_{\Bbb R^4} e^{ -\alpha_q^2 (x^2 +y^2) - \beta_q^2 ( x +y)^2} d^2xd^2y } &(56) \cr}$$ The denominator equals ${\pi^2\over \alpha_q^2(\alpha_q^2 +2\beta_q^2)}$ and the numerator gives ${\pi^2\over \alpha_q^2(\alpha_q^2 +2\beta_q^2)}{1\over\lambda}\left( {\alpha_q^2+\beta_q^2\over \alpha_q^4+2\alpha^2_q\beta_q^2}-1\right)$ which proves (49). \par \noindent{\bf c)} We had (33) $$\alpha_q^2={\ts {\lambda\over \beta L^d}}\sum_k{\ts {q_0^2+(e_\k-e_{\k-\q})^2 \over (k_0^2+E_\k^2)[(k_0-q_0)^2+E_{\k-\q}^2]}}$$ If one substitues the Riemannian sum over $k_0$ by an integral, a residue computation gives (for $\beta=\infty$) $$\alpha_q^2={\ts {\lambda\over L^d}}\sum_\k{\ts {q_0^2+(e_\k-e_{\k-\q})^2 \over E_\k(q_0^2+[E_\k+E_{\k-\q}]^2) }}\le {\ts {\lambda\over L^d}}\sum_\k\ts {q_0^2+(\k\q-\q^2/2)^2\over E_\k^3} \le K{q_0^2+\q^2\over \Delta^2}$$ and $\Lambda(q)\ge {1\over\lambda} \left( {{1\over 2} \alpha_q^2 +\beta_q^2\over \alpha_q^2(\alpha_q^2 +2\beta_q^2)}-1 \right)={1\over 2\lambda \alpha_q^2}-{1\over\lambda}$ $ \blacksquare$ \end ---------------9810140415368--