Content-Type: multipart/mixed; boundary="-------------0302110110313" This is a multi-part message in MIME format. ---------------0302110110313 Content-Type: text/plain; name="03-46.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-46.keywords" Maxwell equations, heat kernel expansion, Casimir effect ---------------0302110110313 Content-Type: application/x-tex; name="bgh.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bgh.tex" \documentclass[12pt]{article} \usepackage{amsthm,amsmath,amsfonts,amssymb,cases} %\usepackage{amsmath,amsfonts,amsthm} \setlength{\textwidth}{16cm} \setlength{\textheight}{23cm} \setlength{\topmargin}{-1.5cm} \addtolength{\evensidemargin}{-1.5cm} \addtolength{\oddsidemargin}{-1.5cm} %\numberwithin{equation}{section} \newcommand{\R}{{\mathord{\mathbb R}}} \newcommand{\Z}{{\mathord{\mathbb Z}}} \newcommand{\N}{{\mathord{\mathbb N}}} \newcommand{\C}{{\mathord{\mathbb C}}} \newcommand{\I}{{\mathord{\mathbb I}}} \newcommand{\HH}{\mathcal{H}} %----------------------------------------------------------------------- \newcommand{\ud}{\mathrm{d}} \newcommand{\hl}{{L^2(\Omega,\mathbb{R}^3)}} \newcommand{\hs}{{L^2(\Omega)}} \newcommand{\ran}{ \textrm{Im }} \newcommand{\Ran}{ \textrm{Ran }} \newcommand{\krn}{ \textrm{Ker }} \newcommand{\grad}{ \textrm{grad\,}} \newcommand{\rot}{ \textrm{rot\,}} \newcommand{\dive}{\textrm{div\,}} \newcommand{\hc}{\mathcal{H}} \newcommand{\E}{\textbf{E}} \newcommand{\B}{\textbf{B}} \newcommand{\V}{\textbf{V}} \newcommand{\U}{\textbf{U}} \newcommand{\n}[1]{\textbf{#1}} \newcommand{\p}{\partial} %\newcommand{\Tr}{\textrm{Tr}} \newcommand{\uc}{\overset{\wedge}{=}} %----------------------------------------------------------------------- \def\tr{\operatorname{tr}} \def\Tr{\operatorname{Tr}} \def\const{{\rm const}\,} \def\dist{{\rm dist}\,} \def\supp{\operatorname{supp}} \def\e{{\rm e}} \def\i{{\rm i}} %\def\d{{\rm d}} \def\dn{|\hskip -1.2pt|\hskip -1.2pt|} \def\Im{\operatorname{Im}} \def\Re{\operatorname{Re}} \def\ad#1#2#3{{\operatorname{ad}}^{(#1)}_{#2}({#3})} \def\O#1{{\mathrm{O}(#1)}} \def\o1{{\mathrm{o}(1)}} %\def\slim{\mathop{\mathrm{s-}}\!\lim} %\def\Lim{\mathop{\mathrm{Lim}}} \setcounter{secnumdepth}{1} %\newtheorem{cor}[thm]{Corollary} %\newtheorem{prop}[thm]{Proposition} %\newtheorem{lemm}{Lemma}[section] %\newtheorem{theo}[lemm]{Theorem} %\newtheorem{defi}{Definition}[section] %\newtheorem{exa}[thm]{Example} %\newtheorem{rem}[thm]{Remark} \title{The heat kernel expansion for the electromagnetic field in a cavity} \author{\hspace{-.2 cm} F. Bernasconi${}^{(a)}$, G.M. Graf${}^{(b)}$, D. Hasler${}^{(c)}$\\ \normalsize\it \hspace{-.5 cm}\\ \hspace{-.5 cm}\normalsize\it ${}^{(a)}$ Department of Mathematics, ETH-Zentrum, 8092 Z\"urich, Switzerland\\ \normalsize\it \hspace{-.5 cm}${}^{(b)}$ Theoretische Physik, ETH-H\"onggerberg, 8093 Z\"urich, Switzerland\\ \normalsize\it \hspace{-.5 cm}${}^{(c)}$ Department of Mathematics, University of Copenhagen, 2100 Copenhagen, Denmark} \begin{document} \maketitle \vspace{0.4cm} \begin{abstract} We derive the first six coefficients of the heat kernel expansion for the electromagnetic field in a cavity by relating it to the expansion for the Laplace operator acting on forms. As an application we verify that the electromagnetic Casimir energy is finite. \end{abstract} % \section{Introduction}\label{INTRO} The modes of an electromagnetic field in a cavity, taken together with their unphysical, longitudinal counterparts, can be mapped onto the eigenstates of the Laplacian acting on the de Rham complex of a 3-manifold with boundary. The electric and magnetic fields are thereby associated to forms of degree $p=1$ and $p=2$ respectively. In this correspondence transverse modes are associated with coexact, resp. exact forms, which permits to further map longitudinal modes to forms of degree $p=0$ and $p=3$. We will use this observation, which is explained in detail in Sect.~\ref{proofs} below, to compute the first six coefficients of the heat kernel expansion for the electromagnetic field in a cavity. The result is used to show in a simple way that the Casimir energy in an arbitrary cavity with smooth boundaries is finite, a conclusion which has been reached previously \cite{BD2}. In an appendix the derivation of the numerical coefficients of the expansion is presented.\\ We shall present a Hilbert space formulation of the classical Maxwell equations in a cavity $\Omega \subset \mathbb{R}^3$. In a preliminary Hilbert space $L^2(\Omega,\mathbb{R}^3)$ we define the dense subspaces % \begin{eqnarray*} \mathcal{R}&=& \left \{ \V \in L^2(\Omega,\mathbb{R}^3)\mid \rot \V \in L^2(\Omega,\mathbb{R}^3) \right \}\;,\\ \mathcal{R}_{0} &=& \left \{\V \in \mathcal{R}\mid \langle \U,\rot \V \rangle = \langle \rot \U, \V \rangle,\, \forall \U \in \mathcal{R}\right \} \end{eqnarray*} % and the (closed) operator % \[ R=\rot \qquad \textrm{with domain}\quad \mathcal{D}(R)=\mathcal{R}_{0}\;. \] % Its adjoint is then given as $R^{*}= \rot$ with $\mathcal{D}(R^{*})=\mathcal{R}$. We remark that $R$, resp. $R^*$, is also the closure of $\rot$ defined on smooth vector fields $\V$ with boundary condition $\V_\parallel =0$ on the smooth boundary $\p \Omega$, resp. without boundary conditions. This is what is meant when we later simply say that a differential operator is defined with (or without) a certain boundary condition.\\ The subspace % \begin{equation}\label{uno} \hc = \left \{\V \in L^2(\Omega,\mathbb{R}^3)\mid\dive \V = 0 \right \} \end{equation} % and its orthogonal complement in $L^2(\Omega,\mathbb{R}^3)$ are preserved by $R$ and, therefore, by $R^*$. We will thus view them as operators on the physical Hilbert space $\hc$. The Maxwell equations with boundary condition $\E_\parallel=0$ on the ideally conducting shell $\p \Omega$ can now be written as % \begin{equation}\label{maxwell} \i \frac{\p}{\p t} \left (\begin{array}{l}\E\\\B \end{array}\right)= M\left (\begin{array}{l}\E\\\B \end{array}\right) \end{equation} % with % \[ M =\left (\begin{array}{cc} 0 & \i R^*\\-\i R & 0\end{array}\right ) = M^*\qquad \textrm{ on } \hc \oplus \hc\;, \] % cf. \cite{L}. Since no boundary condition has been imposed on $\B$, we have $M(0,\B)=0$ for all $\B=\nabla\psi$ with $\psi$ harmonic, and hence % \begin{equation}\label{due} \textrm{dim } \krn M = \infty \;. \end{equation} We shall compute the heat kernel trace % \[ \Tr'_{\hc \oplus \hc}(\e^{-tM^2}) = \sideset{}{'}\sum_{k}\e^{-t\omega_k^2} \;, \] % where $'$ means that the contributions of zero-modes, i.e., of eigenvalues $\omega_k = 0$ of $M$, have been omitted. This is necessary in view of (\ref{due}), but a more physical justification, tied to the application to the Casimir effect to be discussed later, is that zero-modes are not subject to quantization.\\ The square of $M$ is % \begin{equation} M^2 = \left (\begin{array}{cc}R^*R & 0\\0 & RR^*\end{array}\right) = \left(\begin{array}{cc}-\Delta_{\E} & 0\\0 & -\Delta_{\B} \end{array}\right) \;, \label{m2} \end{equation} % where $\Delta_{\E}$, resp. $\Delta_{\B}$, is the Laplacian on $\hc$ with boundary conditions % \begin{equation} \label{dueprimo} \E_\parallel = 0\;,\qquad \textrm{resp.}\quad (\rot\B)_\parallel = 0\;. \end{equation} % The operators $RR^*$ and $R^*R$ have the same spectrum, including multiplicity, except for zero-modes. Incidentally, we note that eigenfunctions $(\E,\B)$ corresponding to $\omega_k \ne 0$ satisfy $\B = -\i\omega_k^{-1} \rot \E$ and hence, by Stokes' theorem, the boundary condition $\B_\perp = 0$, which we did not impose, but which is usually also associated with ideally conducting shells. Since $\p_t^2+M^2=(\i\p_t-M)(-\i\p_t-M)$, each pair of non-zero eigenvalues of $R^*R$ and $RR^*$ corresponds to a single oscillator mode for (\ref{maxwell}). We will thus discuss the heat kernel asymptotics for % \begin{numcases}{ \frac{1}{2} \Tr'_{\hc \oplus \hc}(\e^{-tM^2})=} \Tr'_{\hc}\e^{t\Delta_{\E}} \label{eq:trea}\\ \Tr'_{\hc}\e^{t\Delta_{\B}} \label{eq:treb} \end{numcases} \begin{equation} \hskip 5cm \cong\sum_{n=0}^\infty a_n t^{\frac{n-3}{2}} \;, \qquad ( t \downarrow 0)\;. \label{eq:asexp} \end{equation} % The coefficients $a_{n}$ are known, see e.g. \cite{BGKV}, for general operators of Laplace type. The direct application of such results is prevented by the divergence constraint in $\hc$, see~(\ref{uno}). In the next section we indicate how to remove it. First however we present the main result.\\ %\section{Statement of the result} % Let % \begin{equation*} L_{ab}= (\nabla_{\n{e}_a}\n{e}_b,\n{n} )\;,\qquad (a,b=1,2)\;, \end{equation*} % be the second fundamental form on the boundary $\p\Omega$ with inward normal $\n{n}$ and local orthonormal frame $\{\n{e}_1,\n{e}_2,\n{n}\}$. We denote by $|\Omega|$ the volume of $\Omega$ and set % \[ f[\p\Omega] = \int_{\p\Omega} f(y)\ud y\;, \] % where $\ud y$ is the (induced) Euclidean surface element on $\p\Omega$. The corresponding Laplacian on $\p\Omega$ is denoted by $\nabla^2$. % \newtheorem{teo1}{Theorem} \begin{teo1}\label{teo1} Let $\Omega \subset \mathbb{R}^3$ be a compact, connected domain with smooth boundary $\p\Omega$ consisting of $n$ components of genera $g_1, g_2,\dots,g_n$. Then % \begin{eqnarray} a_0&=& 2(4\pi)^{-\frac{3}{2}}|\Omega|\;,\nonumber \\ a_1&=& 0 \;, \nonumber \\ a_2&=& -\frac{4}{3}(4\pi)^{-\frac{3}{2}}(\tr L)[\p\Omega]\;,\nonumber \\ a_3&=& \frac{1}{64}(4\pi)^{-1} \bigl(3(\tr L)^2-4\det L\bigr)[\p\Omega]- \frac{1}{2}\sum_{i=1}^n(1+g_i) + 1\;, \label{eq:a3}\\ a_4&=& \frac{16}{315}(4\pi)^{-\frac{3}{2}} \bigl(2(\tr L)^3- 9\tr L\cdot\det L \bigr)[\p\Omega]\;,\nonumber \\ a_5&=& \frac{1}{122880}(4\pi)^{-1} \bigl(2295(\tr L)^4 - 12440(\tr L)^2\det L + \nonumber \\ &&\qquad\qquad \qquad +13424(\det L)^2 + 1200\tr L\cdot\nabla^2\tr L\bigr)[\p\Omega]\;.\nonumber \end{eqnarray} % \end{teo1} % We will give two partially independent proofs, based on (\ref{eq:trea}), resp. (\ref{eq:treb}). Their agreement is related to the index theorem, as it may be seen from (\ref{m2}). A further, partial check of these coefficients has been made on the basis of general cylindrical domains and of the sphere, where a separation into TE and TM modes is possible. The coefficient $a_0$ was computed in \cite{W} (except for the factor $2$ replaced by $3$, as the divergence condition (\ref{uno}) was ignored), $a_1,\,a_2$ in \cite{BB}. The coefficient $a_3$ is closely related to a result of \cite{BD2}, as discussed in Sect.~\ref{casi}.\\ \section{Proofs}\label{proofs} We consider the space of (square integrable) forms, $\Lambda(\Omega)= \bigoplus_{p=0}^n \Lambda_p(\Omega)$, on the manifold $\Omega$ with boundary, together with the exterior derivative $d_{p+1}: \Lambda_p(\Omega)\rightarrow \Lambda_{p+1}(\Omega)$ defined with relative boundary condition (\cite{G}, Sect.~2.7.1) % \[ \omega \big\rvert_{\p\Omega} = 0\;, \] % as a form $\omega\rvert_{\p\Omega}\in \Lambda_p(\p\Omega)$. For later use we recall that by the de Rahm theorem for manifolds with boundary (\cite{D} or \cite{G}, Thm.~2.7.3) we have \begin{equation} \label{cinque} H_r^p(\Omega)\cong H_{n-p}(\Omega)\cong H_p (\Omega,\p\Omega)\;, \end{equation} where $H_r^p(\Omega)= \krn d_{p+1}/ \ran d_p$ is the $p$-th relative cohomology group, $H_p(\Omega)$ is the $p$-th homology group, and $H_p(\Omega,\p\Omega)$ is the $p$-th relative homology group, i.e., the homology based on chains mod $\p\Omega$. We shall henceforth restrict to $\Omega \subset \mathbb{R}^3$ as in Theorem~\ref{teo1}. Using either homology (\ref{cinque}), the dimension of $H_r^p(\Omega)$ is seen to be % \begin{equation}\label{sei}\begin{aligned} 0 &&\qquad\qquad&&(p=0)\;,\\ n-1 && &&(p=1)\;,\\ \sum_{i=1}^n g_i && &&(p=2)\;,\\ 1 && &&(p=3)\;. \end{aligned}\end{equation} % These are also the dimensions of the spaces of harmonic $p$-forms.\\ The space $\Lambda(\Omega)= \bigoplus_{p=0}^3 \Lambda_p(\Omega)$ may be identified as % \begin{equation*} \Lambda(\Omega) = L^2(\Omega)\oplus L^2(\Omega,\mathbb{R}^3)\oplus L^2(\Omega,\mathbb{R}^3)\oplus L^2(\Omega)\ni (\phi,\E,\B,\psi)\;, \end{equation*} % where $d : \Lambda(\Omega) \rightarrow \Lambda(\Omega)$ acts as % \begin{gather*} d :L^2(\Omega)\underset{\text{grad}}{\longrightarrow} L^2(\Omega,\mathbb{R}^3)\underset{\text{rot}}{\longrightarrow} L^2(\Omega,\mathbb{R}^3)\underset{\text{div}}{\longrightarrow} L^2(\Omega){\longrightarrow}0 \intertext{with boundary conditions $\phi = 0,\,\E_\parallel = 0,\, \B_\perp = 0$ on $\p\Omega$. Then} d^* :0 \longleftarrow L^2(\Omega)\underset{-\text{div}}{\longleftarrow} L^2(\Omega,\mathbb{R}^3)\underset{\text{rot}}{\longleftarrow} L^2(\Omega,\mathbb{R}^3)\underset{-\text{grad}}{\longleftarrow} L^2(\Omega)\; \end{gather*} % without any boundary conditions. The Laplace-Beltrami operator on forms, % \[ -\Delta = \bigoplus_{p=0}^3(-\Delta_p)= dd^* + d^*d\;, \] % is seen to correspond to the Euclidean Laplacian with boundary conditions \begin{equation}\label{sette}\begin{aligned} {}&{\phi} = 0 &&\qquad\qquad&&(p=0)\;,\\ {}&{\E}_\parallel = 0\;,\quad \dive{\E} = 0 && &&(p=1)\;,\\ {}&{\B}_\perp = 0\;,\quad (\rot {\B})_\parallel = 0 && &&(p=2)\;,\\ {}&(\grad \psi)_\perp = 0 && &&(p=3)\;. \end{aligned}\end{equation} % Each of the four problems admits a heat kernel expansion, % \begin{equation} \Tr_{\Lambda_p(\Omega)}\e^{\Delta_pt}\cong \sum_{n=0}^{\infty} a_n^{(p)}t^{\frac{n-3}{2}}\;, \label{eq:hke} \end{equation} % whose coefficients have been computed ($n = 0,\dots,3$) \cite{BBG} or can be computed using existing results ($n=4,5$) \cite{BGKV}. To this end we note that the boundary conditions for $p=1,2$ can be formulated equivalently as % \begin{equation}\begin{aligned} \E_\parallel= 0\;,&\quad \frac{\p \E_\perp}{\p n} - (\tr L)\E_\perp = 0 &\qquad\qquad&(p=1)\;,\\ \B_\perp = 0\;,&\quad \frac{\p \B_\parallel}{\p n} - L\B_\parallel = 0 &\qquad\qquad&(p=2)\;. \end{aligned} \label{bdry} \end{equation} %\subsection*{Approach 1} \noindent {\bf First approach.} We will compute (\ref{eq:trea}). We observe that $-\Delta_{\E}$ is just the restriction of $-\Delta_1$ to its invariant subspace % \[ \hc =\left\{\E \in L^2(\Omega,\mathbb{R}^3)\mid \dive \E = 0 \right \}= \krn d_1^*\;. \] % Hence % \[ \Tr'_{\hc}e^{t\Delta_{\E}} = \Tr'_{L^2(\Omega,\mathbb{R}^3)}\e^{t\Delta_1}- \Tr'_{\hc^\perp}e^{t\Delta_1}\;, \] % where the orthogonal complement of $\hc$ in $L^2(\Omega,\mathbb{R}^3)$ is % \[ \hc^\perp = \overline{\Ran d_1} = \Ran d_1 = \left\{ \nabla \phi \in L^2(\Omega,\mathbb{R}^3)\mid \phi = 0 \text{ on } \p\Omega\right\}\;, \] % ($\Ran d$ is closed by the Hodge decomposition, see e.g.~\cite{CFKS, G}). By $ d\Delta = \Delta d$, the operators $(-\Delta_1)\restriction_{\hc^\perp}$ and $-\Delta_0$ have the same spectrum (in fact $\nabla\phi = 0$ implies $\phi=0$ by the boundary condition). Thus, using also (\ref{sei}), we find % \begin{align*} \Tr'_\hc \e^{t\Delta_{\E}} &= \Tr'_\hl \e^{t\Delta_1} - \Tr'_\hs \e^{t\Delta_0}\\ &=\Tr_\hl \e^{t\Delta_1} - \Tr_\hs \e^{t\Delta_0} - (n-1)\;, \intertext{i.e.,} a_k &= a_k^{(1)} - a_k^{(0)} \;, \qquad (k \ne 3)\;,\\ a_3 &= a_3^{(1)} - a_3^{(0)} - n + 1 \;. \end{align*} % These relations, together with the values of $a_k^{(p)}$ computed in the Appendix, yield the values of the coefficients stated in the Theorem~\ref{teo1}. In particular, we will obtain % \[ a_3^{(1)} - a_3^{(0)} = \frac{1}{64}(4\pi)^{-1}\bigl( 3 (\tr L)^2 + 28\det L \bigr)[\p\Omega]\;. \] % This matches the stated value of $a_3$ because of % \[ n = \frac{1}{2}\sum_{i=1}^{n}(1 + g_i) + \frac{1}{2}\sum_{i=1}^{n}(1 - g_i) \] % and of the Gauss-Bonnet theorem, % \begin{equation} \label{otto} \frac{1}{2}\sum_{i=1}^{n}(1 - g_i) = \frac{1}{2} (4\pi)^{-1}(\det L)[\p\Omega]\;. \end{equation} % %\subsection*{Approach 2} \noindent {\bf Second approach.} We now compute (\ref{eq:treb}). As has been noted in the Introduction, eigenmodes of $-\Delta_{\B}$, except for zero-modes, satisfy the boundary condition$\ \B_\perp = 0$, and are thus eigenmodes of $-\Delta_2$ belonging to its invariant subspace $\hc$, cf.~(\ref{dueprimo}, \ref{sette}). The converse is obvious. We conclude that % \[ \Tr'_\hc \e^{t\Delta_{\B}} = \Tr'_\hl \e^{t\Delta_2} - \Tr'_{\hc^\perp} \e^{t\Delta_2}\;. \] % Since % \[ \hc = \{ \B \in \hl\mid \dive \B = 0 \} = \krn d_3 \;, \] % we have % \[ \hc^\perp = \overline{\Ran d_3^*} = \Ran d_3^* = \left\{ -\nabla \psi \in L^2(\Omega,\mathbb{R}^3)\mid \psi \in \hs\right\}\;. \] % Using $d^*\Delta = \Delta d^*$, we see that $(-\Delta_2)\restriction_{ \hc^\perp}$ and $-\Delta_3$ have the same spectrum, except for a single zero-mode (in fact, $-\nabla\psi = 0$ implies $\psi=\const$). We thus find, using (\ref{sei}), \begin{align*} \Tr'_\hc \e^{t\Delta_{\B}} &= \Tr'_\hl \e^{t\Delta_2} - \Tr'_\hs \e^{t\Delta_3}\\ &=\Tr_\hl \e^{t\Delta_2} - \Tr_\hs \e^{t\Delta_3} - \bigl(\sum_{i=1}^{n}g_i\ -1\bigr)\;, \intertext{i.e.,} a_k &= a_k^{(2)} - a_k^{(3)} \;, \qquad (k \ne 3)\;,\\ a_3 &= a_3^{(2)} - a_3^{(3)} -\sum_{i=1}^{n}g_i\ +1 \;. \end{align*} % {From} these relations and from the results of the Appendix we again recover Theorem~\ref{teo1}. In particular, % \[ a_3^{(2)} - a_3^{(3)} = \frac{1}{64}(4\pi)^{-1}\bigl( 3(\tr L)^2 - 36\det L \bigr)[\p\Omega] \] % leads to the claim for $a_3$, because of % \[ \sum_{i=1}^n g_i = \frac{1}{2}\sum_{i=1}^n(1+g_i)-\frac{1}{2}\sum_{i=1}^n(1-g_i) \] % and of (\ref{otto}). \section{Application to the Casimir effect} \label{casi} For the purpose of this discussion we simply define the Casimir energy by the mode summation method, see e.g. \cite{BD2}. In particular, we do not address the issue \cite{C} of whether it is the most appropriate physically. We shall however observe that the Casimir energy is finite -- a conclusion obtained in \cite{BD2}, but questioned in \cite{DC}. \\ Consider the cavity $\Omega \subset \mathbb{R}^3$ enclosed in a large ball $\Omega_0$. As usual we compare the vacuum energy of the electromagnetic field in the domains $\Omega \cup (\Omega_0 \setminus\overline\Omega)$ with that of the reference domain $\Omega_0$. Each eigenmode of either domain contributes a zero-point energy $\omega_k/2$, resp. $\omega_k^0/2$. As a regulator for the eigenfrequencies $\omega_k = \lambda_k^{1/2}$, we choose $\e^{-\gamma \lambda_k }$, ($\gamma > 0$). The corresponding definition of the Casimir energy is % \[ %\label{CaEn} E_C = \frac{1}{2} \lim_{\Omega_0 \rightarrow \infty}\ \lim_{\gamma \downarrow 0}\left ( \sum_k \lambda_k^{\frac{1}{2}}\e^{-\gamma \lambda_k} \ -\ \sum_k (\lambda_k^0)^{\frac{1}{2}}\e^{-\gamma \lambda_k^0} \right )\;. \] % We shall prove that the limit $\gamma \downarrow 0$ is finite. It will also be clear that the subsequent limit $\Omega_0 \rightarrow \infty$ exists, though we shall not make the effort to prove that (see however e.g. \cite{CFKS}, Section 12.7 for the necessary tools). Using $$\lambda_k^{\frac{1}{2}} = -\frac{1}{\sqrt{\pi}}\int_0^{\infty}\ud t\ t^{-\frac{1}{2}} \frac{d}{dt}\e^{-t\lambda_k}$$ and (\ref{eq:asexp}) we find for the regularized sum of the eigenfrequencies % \[ \sum_k \lambda_k^{\frac{1}{2}}\e^{-\gamma \lambda_k} \approx - \sum_{n=0}^4 \frac{n-3}{2 \sqrt{\pi}}a_n \int_0^{\delta}\ud t\ t^{-\frac{1}{2}}(t+\gamma)^{\frac{n-5}{2}} \] % as $\gamma \downarrow 0$. Here $\delta > 0$ is arbitrary, but fixed, and ``$\approx$'' means up to terms $O(1)$. Using % \begin{equation*} \int_0^{\delta}\ud t\ t^{-\frac{1}{2}}(t+\gamma)^{\frac{n-5}{2}} \approx \begin{cases} \frac{4}{3}\gamma ^{-2} \qquad \qquad &(n=0)\;,\\ \frac{\pi}{2}\gamma^{-\frac{3}{2}} \qquad \qquad &(n=1)\;,\\ 2\gamma^{-1}\qquad \qquad &(n=2)\;,\\ \pi\gamma^{-\frac{1}{2}}\qquad \qquad &(n=3)\;,\\ - \log \gamma \qquad \qquad &(n=4)\;,\\ \end{cases} \end{equation*} % we find % \begin{equation*} \sum_k \lambda_k^{\frac{1}{2}}\e^{-\gamma \lambda_k} \approx \frac{2}{\sqrt{\pi}} a_0\gamma^{-2}+ \frac{\sqrt{\pi}}{2} a_1 \gamma^{-\frac{3}{2}} + \frac{1}{\sqrt{\pi}}a_2 \gamma^{-1} + 0\cdot a_3\gamma^{-\frac{1}{2}}+ \frac{1}{2\sqrt{\pi}}a_4\log\gamma\;. \end{equation*} % Hence a finite Casimir energy requires (cf. \cite{CVZ}) that $a_0,a_1,a_2,a_4$ (but not necessarily $a_3$!) agree for $\Omega \cup (\Omega_0 \setminus\overline \Omega)$ and for the reference domain $\Omega_0$. This is indeed so for $a_0 = 2 (4\pi)^{-\frac{3}{2}}|\Omega_0|$ and for $a_1 = 0$, but also for $a_2,\,a_4$ as the contribution from the two sides of $\p\Omega$ cancel. The same conclusion is obtained if the regulator $\e^{-\gamma \lambda_k}$ is replaced by $\e^{-(\gamma \lambda_k)^{1/2}}$ (see \cite{CVZ}, Eq.~(27)): \begin{equation*} \sum_k \lambda_k^{\frac{1}{2}}\e^{-(\gamma \lambda_k)^{1/2}}\approx \frac{24}{\sqrt{\pi}} a_0\gamma^{-2}+4 a_1 \gamma^{-\frac{3}{2}} + \frac{2}{\sqrt{\pi}}a_2 \gamma^{-1} + 0\cdot a_3\gamma^{-\frac{1}{2}}+ \frac{1}{\sqrt{\pi}}a_4\log\gamma\;. \end{equation*} Since no renormalization is necessary, the value of $E_C$ agrees with that obtained by means of the zeta function.\\ In the rest of this section we compare our results with those of \cite{BD, BD2}. To the extent the comparison is done we will find agreement. An important tool there is the mode generating function, Eq. (4.5) in \cite{BD}, % \begin{equation}\label{nove} \begin{split} \Phi(k) &\doteq \frac{1}{2} \Tr\left(\frac{-\Delta_{\E}}{-\Delta_{\E}-k^2} + \frac{-\Delta_{\B}}{-\Delta_{\B}-k^2}\right)\\ &\doteq \frac{k^2}{2}\Tr'\Bigl( (-\Delta_{\E}-k^2)^{-1} + (-\Delta_{\B}-k^2)^{-1}\Bigr)\;,\qquad (k \in \mathbb{C}\setminus \mathbb{R})\;, \end{split} \end{equation} % where ``$\doteq$'' means equality ``within addition of some polynomial in $k^2$''. Since the resolvents in (\ref{nove}) are not trace class, but their squares are, we first consider that replacement. Using $(A+\mu)^{-2} = \int_0^{\infty} \ud t \ t\ \e^{-t(A+\mu)}$ we obtain, as $\mu \rightarrow \infty$, % \begin{equation*} \frac{1}{2}\Tr'\Bigl( (-\Delta_{\E}+\mu)^{-2} + (-\Delta_{\B}+\mu)^{-2}\Bigr) \cong\sum_{n=0}^{\infty}a_n \int_{0}^{\infty}\ud t \cdot t^{\frac{n-3}{2}}\e^{-t\mu} =\sum_{n=0}^{\infty} \Gamma(\mbox{$\frac{n+1}{2}$}) a_n\mu^{-\frac{n+1}{2}} \end{equation*} % with coefficients $a_n$ given in Theorem \ref{teo1}. Integrating w.r.t. $\mu$ we find % \begin{equation*} \frac{1}{2}\Tr'\Bigl( (-\Delta_{\E}+\mu)^{-1} + (-\Delta_{\B}+\mu)^{-1}\Bigr)\doteq \sum_{\substack{n=0\\n\ne1}}^\infty \Gamma(\mbox{$\frac{n-1}{2}$})a_n\mu^{-\frac{n-1}{2}}\ -a_1 \log\mu \end{equation*} % and hence, with $\mu^{1/2}= -\i k$, % \begin{equation*} \Phi(k) \doteq 2 \sqrt{\pi}a_0\i k^3-\sqrt{\pi}a_1k^2\ln(-k^2) + \i\sqrt{\pi}a_2k - a_3+O(k^{-1})\;. \end{equation*} % Upon insertion of the mentioned values for $a_0,\dots,a_3$ this agrees with Eq.~(4.40) in \cite{BD}, except for $a_3$ which is there replaced by its local part, see (\ref{eq:a3}), % \begin{equation*} \tilde{a}_3= \frac{1}{64}(4\pi)^{-1}\bigl(3(\tr L)^2-4\det L \bigr)[\p\Omega] = \frac{1}{64}\int_{\p\Omega}\ud \sigma \Bigl(\frac{3}{4}(\kappa_1^2+\kappa_2^2)-\kappa_1\kappa_2 \Bigr)\;, \end{equation*} % where $\kappa_1,\kappa_2$ are the principal curvatures. Note however that this discrepancy is implicit in the definition of ``$\doteq$''. It is resolved in \cite{BD2} by first considering $\delta\Phi (k)$, i.e., the difference of the mode generating functions corresponding to the configurations $\Omega \cup (\Omega_0 \setminus\overline \Omega)$ and $\Omega_0$. Thus % \begin{equation*} \delta\Phi (k) = -2\tilde{a}_3 + O(k^{-1}) \;, \end{equation*} % since the contributions to $a_0,\,a_2$ cancel, and those to $\tilde{a}_3$ double the value. Not ambiguous then is ``the number of additional modes of finite frequency created by introducing the conducting surface $\p \Omega$'': % \begin{equation*} \mathcal{C} = \psi(0+)-\psi(\infty) \;, \end{equation*} % where $\psi(y) = \delta\Phi(\i y)$. For a connected boundary $\p\Omega$ of genus $g$ the value of $\psi(0+)$ has been established as $\psi(0+)= -g$ (see \cite{BD2}, Eq.~(5.8)), resulting in % \begin{equation}\label{dieci} \mathcal{C} = 2\tilde{a}_3 - g\;. \end{equation} % This result agrees with Theorem \ref{teo1}: the non-local terms in (\ref{eq:a3}) take the values $-\frac{1}{2}(g-1),\, -\frac{1}{2}g,\, \frac{1}{2}$ for $\Omega,\, \Omega_0 \setminus\overline\Omega$ and $\Omega_0$ respectively. Thus, % \[ \delta a_3 = 2\tilde{a}_3 - g\;, \] % in agreement with (\ref{dieci}). %\newpage %\numberwithin{equation}{section} \appendix \section{Appendix} In this appendix we compute the heat kernel coefficients in (\ref{eq:hke}) for $p=0,\dots,3$ and $n=0,\dots,5$ on the basis of Theorems 1 and 4 in \cite{BGKV}. We use the same notation, together with $P = \n{n} \otimes \n{n}$ denoting the normal projection at the boundary. The vector bundle is $V = \Omega \times \mathbb{R}$ for $p = 0,3$, resp. $V = T\Omega$ for $p=1,2$, equipped with the Euclidean connection. The decompositions of $V |_{\p\Omega} = V_N \oplus V_D \ni (\phi^N, \phi^D)$ (with projections $\Pi_+$, resp. $\Pi_-$) and boundary conditions $\phi_{;n}^N + S\phi^N = 0$, resp. $\phi^D = 0$, are specified as follows, cf. (\ref{bdry}) and \cite{BGKV}: % \begin{equation}\label{auno} \begin{aligned}p&=0: &&\qquad\begin{cases}\Pi_+ = 0\;,\\ \Pi_- = 1\;, \end{cases} \\ p&=1: &&\qquad\begin{cases}\Pi_+ = P\;,\qquad S=-L_{aa}P\;,\\ \Pi_- = 1-P\;,\end{cases}\\ p&=2: &&\qquad\begin{cases}\Pi_+ = 1- P\;,\qquad S=-L\;,\\ \Pi_- = P\;,\end{cases}\\ p&=3: &&\qquad\begin{cases}\Pi_+ = 1\;,\qquad S=0\;,\\ \Pi_- = 0\;.\end{cases} \end{aligned} \end{equation} % The result is % \begin{align*}a_0^{(p)} &= (4\pi)^{-\frac{3}{2}} c_0^{(p)}|\Omega|\;,\\ a_1^{(p)} &= \frac{1}{4}(4\pi)^{-1} c_1^{(p)}|\p\Omega|\;,\\ a_2^{(p)} &= \frac{1}{3}(4\pi)^{-\frac{3}{2}} c_2^{(p)}(\tr L)[\p\Omega]\;,\\ a_3^{(p)} &= \frac{1}{384}(4\pi)^{-1}\bigl( c_{31}^{(p)} (\tr L)^2+ c_{32}^{(p)}(\det L)\bigr)[\p\Omega]\;,\\ a_4^{(p)} &= \frac{1}{315}(4\pi)^{-\frac{3}{2}} \bigl(c_{41}^{(p)}(\tr L)^3+c_{42}^{(p)}\tr L\cdot\det L\bigr)[\p\Omega]\;,\\ a_5^{(p)} &= \frac{1}{245760}(4\pi)^{-1} \bigl( c_{51}^{(p)} (\tr L)^4+c_{52}^{(p)}(\tr L )^2\det L + c_{53}^{(p)}(\det L)^2 + c_{54}^{(p)}\tr L \cdot \nabla ^2 \tr L \bigr)[\p\Omega] \; \end{align*} % with coefficients given by \medskip \begin{tabular}{c@{\bigg |}cccc} & $\quad p=0\quad$ & $\quad p=1\quad$ & $\quad p=2\quad$ & $\quad p=3\ $ \\\hline $ c_0^{(p)}$ & $ 1$ & $3$ & $3$ & $1$ \\ $ c_1^{(p)}$ & $-1$ & $-1$ & $1$ & $1$ \\ $ c_2^{(p)}$ & $1$ & $-3$ & $-3$ & $1$ \\ $ c_{31}^{(p)}$ & $3$ & $21$ & $33$ & $15$ \\ $ c_{32}^{(p)}$ & $-20$ & $148$ & $-220$ & $-4$ \\ $ c_{41}^{(p)}$ & $4$ & $36$ & $60$ & $28$ \\ $ c_{42}^{(p)}$ & $-18$ & $-162$ & $-186$ & $-42$ \\ $ c_{51}^{(p)}$ & $555$ & $5145$ & $8625$ & $4035$ \\ $ c_{52}^{(p)}$ & $- 2840$ & $-27720$ & $-35720$ & $- 10840$ \\ $ c_{53}^{(p)}$ & $2224$ & $29072$ & $29712$ & $2864$ \\ $ c_{54}^{(p)}$ & $120$ & $2520$ & $4680$ & $2280$ \\ \end{tabular} \bigskip\noindent These values imply Theorem \ref{teo1}, as explained in its proof.\\ The computation of the table is based on the general result of \cite{BGKV}, which has been applied to (\ref{auno}) using the following identities: % \begin{align*} \Tr(P_{:a}P_{:b}) &= 2 (L^2)_{ab}\;,\\ \Tr(P_{:a}P_{:a}P_{:b}P_{:b}) &= (L^4)_{aa} + (L^2)_{aa}(L^2)_{bb}\;,\\ \Tr(P_{:a}P_{:b}P_{:a}P_{:b}) &= 2(L^4)_{aa}\;,\\ \Tr(P_{:aa}P_{:bb}) &= 2 L_{ac:a}L_{bc:b} + 4 (L^4)_{aa} + 4 (L^2)_{aa}(L^2)_{bb}\;,\\ \Tr(P_{:ab}P_{:ab}) &= 2 L_{ab:c}L_{ab:c} + 6 (L^4)_{aa} + 2 (L^2)_{aa}(L^2)_{bb}\;. \end{align*} % They can be derived by using $\nabla_{\n{e}_a}\n{n}=-L_{ab}\n{e}_b$, so that \begin{equation*} P_{:a} = -L_{ac} (\n{e}_c \otimes \n{n} + \n{n} \otimes \n{e}_c)\;, \end{equation*} and by assuming without loss that $\nabla_{\n{e}_a}\n{e}_b$ has no component parallel to $T_p\p\Omega$ at the point $p$ of evaluation, i.e., $\nabla_{\n{e}_a}\n{e}_b = L_{ab}\n{n}$. Then % \begin{equation*} P_{:ab} = -L_{ac:b} (\n{e}_c \otimes \n{n} + \n{n} \otimes \n{e}_c) - 2 (L^2)_{ab}P+(L_{ac}L_{bd} + L_{ad}L_{bc})\n{e}_c\otimes \n{e}_d \;, \end{equation*} % from which the above traces follow. In turn they allow the computation of similar traces with $P$ replaced by $\chi = \Pi_+ - \Pi_-$, i.e., by $\chi = \pm (2P - 1)$ in the cases $p=1,2$. In these two cases we also have % \begin{align*} \Tr S_{:a} &= -L_{bb:a} \;,\\ \Tr S_{:ab} &= -L_{cc:ab} \;, \end{align*} % and, moreover, for $p=1$, % \begin{align*} \Tr(S_{:a}S_{:a}) &= L_{bb:a} L_{cc:a} + 2 L_{bb}L_{cc}(L^2)_{aa}\;,\\ \Tr(P_{:a}S_{:b}) &= -2 (L^2)_{ab}L_{cc}\;,\\ \Tr(P S_{:a}S_{:a}) &= L_{bb:a} L_{cc:a} + L_{bb}L_{cc}(L^2)_{aa}\;, \end{align*} % resp. for $p=2$, % \begin{align*}\Tr(S_{:a}S_{:a}) &= L_{ab:c} L_{ab:c} + 2(L^4)_{aa}\;,\\ \Tr(P_{:a}S_{:a}) &= 2 (L^3)_{aa}\;,\\ \Tr(P S_{:a}S_{:a}) &= (L^4)_{aa}\;. \end{align*} Furthermore, traces of $L^k$, $(k \geq 2)$, were reduced to $\tr L,\,\det L$ by means of $L^2 - (\tr L) L + \det L = 0$. Finally, we used the Codazzi equation, $L_{ab:c} = L_{ac:b}$, as well as % \[ L_{ab:ca}-L_{ab:ac}= L_{aa}(L^2)_{bc} - (L^2)_{aa}L_{bc} \;, \] % which follows from the Gauss equation. \bigskip\noindent {\bf Acknowledgement.} We thank M. Levitin and G. Scharf for discussions. The research of D. Hasler was supported in part under the EU-network contract HPRN-CT-2002-00277. \begin{thebibliography}{99} \bibitem{BB} R. Balian, C. Bloch, Distribution of eigenfrequencies for the wave equation in a finite domain. II. Electromagnetic field. Riemannian spaces, Ann. Phys. (N.Y.) {\bf 64}, 271 (1971); Errata, ibid. {\bf 84}, 559 (1974). \bibitem{BD} R. Balian, B. Duplantier, Electromagnetic waves near perfect conductors, I. Multiple scattering expansions. Distribution of modes, Ann. Phys. (N.Y.) {\bf 104}, 300 (1977). \bibitem{BD2} R. Balian, B. Duplantier, Electromagnetic waves near perfect conductors, II. Casimir effect, Ann. Phys. (N.Y.) {\bf 112}, 165 (1978). \bibitem{BBG} N. Bla\v zi\'c, N. Bokan, P.B. Gilkey, Spectral geometry of the form valued Laplacian for manifolds with boundary, Indian J. Pure Appl. Math. {\bf 23}, 103 (1992). \bibitem{BGKV} T.P. Branson, P.B. Gilkey, K. Kirsten, D.V. Vassilevich, Heat kernel asymptotics with mixed boundary conditions, Nucl. Phys. B{\bf 563}, 603 (1999). \bibitem{C} P. Candelas, Vacuum energy in the presence of dielectric and conducting surfaces, Ann. Phys. (N.Y.) {\bf 143}, 241 (1982). \bibitem{CVZ} G. Cognola, L. Vanzo, S. Zerbini, Regularization dependence of vacuum energy in arbitrarily shaped cavities, J. Math. Phys. {\bf 33}, 222 (1992). \bibitem{CFKS} H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, {\it Schr\"odinger operators\/}, Springer (1987). \bibitem{D} G.F.G. Duff, Differential forms in manifolds with boundary, Ann. Math. {\bf 56}, 115 (1952). \bibitem{DC} D. Deutsch, P. Candelas, Boundary effects in quantum field theory, Phys. Rev. D{\bf 20}, 895 (1978). \bibitem{G} P.B. Gilkey, {\it Invariance theory, the heat equation, and the Atiyah-Singer index theorem\/}, CRC (1995). \bibitem{L} R. Leis, {\it Initial boundary value problems in mathematical physics\/}, Teubner/Wiley (1986). \bibitem{W} H. Weyl, \"Uber die Randwertaufgabe der Strahlungstheorie und asymptotische Spektralgesetzte, J. f. reine u. angew. Math. {\bf 143}, 177 (1913). \end{thebibliography} \end{document} ---------------0302110110313 Content-Type: application/x-tex; name="cases.sty" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="cases.sty" % C A S E S . S T Y ver 2. June 1994 % % Copyright (C) 1993,1994 by Donald Arseneau % These macros may be freely transmitted, reproduced, or modified % provided that this notice is left intact. Sub-equation numbering % is based on subeqn.sty by Stephen Gildea; most of the rest is based % on LaTeX's \eqnarray by Leslie Lamport and the LaTeX3 team. % % LaTeX environment {numcases} to produce multi-case equations with % a separate equation number for each case. 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