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\controldates{11-JAN-2005,11-JAN-2005,11-JAN-2005,11-JAN-2005}
 
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\begin{document}
\title[Decomposition of spectral invariants]
{On gluing formulas for the spectral invariants of Dirac type
operators}
\author{Paul Loya}
\address{Department of Mathematics, Binghamton University,
Binghamton, New York 13902}
\email{paul@math.binghamton.edu}
\author{Jinsung Park}
\address{Mathematisches Institut, Universit\"at Bonn,
Beringstra{\ss}e 1, D-53115 Bonn, Germany}
\email{jpark@math.uni-bonn.de}
\commby{Michael Taylor}
\date{October 6, 2004}
\subjclass[2000]{Primary 58J28, 58J52}

\begin{abstract}
In this note, we announce gluing and comparison formulas for the
spectral invariants of Dirac type operators on compact manifolds
and manifolds with cylindrical ends. We also explain the central
ideas in their proofs.
\end{abstract}

\maketitle

\section{The gluing problem for the spectral invariants}\label{sec-glue}

Since their inception, the eta invariant and the $\z$-determinant
of Dirac type operators have influenced mathematics and physics in
innumerable ways. Especially with the development of quantum field
theory, the behavior of these spectral invariants under gluing of
the underlying manifold has become an increasingly important
topic. However, the gluing formula for the $\z$-determinant of a
Dirac Laplacian has remained an open question due to the nonlocal
nature of this invariant. In fact, Bleecker and Booss-Bavnbek
stated that \cite[p.\ 89]{BB} ``no precise pasting formulas are
obtained but only adiabatic ones.'' In \cite{LP2, LP3}, we
give precise gluing formulas for $\z$-determinants of Dirac type
operators on compact manifolds and manifolds with cylindrical
ends, respectively, and moreover we present new and unified
derivations of the gluing formulas for \emph{both} invariants. The
purpose of this note is to announce these gluing formulas for the
spectral invariants and to indicate the main ideas in their
proofs. We also announce a relative invariant formula proved in
\cite{LP1, LP3}.

We begin with describing the gluing problem for compact manifolds.
Let $\Dd$ be a Dirac type operator acting on $C^{\infty}(M,S)$
where $M$ is a closed compact Riemannian manifold of arbitrary
dimension and $S$ is a Clifford bundle over $M$. Let $Y$ be an
embedded hypersurface in $M$ and let $M = M_- \cup M_+$ be
decomposition of $M$ into manifolds with boundary such that $
\partial M_- =
\partial M_+ = Y$. We assume that all geometric structures are of
product type over a tubular neighborhood $N = [-1,1] \times Y$ of
$Y$ where the Dirac operator takes the product form $\Dd =
G(\partial_u +D_Y)$, where $G$ is a unitary operator on
$S_0:=S|_Y$ and $D_Y$ is a Dirac type operator over $Y$ satisfying
$G^2 = -\Id$ and $D_Y G = -GD_Y$. Recall that the eta function of
$\Dd$ and the zeta function of $\Dd^2$ are defined through the
heat operator $e^{- t \Dd^2}$ via
\begin{equation}\label{etadef}
\eta_{\Dd}(s) = \frac{1}{\Gamma(\frac{s+1}{2})} \Big( \int_0^1 +
\int_1^\infty \Big) \, t^{\frac{s-1}{2}} \Tr(\Dd e^{-t \Dd^2}) \ dt,
\end{equation}
\pagebreak
\begin{equation}
\z_{\Dd^2}(s) = \frac{1}{\Gamma(s)} \Big( \int_0^1 + \int_1^\infty
\Big) t^{s-1} \Tr(e^{-t \Dd^2}) \, dt,
\label{detdef}
\end{equation}
where the first integrals in \eqref{etadef} and \eqref{detdef} are
defined a priori for $\RE s \gg 0$ and the second ones for $\RE s
\ll 0$, and both extend to meromorphic functions on
$\C$ that are regular at $s = 0$; actually, the second integral in
\eqref{etadef} is entire, but we present these general definitions
because they work later for $b$-spectral invariants on manifolds
with cylindrical ends (see Section \ref{sec-mwce}). The eta
invariant $\eta(\Dd)$ is by definition the value $\eta_{\Dd}(0)$,
the reduced eta invariant is $\tilde{\eta}(\Dd):= (\eta(\Dd) +
\dim \ker (\Dd))/2$, and the $\z$-determinant is by definition
\[
\zd \Dd^2 : = \exp\Big(-\frac{d}{ds}\Big|_{s=0}
\z_{\Dd^2}(s)\Big).
\]
The eta invariant was introduced in the paper
\cite{APS} by Atiyah, Patodi, and Singer as the boundary
correction term in their index formula for manifolds with boundary.
The $\z$-determinant was introduced by Ray and Singer in the
paper \cite{RS} on the analytic torsion. Since their
introductions, these invariants have impacted geometry, topology,
and physics in incredible ways; cf.\ Singer \cite{S1, S2}.

 By restriction, $\Dd$ induces Dirac type
operators $\Dd_+$ over $M_+$ and $\Dd_-$ over $M_-$. For these
operators, we choose orthogonal projections $\Pp_+$, $\Pp_-$ over
$L^2(Y,S_0)$ that provide us with \emph{well-posed boundary
conditions} for $\Dd_+,\Dd_-$ in the sense of Seeley \cite{Se69}.
Then the operators
\begin{equation} \label{DdPpm}
\Dd_{\Pp_{\pm}} : \dom(\Dd_{\Pp_{\pm}}) \to L^2(M_{\pm},S),
\end{equation}
where
\[
\dom(\Dd_{\Pp_{\pm}}):= \{\, \phi\in H^1(M_{\pm},S) \ | \
\Pp_{\pm}( \phi|_Y)=0 \, \},
\]
share many of the analytic properties of $\Dd$; in particular,
they are Fredholm and have discrete spectra, but are not
necessarily self-adjoint. Amongst such projectors are the
(orthogonalized) Calder\'on projectors $\Cc_\pm$ \cite{C}, which
are projectors defined intrinsically as the unique orthogonal
projectors onto the closures in $L^2(Y,S_0)$ of the
infinite-dimensional \emph{Cauchy data spaces} of $\Dd_\pm$:
\[
\{\, \phi|_Y\ |\ \phi \in C^\infty(M_\pm,S),\ \Dd_\pm \phi = 0\,
\} \subset C^\infty(Y,S_0).
\]
In order to have well-defined eta invariants and
$\z$-determinants, it is necessary to restrict to a subclass of
projectors. A natural class is formed by those in the \emph{smooth,
self-adjoint Grassmannian} $Gr^*_\infty(\Dd_\pm)$, which consists
of orthogonal projections $\Pp_\pm$ such that $\Pp_\pm - \Cc_\pm$ are 
smoothing operators and $G \Pp_\pm = (\Id -
\Pp_\pm) G$.
Examples of such projectors are the generalized APS spectral
projectors $\Pp_- = \Pi_< + \frac{1+\sigma_1}{2}\Pi_0$, $\Pp_+ =
\Pi_> + \frac{1+\sigma_2}{2}\Pi_0$, where $\Pi_<$, $\Pi_>$, $\Pi_0$
are the orthogonal projectors onto the negative, positive, and
zero eigenspaces  of $D_Y$, respectively, and the $\sigma_i$'s are
involutions on $\ker(D_Y)$ anticommuting with $G$. For $\Pp_\pm
\in Gr^*_\infty(\Dd_\pm)$, the invariants $\eta(\Dd_{\Pp_{\pm}})$
and $\zd \Dd_{\Pp_{\pm}}^2$ can be defined via the formulas
\eqref{etadef} and \eqref{detdef}; see Grubb
\cite{Gr01, Gr03}, Loya and Park \cite{LP1}, and
Wojciechowski \cite{WoK99}.



We now have all the ingredients to state the gluing problem. The
\emph{gluing problem} is to describe the ``defects''
\[
 \frac{\zd
\Dd^2}{\zd \Dd_{\Pp_+}^2 \cdot \zd \Dd_{\Pp_-}^2} = \ \boxed{?}\,
,\qquad \tilde{\eta}(\Dd) - \tilde{\eta} (\Dd_{\Pp_+}) -
\tilde{\eta}(\Dd_{\Pp_-}) = \ \boxed{?}
\]
in terms of recognizable data.  The gluing problem for the eta
invariant has been solved by several authors. Moreover, because
the variation of the eta invariant is local, a variety of formulas
and proofs have appeared (many modulo $\mathbb Z$); see, for
instance, Br\"uning and Lesch \cite{BL98}, Bunke \cite{Bu95}, Dai
and Freed \cite{Dfr94}, Hassell, Mazzeo, and Melrose
\cite{HMaM95}, Kirk and Lesch \cite{KL01}, Mazzeo and Melrose
\cite{MM95}, M\"uller \cite{Mu96}, Park and Wojciechowski
\cite{PWII}, Wojciechowski \cite{KPW95,WoK99}; cf.\ the superb
survey articles by Mazzeo and Piazza \cite{MP98} or Bleecker and
Booss-Bavnbek \cite{BB}. Because of the highly nonlocal nature
of the $\z$-determinant and its variation, according to Bleecker
and Booss-Bavnbek \cite[p.\ 90]{BB} the $\z$-determinants (of a
Dirac operator and its square) are ``the most subtle and the most
fascinating objects of our study.'' Nonetheless, the gluing
problem for the $\z$-determinant of Laplace type operators with
\emph{local} boundary conditions was solved by Burghelea,
Friedlander, and Kappeler \cite{BFK} and has been further extended
by Carron \cite{Car}, Hassell \cite{Ha98}, Hassell and Zelditch
\cite{HZ}, Lee \cite{Lee}, Loya and Park \cite{LP0}, Vishik
\cite{V}, and many others. However, the gluing formula for the
$\z$-determinant of a Dirac Laplacian has remained an open
question, partly due to the nonlocal nature of the
$\z$-determinant and its variation and the technical aspects
inherent in the \emph{global} pseudodifferential boundary
problems required for Dirac type operators. We now describe our
solution to the gluing problem.



To state our main theorem, we recall that the Calder\'on
projectors $\Cc_{\pm}$ have the matrix forms
\begin{equation} \label{Cc}
\Cc_{\pm} = \frac12 \begin{pmatrix} \Id & \kp_{\pm}^{-1} \\
                                    \kp_{\pm} & \Id \end{pmatrix}
\end{equation}
with respect to the decomposition $L^2(Y,S_0) = L^2(Y,S^+)\oplus
L^2(Y,S^-)$, where $S^{\pm} \subset S_0$ are the $(\pm
i)$-eigenspaces of $G$. The maps $\kp_{\pm}:L^2(Y,S^+)\to
L^2(Y,S^-)$ are isometries, so that $U:=-\kp_-\kp_+^{-1}$ is a
unitary operator over $L^2(Y,S^-)$, which is moreover of Fredholm
determinant class. We denote by $\widehat{U}$ the restriction of
$U$ to the orthogonal complement of its $(-1)$-eigenspace. We also
put
\begin{equation} \label{Ll}
\Ll:=  \sum_{k=1}^{h_M} \g_0 U_k \otimes \g_0 U_k
= \sum_{k=1}^{h_M}\, \langle \ \cdot \ , \gamma_0 U_k
\rangle_{{}_{\hspace{-.1em} L^2(Y,S_0)}} \gamma_0 U_k,
\end{equation}
where $h_M=\dim \ker(\Dd)$, $\g_0$ is the restriction map from $M$
to $Y$, and $\{ U_k \}$ is an orthonormal basis of $\ker(\Dd)$.
Then $\Ll$ is a positive operator on the finite-dimensional vector
space $\g_0(\ker (\Dd))$. The following theorem is the main result
of \cite{LP2}.



\begin{theorem}[\cite{LP2}] \label{t:glue} The following gluing formulas
hold:
\begin{align*}
& \frac{\zd \Dd^2}{\zd \Dd^2_{\Cc_+}\cdot \zd \Dd_{\Cc_-}^2}  =
2^{-\z_{D_Y^2}(0)-h_Y} (\det {\Ll})^{-2}\, {\det}_F \Big(
\frac{2\,\Id+\widehat{U}+\widehat{U}^{-1}}{4} \Big),\\
&\qquad \tilde{\eta}(\Dd) - \tilde{\eta}(\Dd_{\Cc_+}) -
\tilde{\eta}(\Dd_{\Cc_-}) = \frac{1}{2 \pi i} \, \Log \detF U
\ (\operatorname{mod} \Z),
\end{align*}
where $h_Y=\dim \ker(D_Y)$, $\det_F$ denotes the Fredholm
determinant, $\Log$ is the principal value of the logarithm, and the integer
defect in the eta formula is given in terms of the winding numbers
of the function $\detF (K(\la) K^c(\la)^{-1})$ appearing in
Section \ref{sec-pf}.
\end{theorem}

Let us remark that the $\z$-determinant formula in Theorem
\ref{t:glue} is a new result, and only its adiabatic limit form
\cite{PWI}, \cite{PWII} has been proved hitherto. The formulation
of the integer defect in the eta formula in Theorem \ref{t:glue}
is also new. The eta formula, without integer ambiguity, 
in terms of $\Tr \log U$ was proved
by Kirk and Lesch \cite[Th.\ 5.10]{KL01} using spectral flow, the
Scott-Wojciechowski comparison theorem \cite{ScKW99}, and the
rotating boundary condition technique from Br\"uning and Lesch
\cite{BL98}. Our proof of the eta formula is different from and
independent of their proof, and is obtained ``simultaneously'' as a
byproduct from the proof for the $\z$-determinant.



We can generalize the gluing formulas in Theorem \ref{t:glue} in
terms of other boundary conditions by using Theorem \ref{t:comp}
(see Section \ref{sec-comp}) proved in \cite{LP1}. Let $\Pp_1 \in
Gr^*_\infty(\Dd_{-})$ and $\Pp_2 \in Gr^*_\infty(\Dd_{+})$. Then
$\Pp_1$ and $\Pp_2$ determine maps $\kp_1$ and $\kp_2$ as in
\eqref{Cc}, and we can define $U_1:=\kp_-\kp_1^{-1}$,
$U_2:=\kp_2\kp_+^{-1}$ and $U_{12}:=-\kp_1\kp_2^{-1}$ over
$L^2(Y,S^-)$. As before, we let $\widehat{U}_i$ denote the
restriction of $U_i$ to the orthogonal complement of its
$(-1)$-eigenspace. We define the operator $\Ll_1$ over the
finite-dimensional vector space $\IM(\Cc_-)\cap \IM(\Id-\Pp_1)$ by
\begin{equation} \label{Ll-}
\Ll_1=-P_1\, G\, \Rr_{-}^{-1}\, G\, P_1,
\end{equation}
where $\Rr_{-}$ is the sum of the Dirichlet to Neumann maps on the
double of $M_-$, which was introduced by Burghelea, Friedlander,
and Kappeler in \cite{BFK}, and $P_1$ is the orthogonal projection
onto $\IM(\Cc_-)\cap \IM(\Id-\Pp_1)$. Then $\Ll_1$ is a positive
operator \cite{LP1}. We define $\Ll_2$ in a similar way. We can
now state the general gluing formulas for the spectral invariants.

\begin{theorem} \label{t:g-main thm}
The following general gluing formulas hold:
\begin{align*}
\frac{\zd \Dd^2}{ \zd \Dd^2_{\Pp_1} \cdot \zd \Dd^2_{\Pp_2}}& =
2^{-\z_{D_Y^2}(0)-h_Y} (\det {\Ll})^{-2}\, {\det}_F \Big(
\frac{2\Id+\widehat{U}+\widehat{U}^{-1}}{4} \Big)\\
&\quad \times \prod_{i=1}^2 (\det\Ll_{i})^{-2} \cdot
{\det}_F \Big(\frac{2\Id+\widehat{U}_i +\widehat{U}_i^{-1}}{4}
\Big)^{-1},\\
\tilde{\eta} (\Dd) - \tilde{\eta} (\Dd_{\Pp_1}) - \tilde{\eta} (&
\Dd_{\Pp_2} ) \ =\  \frac{1}{2 \pi i} \, \Log \detF U_{12} \
(\operatorname{mod} \Z),
\end{align*}
where the integer defect in the eta formula can be identified
exactly in terms of winding numbers of specific naturally defined
operators.
\end{theorem}


Let us remark that the adiabatic decomposition formulas presented
in \cite{PWI}, \cite{PWII}, which are proved mainly using the
Duhamel principle and scattering theory, can be derived from the
$\z$-determinant formula in Theorem \ref{t:g-main thm} (cf.\
\cite{LPasymp}).
\section{The gluing problem on manifolds with cylindrical ends}\label{sec-mwce}

We now describe the solution to the gluing problem on manifolds
with cylindrical ends presented in \cite{LP3}. Assume now that
instead of $M_+$ being compact, it is a noncompact half-infinite
cylinder: $M_+ = [0,\infty) \times Y$, over which $\Dd$ is of
product type. We still assume that $M_-$ is compact. Then $M$ is
called a manifold with cylindrical end. For a noncompact
manifold $M$ with cylindrical end, the heat operators $\Dd e^{-t
\Dd^2}$ and $e^{-t \Dd^2}$ are not of trace class. In particular,
the definitions \eqref{etadef} and \eqref{detdef} cannot be used
as in the compact case. There are two principal ways to make sense
of these invariants. One way is to define so-called \emph{relative
invariants} as in Bruneau \cite{Br}, Carron \cite{Car}, 
M\"uller \cite{Mu98}, and others, whereby we subtract off certain
operators that make the difference of the heat operators of trace
class, and the other way is to use Melrose's $b$-trace
\cite{BMeR93}, $\bTr$, which is a natural substitute for the
trace. In particular, $\Dd e^{-t \Dd^2}$ and $e^{-t \Dd^2}$ are
$b$-trace class. Moreover, $\bTr (\Dd e^{-t \Dd^2})$ and $\bTr
(e^{-t \Dd^2})$ have asymptotic expansions in half-integer powers
of $t$ as $t \to 0$ and $t \to \infty$. It follows that the
$b$-eta function $\be_{\Dd}(s)$ and the $\bz$-function
$\bz_{\Dd^2}(s)$ can be defined exactly as in formulas
\eqref{etadef} and \eqref{detdef}, respectively, where we replace
$\Tr$ with $\bTr$. Furthermore, $\be_{\Dd}(s)$ and
$\bz_{\Dd^2}(s)$ extend to define meromorphic functions on $\C$
that are regular at $s = 0$, so we can define the $b$-eta
invariant of $\Dd$ by $\be (\Dd): = \be_{\Dd}(0)$, and the
$b$-determinant of $\Dd^2$ by $\bzd \Dd^2 := \exp (-
\bz_{\Dd^2}'(0))$.

In order to discuss the gluing formula for the $b$-spectral
invariants into the decomposition $M = M_- \cup M_+$, we need to
impose a boundary condition on $M_+$. Note that $M_-$ is compact
so $\Dd_{\Cc_-}$ is defined as before. Hence, we need a
``Calder\'on projector'' on the noncompact manifold $M_+$, which
is found by looking at the Cauchy data space of $\Dd$ over the
whole manifold $M$:
\[
\{\, \phi|_Y\ |\ \phi \in C^\infty(M,S),\ \Dd \phi = 0\, \}
\subset C^\infty(Y,S_0).
\]
Atiyah, Patodi, and Singer \cite{APS} (cf.\ \cite{BMeR93,Mu94})
showed that an element $\psi$ of this space is the restriction
$\psi = \phi|_Y$ of an $L^2$ section $\phi$ over $M$ if and only
if $\big(\Pi_< + \frac{1 + \sigma}{2} \Pi_0\big) \psi = 0$, where
$\sigma$ is \emph{the} unitary map on $\ker (D_Y)$ such that
$\sigma^2 = \Id$ and $\sigma G = - G \sigma$ determined by the
\emph{scattering matrix}. For this reason, the natural projection on $M_+$ is
\begin{equation} \label{Caldcylin}
\Cc_+ : = \Pi_> + \frac{1 - \sigma}{2} \Pi_0.
\end{equation}
We can now define $\Dd_{\Cc_+}$ exactly as before (see
\eqref{DdPpm}) in the case when $M_+$ was compact. However, since
$M_+$ is a manifold with cylindrical end, we need to use the
$b$-trace to define the corresponding $b$-spectral invariants,
$\be(\Dd_{\Cc_+})$ and $\bzd \Dd_{\Cc_+}^2$.

We now have all the ingredients to state the gluing problem for
manifolds with cylindrical end. The ($b$-)\emph{gluing problem} is
to describe the ``defects''
\[
\frac{\bzd \Dd^2 }{\zd \Dd_{\Cc_-}^2 \cdot \bzd \Dd_{\Cc_+}^2}=
\boxed{?} \, , \quad \tbe (\Dd)- \tilde{\eta}(\Dd_{\Cc_-}) -
\tbe(\Dd_{\Cc_+}) = \boxed{?}
\]
in terms of recognizable data. Just as before, the projectors
$\Cc_{\pm}$ can be written as \eqref{Cc} for unitary operators
$\kp_{\pm}$, and we denote by $\widehat{U}$ the restriction of $U
:= - \kp_- \kp_+^{-1}$ to the orthogonal complement of its
$(-1)$-eigenspace. Also, we define $\Ll$ as in \eqref{Ll}, using
the orthonormal basis $\{ U_k \}$ of $\ker_{L^2} (\Dd)$. Then
$\Ll$ is a positive operator on the finite-dimensional vector
space $\g_0( \ker_{L^2} (\Dd))$. The following theorem is proved
in \cite{LP3}.
\begin{theorem}[\cite{LP3}] \label{t:main1} The following gluing 
formulas hold:
\begin{align*}
& \frac{\bzd \Dd^2 }{\zd \Dd_{\Cc_-}^2 \cdot \bzd \Dd_{\Cc_+}^2} =
2^{-\z_{D_Y^2}(0)-h_Y}\!(\det\Ll)^{-2} \, {\det}_F \Big(
\frac{2\,\Id+\widehat{U}+\widehat{U}^{-1}}{4} \Big),\\
&  \tbe (\Dd)- \tilde{\eta}(\Dd_{\Cc_-}) - \tbe(\Dd_{\Cc_+}) \ =\
 \tbe (\Dd)- \tilde{\eta}(\Dd_{\Cc_-}) \ = \
 \frac{1}{2 \pi i} \, \Log \detF U \ (\operatorname{mod} \Z),
\end{align*}
where $h_Y:= \dim \ker (D_Y)$ and $\det_F$ denotes the Fredholm
determinant.
\end{theorem}

One can show that $\bzd \Dd^2/\bzd\Dd^2_{\Cc_+} =
\zd(\Dd^2,\Dd^2_{\Cc_+})$, the relative determinant of the pair
$(\Dd^2,\Dd^2_{\Cc_+})$, and $\tbe (\Dd) =
\tilde{\eta}(\Dd,\Dd_{\Cc_+})$, the reduced relative eta invariant
of the pair $(\Dd,\Dd_{\Cc_+})$, so this theorem can be written
with relative spectral invariants on the left-hand side. Also,
some computations show that  $\bzd \Dd_{\Cc_+}^2 = 2^{\frac12
\z_{D_Y^2}(0)}$, so after substituting this expression into our
theorem, we can get another version of the $\z$-determinant
formula without the $\bzd \Dd_{\Cc_+}^2$ term.
\section{The comparison, or relative invariant, problem}\label{sec-comp}

The \emph{comparison problem} for the spectral invariants on $M_-$
can be stated as follows: For $\Pp_- \in Gr_\infty^*(\Dd_-)$, find
formulas for the ``defects''
\[
\frac{\zd \Dd_{\Pp_-}^2}{\zd \Dd_{\Cc_-}^2} =  \boxed{?}\,
,\qquad \tilde{\eta}(\Dd_{\Pp_-}) - \tilde{\eta}(\Dd_{\Cc_-}) =
\boxed{?}
\]
in terms of recognizable data. (Of course, one can consider $M_+$
too.) For generalized APS spectral projectors, the comparison
problem for the eta invariant was first solved modulo $\mathbb{Z}$
by Lesch and Wojciechowski \cite{LW96} and M\"{u}ller \cite{Mu94}; 
later, this integer
ambiguity was removed by the first author \cite{L}. For $\Pp_- \in
Gr^*_\infty(\Dd_-)$ and assuming the invertibility of
$\Dd_{\Pp_-}$, the comparison problems for the eta invariant and
the $\z$-determinant were solved by Scott and Wojciechowski
\cite{ScKW99} and Scott \cite{ScS02}; cf.\ Forman \cite{F}. The
comparison problem for the eta invariant was later solved for
$\Pp_- \in Gr^*_\infty(\Dd_-)$ without integer ambiguities by Kirk
and Lesch \cite{KL01} using the Scott-Wojciechowski comparison
theorem \cite{ScKW99}.  In \cite{LP1}, we remove the invertibility
assumption on $\Dd_{\Pp_-}$, and we present a new and unified
derivation of the comparison formulas for both the eta invariant
and $\z$-determinant. For the eta invariant, we obtain a new
formulation of the integer defect, and the $\z$-determinant formula
contains an additional term that is absent in the case of
invertible $\Dd_{\Pp_-}$ dealing with $\ker (\Dd_{\Pp_-})$ and the
Dirichlet to Neumann map on the double of $M_-$.

Recall that the Calder\'on projector $\Cc_{-}$ can be written as
\eqref{Cc} for a unitary operator $\kp_{-}$, and $\Pp_-$ has a
similar decomposition with a unitary operator $\kp_{\Pp_-}$. We
denote by $\widehat{U}_{\Pp_-}$ the restriction of $U_{\Pp_-} :=
\kp_- \kp_{\Pp_-}^{-1}$, which is a unitary operator over
$L^2(Y,S^-)$, to the orthogonal complement of its
$(-1)$-eigenspace. We define $\Ll_{\Pp_-} := -P_- \, G\,
\Rr_-^{-1} G\, P_-$ just as in \eqref{Ll-}, where $P_-$ is the
orthogonal projection onto the finite-dimensional vector space
$\IM(\Cc_-)\cap \IM(\Id-\Pp_-)$. Then $\Ll_{\Pp_-}$ is a positive
operator on this space \cite{LP1}, so $\det\Ll_{\Pp_-}$ is a
positive number. The following theorem solves the comparison
problem (there is a corresponding formula for the comparison
problem on $M_+$).
\begin{theorem}[\cite{LP1}] \label{t:comp}
The following comparison formulas hold:
\begin{align*}
 &\frac{\zd \Dd_{\Pp_-}^2}{\zd \Dd_{\Cc_-}^2} = (\det
\Ll_{\Pp_-})^{2} \cdot {\det}_F \Big(\frac{2 \Id
+\widehat{U}_{\Pp_-} +\widehat{U}_{\Pp_-}^{-1}}{4} \Big),\\
& \tilde{\eta}(\Dd_{\Pp_-}) - \tilde{\eta}(\Dd_{\Cc_-}) =
\frac{1}{2 \pi i} \, \Log \detF U_{\Pp_-} \ (\operatorname{mod}
\Z),
\end{align*}
where the integer defect in the eta formula can be identified
exactly in terms of winding numbers of a specific naturally
defined operator.
\end{theorem}

Assume now that $M_+$ is an infinite cylinder as in Section
\ref{sec-mwce}. Recall that $\Cc_+ := \Pi_> + \frac{1 - \sigma}{2}
\Pi_0$ (see \eqref{Caldcylin}) is the natural Calder\'on projector
on the cylinder $M_+$. This suggests that the natural boundary
projector on $M_-$ taking into consideration the infinite cylinder
should be $\Ss_- : = \Pi_< + \frac{1 + \sigma}{2} \Pi_0$, instead
of the Calder\'on projector $\Cc_-$. The projector $\Ss_-$ is
called the \emph{augmented APS spectral projector} (cf.\
\cite{HMaM97}). Then $\Ss_- \in Gr_\infty^*(\Dd_-)$, so we can
define the spectral invariants of $\Dd_{\Ss_-}$ on $M_-$. The
($b$-)\emph{comparison problem} is to describe the ``defects''
\[
\frac{\bzd \Dd^2 }{\zd \Dd^2_{\Ss_-} } = \boxed{?} \,, \qquad
\be(\Dd) - \eta(\Dd_{\Ss_-}) = \boxed{?}
\]
in terms of recognizable data. Thus, the comparison problem in
this context is the comparison of the $b$-spectral invariants on
$M$ with those on $M_-$ corresponding to the ``scattering
Calder\'on projector'' $\Ss_-$. The following theorem solves the
comparison problem on manifolds with cylindrical ends.
\begin{theorem}[\cite{LP3}] \label{t:main2}
The following comparison formulas hold:
\[
\frac{\bzd \Dd^2 }{\zd \Dd^2_{\Ss_-} } = 2^{-\frac12\z_{D_Y^2}(0)
- h_Y} \hspace{-.2em} \, \bigg( \frac{\det \Ll}{\det \Ll_{-}}
\bigg)^{\hspace{-.2em} -2} , \qquad \be(\Dd) =
{\eta}(\Dd_{\Ss_-})\ (\operatorname{mod} 2\Z), 
\]
where $\Ll_{-}$ is defined by \eqref{Ll} but with $\{U_k\}$ an
orthonormal basis for $\ker (\Dd_{\Ss_-})$.
\end{theorem}

Thus, the $b$-eta invariant of $\Dd$ and the eta invariant of
$\Dd_{\Ss_-}$ are identical modulo $2 \Z$, while the
$\z$-determinants differ by data over $Y$ and global data. Because
$\z$-determinants are highly nonlocal compared to eta invariants,
one would not expect the ratio to be unity. This is indeed the
case as shown in Theorem \ref{t:main2}. The $\z$-determinant
formula is new, and a similar formula for the eta
invariants of \emph{compatible} Dirac operators (without integer
ambiguities and with the
augmented APS projector $\Pi_< + \frac{1 - \sigma}{2} \Pi_0$) was
first proved by M\"uller \cite{Mu94} using a completely different
method.
\section{Brief sketch of the proof of Theorem \ref{t:glue}}\label{sec-pf}

In this last section, we briefly sketch the proof of Theorem
\ref{t:glue}; we refer to Section \ref{sec-glue} for the notation
and to our preprint \cite{LP1} for all the details. We remark that
the proofs of Theorems \ref{t:main1} and \ref{t:comp} are similar
in spirit, but the proof of Theorem \ref{t:main1} has surprising
twists due to the presence of continuous spectrum. For $\la$ not
in the spectrum of $\Dd$, the \emph{Calder\'on projectors} of
$\Dd_{\pm} - \la$ are defined by
\begin{equation} \label{defPpm}
P_{\pm}(\la) := {\pm} \g_{0^{\pm}}(\Dd - \la)^{-1}\g_0^*G : 
L^2(Y, S_0) \to L^2(Y,S_0),
\end{equation}
where $\g_{0^{\pm}}:=\lim_{\ep\to0^{\pm}}\g_{\ep}$ with $\g_{\ep}$ the
restriction map from $M$ to $\{\ep\}\times Y$, and $\g_0^*$ is the
adjoint map of $\g_0$, the restriction to $\{0\} \times Y$. We now
introduce an \emph{auxiliary model problem} over $N = [-1,1]
\times Y$. To do so, fix an involution $\omega$ over $\ker(D_Y)$
that anticommutes with $G$. We then impose boundary conditions
$\Cc^c_{-} = \Pi_< + \frac{1 - \omega}{2} \Pi_0$ at $\{1\}\times
Y$, $\Cc^c_{+} = \Pi_> + \frac{1 + \omega}{2} \Pi_0$ at
$\{-1\}\times Y$. Let us denote the operator $G(\partial_u + D_Y)$
with these boundary conditions by $\Dd^c$. Then we can define the
Calder\'on projectors $P^c_{\pm}(\la)$ for the restrictions
$\Dd^c_{\pm} - \la$ of $\Dd^c - \la$ to $M_{\pm} \cap N$ exactly
as in \eqref{defPpm}. Let $V$ be a unitary operator on
$L^2(Y,S_0)$ that satisfies $V(\Id-\Cc_{+}) V^{-1}=\Cc_{-}$. Then
we define operators $K(\la)$ and $K^c(\la)$ on $L^2(Y,S_0)$ by
defining their inverses as
\begin{align*}
K(\la)^{-1} &= \Cc_+ P_+(\la) +
(\Id-\Cc_+)V^{-1} \Cc_- P_-(\la)  , \\
K^c(\la)^{-1} &= \Cc_+^c P^c_+(\la) + \Cc^c_- P^c_-(\la).
\end{align*}
(The operators on the right-hand side do turn out to be
invertible.) From these formulas, we see that $K(\la)$ and
$K^c(\la)$ ``link'' the Calder\'on projectors of $\Dd_{\pm} - \la$
and $\Dd_{\pm}^c - \la$, respectively. Moreover, these operators
have quite remarkable properties as we shall see. First, $K(\la)
K^c(\la)^{-1}$ is of Fredholm determinant class, so $\detF (K(\la)
K^c(\la)^{-1})$ makes sense. We henceforth fix a simply connected
region of the plane consisting of the upper and lower half-planes
and an interval on the real axis and fix a corresponding logarithm
$\log \detF (K(\la)K^c(\la)^{-1})$ depending holomorphically on
$\la$ in this region. Let us define $\Dd_{\Cc} = \Dd_{\Cc_+}\sqcup
\Dd_{\Cc_-}$ and $\Dd^c_{\Cc^c}=\Dd^c_{\Cc^c_+}\sqcup
\Dd^c_{\Cc^c_-}$. Then we prove that
\begin{multline}\label{variation}
\partial_{\la}  \log \detF( K(\la)K^c(\la)^{-1}) \\
= - \Tr\big((\Dd - \la)^{-1} - (\Dd_{\Cc} - \la)^{-1} -((\Dd^c -
\la)^{-1} - (\Dd^c_{\Cc^c} - \la)^{-1})
 \big) .
\end{multline}
The operator in parentheses turns out to be trace class! In fact,
the auxiliary model problem was introduced for exactly this
reason: to overcome certain \emph{trace class issues}, and the
fact that the gluing problem on the model problem can be solved
exactly \cite{LPnote}. The second remarkable property of $K(\la)$
and $K^c(\la)$ is that, using \eqref{variation} and deriving
formulas for the spectral invariants in terms of resolvents, we
prove that for $\nu\in\mathbb{R}$
the spectral invariants can be expressed as
\begin{multline} \label{logdetC}
\frac{\zd (\Dd^2 + \nu^2, \Dd^2_{\Cc}+\nu^2)}{\zd
((\Dd^c)^2+\nu^2, (\Dd^c_{\Cc})^2+\nu^2)} \\
= C \cdot \detF (K(i \nu)K^c(i \nu)^{-1}) \cdot \detF( K(-i
\nu)K^c(-i \nu)^{-1}),
\end{multline}
for a constant $C$, where $\zd (\Dd^2+\nu^2, \Dd^2_{\Cc}+\nu^2) :=
\zd (\Dd^2+\nu^2) \cdot (\zd (\Dd^2_{\Cc}+\nu^2))^{-1}$ with a
similar formula for the operators on the model cylinder, and
\begin{multline}
\eta(\Dd,\Dd_{\Cc})\\ =  
-\frac{1}{\pi i} \Big(
\lim_{\nu\to\infty} \hspace{-.3em} \big( \log \detF
(K(i\nu)K^c(i\nu)^{-1}) - \log \detF (K(-i\nu)K^c(-i\nu)^{-1})
\big)\\
\label{eta} -\hspace{-.2em} \lim_{\nu\to 0^+} \hspace{-.3em} \big( \log
\detF (K(i\nu)K^c(i\nu)^{-1}) \hspace{-.2em} -\hspace{-.2em}  \log \detF
(K(-i\nu)K^c(-i\nu)^{-1}) \big) \Big),
\end{multline}
where $\eta(\Dd,\Dd_\Cc) : = \eta(\Dd) - \eta(\Dd_{\Cc})$. Third,
as $\nu \to \pm \infty$, we prove that
\begin{equation}\label{limdet}
\lim_{\nu\to \pm \infty} \frac{\zd (\Dd^2 + \nu^2,
\Dd^2_{\Cc}+\nu^2)}{\zd ((\Dd^c)^2+\nu^2,
(\Dd^c_{\Cc})^2+\nu^2)}=1  ,
\end{equation}
\begin{equation}\label{limK}
 \lim_{\nu\to \pm \infty} \detF (
K(i \nu)K^c(i \nu)^{-1} ) = a_{\pm},
\end{equation}
where $a_+ = 1$ and $a_- = \detF U$, and as $\nu \to 0^+$,
we have
\begin{equation} \label{limK0}
\detF ( K(\pm i \nu) K^c(\pm i \nu)^{-1}) = (\pm \nu)^{h_M} \,
(\det {\Ll})^{-1} \, \detF \Big(\frac{\Id+\widehat{U}}{2}\Big)
\big( 1 + o(1) \big).
\end{equation}

Using the aforementioned properties, we can now prove the gluing
formulas ``in one shot''! For the $\z$-determinant, we first find
the constant $C$ in \eqref{logdetC}. To do so, we take $\nu \to
\infty$ on both sides of \eqref{logdetC}, and use the limits in
\eqref{limdet} and \eqref{limK} to obtain $C = \detF U^{-1} =
(-1)^{h_M} \cdot \detF \widehat{U}^{-1}$. Substituting this value
into \eqref{logdetC}, then using \eqref{limK0} and simplifying, we
see that as $\nu \to 0^+$,
\begin{align*}
& \frac{\zd (\Dd^2 + \nu^2, \Dd^2_{\Cc}+\nu^2)}{\zd ((\Dd^c)^2
+\nu^2, (\Dd^c_{\Cc})^2+\nu^2)} \\ &\hspace{5em}= \nu^{2 h_M} \,
(\det \Ll)^{-2} \, \detF \Big(\frac{2 \Id + \widehat{U} +
\widehat{U}^{-1}}{4}\Big) \big( 1 + o(1) \big).
\end{align*}
However, we prove that as $\nu \to 0^+$, the left-hand side has
the asymptotics
\[
\frac{\zd (\Dd^2 + \nu^2, \Dd^2_{\Cc}+\nu^2)}{\zd ((\Dd^c)^2
+\nu^2, (\Dd^c_{\Cc})^2+\nu^2)} = \nu^{2 h_M} \frac{\zd (\Dd^2,
\Dd^2_{\Cc})}{\zd ((\Dd^c)^2, (\Dd^c_{\Cc^c})^2)}  \big( 1 + o(1)
\big).
\]
Combining this expression with the previous equality, then setting
$\nu = 0$, we obtain
\[
\zd (\Dd^2, \Dd^2_{\Cc})= \zd ((\Dd^c)^2,
(\Dd^c_{\Cc^c})^2)\,(\det {\Ll})^{-2}\, {\det}_F \Big(
\frac{2\Id+\widehat{U}+\widehat{U}^{-1}}{4} \Big).
\]
Finally, the main result of \cite{LPnote} tells us that $\zd
((\Dd^c)^2, (\Dd^c_{\Cc^c})^2) = 2^{-\z_{D_Y^2}(0)-h_Y}$, which
completes the proof for the $\z$-determinant.

Before proving the eta invariant formula, recall that if $f(t)$ is
a smooth nonzero complex-valued function on an interval $[a,b]$,
then the \emph{winding number} $W(f) \in \Z$ of $f$ is defined by
the equality
\begin{equation} \label{Wf}
\log f(b) - \log f(a) = \Log f(b) - \Log f(a) + 2 \pi i\, W(f),
\end{equation}
where $\log f(t)$ is any continuous logarithm for $f(t)$ with $t
\in [a,b]$ and $\Log$ denotes the principal value of the logarithm.  In
view of the limits \eqref{limK} and \eqref{limK0}, by definition
of the winding number \eqref{Wf}, we have, modulo $2 \pi i \Z$,
\[
\lim_{\nu\to\infty} \big( \log \detF (K(i\nu) K^c(i\nu)^{-1}) -
\log \detF (K(-i\nu) K^c(-i\nu)^{-1}) \big) \equiv - \Log \detF U,
\]
where the integer defect is just the winding number of $\detF
(K(\la) K^c(\la)^{-1})$ from $-i \infty$ to $i \infty$, and modulo
$2 \pi i \Z$,
\begin{multline*}
\lim_{\nu\to 0^+} \hspace{-.3em} \big( \log \detF
(K(i\nu)K^c(i\nu)^{-1}) - \log \detF (K(-i\nu)K^c(-i\nu)^{-1})
\big) \\ \equiv \begin{cases}
0, & \text{$h_M$ even,}\\
- \pi i, & \text{$h_M$ odd},
\end{cases}
\end{multline*}
where the integer defect is just the winding number of $\detF
(K(\la) K^c(\la)^{-1})$ from $-i \nu$ to $i \nu$ for $\nu > 0$
sufficiently small. Substituting these limits into \eqref{eta}
completes the proof for the eta invariant.

\subsection*{Acknowledgement} The authors thank Gerd Grubb and K. P.
Wojciechowski for all their help and for taking a keen interest in
our mathematical careers.

\bibliographystyle{amsplain}

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to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
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\begin{thebibliography}{1}

\bibitem{APS} M.~F.~Atiyah, V.~K.~Patodi, and I.~M.~Singer,
{\em Spectral asymmetry and Riemannian geometry. I}, 
Math. Proc. Cambridge Philos. Soc. \textbf{77} (1975), 43--69. \MR{0397797 
(53:1655a)}

\bibitem{BB}
D.~Bleecker and B.~Booss-Bavnbek,
{\em Spectral invariants of operators of {D}irac type on partitioned manifolds}, 
Aspects of Boundary Problems in Analysis and Geometry, Birkh\"auser,
Boston, 2004, pp.~1--130. \MR{2072498}

\bibitem{BL98} J.~Br\"uning and M.~Lesch,
{\em On the $\eta$-invariant of certain
nonlocal boundary value problems}, 
Duke Math. J. \textbf{96} (1999), 425--468. \MR{1666570 (99m:58180)}

\bibitem{Br} V.~Bruneau,
{\em Fonctions z\^eta et \^eta en pr\'esence de spectre continu}, 
C. R. Acad. Sci. Paris S\'er. I Math. \textbf{323}, 
no.~5  (1996), 475--480. \MR{1408979 (97j:58154)}

\bibitem{Bu95} U.~Bunke, 
{\em On the gluing formula for the $\eta$-invariant}, 
J. Differential Geometry \textbf{18} (1995), 397--448. \MR{1331973 (96c:58163)}

\bibitem{BFK} D.~Burghelea, L.~Friedlander, and T.~Kappeler,
{\em Mayer-Vietoris type formula for determinants of differential operators}, 
J. Funct. Anal. \textbf{107} (1992), 34--65. \MR{1165865 (93f:58242)}

\bibitem{C}
A.-P.~Calder{\'o}n,
{\em Boundary value problems for elliptic equations}, 
Outlines Joint Sympos. Partial Differential Equations
(Novosibirsk, 1963), Acad. Sci. USSR Siberian Branch, Moscow, 1963,
pp.~303--304. \MR{0203254 (34:3107)}

\bibitem{Car} G.~Carron,
{\em D\'eterminant relatif et la fonction {X}i},
Amer. J. Math. \textbf{124}, no.~2 (2002), 307--352. \MR{1890995 (2003c:58023)}

\bibitem{Dfr94} X.~Dai and D.~Freed,
{\em $\eta$-invariants and determinant lines}, 
J. Math. Phys. \textbf{35} (1994), 5155--5195. \MR{1295462 (96a:58204)}

\bibitem{F} R.~Forman,
{\em Functional determinants and geometry},
Invent. Math. \textbf{88} (1987), 447--493. \MR{0884797 (89b:58212)}

\bibitem{Gr99} G.~Grubb,
{\em Trace expansions for pseudodifferential
boundary problems for Dirac-type operators and more general
systems}, 
Ark. Math. \textbf{37} (1999), 45--86. \MR{1673426 (2000c:35265)}

\bibitem{Gr01} \bysame,
{\em Poles of zeta and eta functions for perturbations of the
Atiyah-Patodi-Singer problem}, 
Comm. Math. Phys. \textbf{215} (2001), 583--589. \MR{1810945 (2002a:58036)}

\bibitem{Gr03} \bysame,
{\em Spectral boundary conditions for generalizations of 
Laplace and Dirac operators}, 
Comm. Math. Phys. \textbf{240} (2003), 243--280. \MR{2004987 (2004f:58030)}

\bibitem{Ha98} A.~Hassell,
{\em Analytic surgery and analytic torsion}, 
Comm. Anal. Geom. \textbf{6}, no.~2 (1998), 255--289. \MR{1651417 (2000c:58061)}

\bibitem{HZ} A.~Hassell and S.~Zelditch,
{\em Determinants of Laplacians in exterior domains}, 
IMRN \textbf{18} (1999), 971--1004. \MR{1722360 (2001a:58045)}

\bibitem{HMaM95} A.~Hassell, R.~R.~Mazzeo, and R.~B.~Melrose,
{\em Analytic surgery and the accumulation of eigenvalues}, 
Comm. Anal. Geom. \textbf{3} (1995), 115--222. \MR{1362650 (97f:58132)}

\bibitem{HMaM97} \bysame,
{\em A signature formula for manifolds with corners of codimension two}, 
Topology \textbf{36}, no.~5 (1997), 1055--1075. \MR{1445554 (98c:58163)}

\bibitem{KL01}
P.~Kirk and M.~Lesch,
{\em The eta invariant, Maslov index, and
spectral flow for Dirac-type operators on manifolds with boundary}, 
Forum Math. {\bf 16} (2004), 553--629. \MR{2044028 (2005b:58029)}

\bibitem{Lee} Y.~Lee,
{\em Burghelea-{F}riedlander-{K}appeler's gluing
formula for the zeta-determinant and its applications to the
adiabatic decompositions of the zeta-determinant and the analytic torsion}, 
Trans. Amer. Math. Soc. \textbf{355}, no.~10 (2003),
4093--4110. \MR{1990576 (2004e:58058)}

\bibitem{LW96}
M.~Lesch and K.~P.~Wojciechowski,
{\em On the $\eta$-invariant of
generalized {A}tiyah-{P}atodi-{S}inger boundary value problems},
Illinois J. Math. \textbf{40}, no.~1 (1996), 30--46. \MR{1386311 (97d:58194)}

\bibitem{L} P.~Loya,
{\em Dirac operators, boundary value problems, and the b-calculus},
Contemp. Math. {\bf 366} (2005), 241--280. 

\bibitem{LP0} P.~Loya and J.~Park,
{\em Decomposition of the $\z$-determinant for the Laplacian on
manifolds with cylindrical end}, 
Illinois J. Math. to appear.

\bibitem{LP2} \bysame,
{\em On the gluing problem for the spectral invariants of Dirac operators}, 
Advances in Math. to appear.

\bibitem{LP3} \bysame,
{\em On the gluing problem for Dirac operators on manifolds with
cylindrical ends}, 
Preprint, 2004.

\bibitem{LP1} \bysame,
{\em The comparison problem for the spectral invariants of Dirac type
operators}, 
Preprint, 2004.

\bibitem{LPnote} \bysame,
{\em The $\z$-determinant of generalized APS boundary problems over
the cylinder}, 
J. Phys. A. {\bf 37}, no. 29 (2004), 7381--7392. \MR{2078962}

\bibitem{LPasymp} \bysame, 
{\em Asymptotics of $\z$-determinant of
Dirac Laplacian under adiabatic process},
In preparation.

\bibitem{MM95} R.~Mazzeo and R.~B.~Melrose,
{\em Analytic surgery and the eta invariant}, 
Geom. Funct. Anal. \textbf{5}, no.~1 (1995), 14--75. \MR{1312019 (96a:58200)}

\bibitem{MP98} R.~Mazzeo and P.~Piazza,
{\em Dirac operators, heat kernels and microlocal analysis. {I}{I}. 
{A}nalytic surgery}, 
Rend. Mat. Appl. (7) \textbf{18}, no.~2 (1998), 221--288. \MR{1659838
(2000a:58065)}

\bibitem{BMeR93} R.~B.~Melrose, 
The {Atiyah}-{Patodi}-{Singer} {index} {theorem}, 
A. K. Peters, Wellesley, 1993. \MR{1348401 (96g:58180)}

\bibitem{Mu94} W.~M{\"u}ller,
{\em Eta invariants and manifolds with boundary},
J. Differential Geometry \textbf{40} (1994),
311--377. \MR{1293657 (96c:58165)}

\bibitem{Mu96} \bysame,
{\em On the {$L^{2}$}-index of {Dirac}
operators on manifolds with corners of codimension two. {I}}, 
J. Differential Geometry \textbf{44} (1996), 97--177. \MR{1420351 (98b:58163)}

\bibitem{Mu98} \bysame,
{\em Relative zeta functions, relative
determinants and scattering theory}, 
Comm. Math. Phys. \textbf{192} (1998), 309--347. \MR{1617554 (99k:58189)}

\bibitem{PWI} J.~Park and K.~P.~Wojciechowski,
{\em Scattering theory and adiabatic decomposition of the
{$\zeta$}-determinant of the {D}irac {L}aplacian}, 
Math. Res. Lett. \textbf{9}, no.~1 (2002), 17--25. \MR{1892311 (2003c:58024)}

\bibitem{PWII} \bysame,
{\em Adiabatic decomposition of the $\z$-determinant and Scattering theory}, 
MPI Preprint, 2002.

\bibitem{RS} D.~B. Ray and I.~M. Singer,
{\em {$R$}-torsion and the {L}aplacian on {R}iemannian manifolds}, 
Advances in Math. \textbf{7} (1971),
145--210. \MR{0295381 (45:4447)}

\bibitem{ScS02} S.~Scott,
{\em Zeta determinants on manifolds with boundary}, 
J. Funct. Anal. \textbf{192}, no.~1 (2002), 112--185. \MR{1918493 (2003g:58051)}

\bibitem{ScKW99} S.~Scott and K.~P.~Wojciechowski,
{\em The $\zeta$-determinant and Quillen determinant for a Dirac
operator on a manifold with boundary},
Geom. Funct. Anal. \textbf{10} (1999), 1202--1236. \MR{1800067 (2001k:58067)}

\bibitem{Se69} R.~T.~Seeley,
{\em Topics in pseudo-differential operators}, 
Pseudo-Diff. Operators (C.I.M.E., Stresa, 1968),  1969,
pp.~167--305. \MR{0259335 (41:3973)}

\bibitem{S1} I~.M.~Singer
{\em The eta invariant and the index}, Mathematical
aspects of string theory, World Scientific, Singapore, 1988,
pp.~239--258. \MR{0915824}

\bibitem{S2} \bysame,
{\em Families of {D}irac operators with applications to physics}, 
Ast\'erisque, Numero Hors Serie, The
mathematical heritage of \'Elie Cartan (Lyon, 1984), 1985,
pp.~323--340. \MR{0837207 (88a:58192)}

\bibitem{V} S.~M.~Vishik,
{\em Generalized {R}ay-{S}inger conjecture. {I}. {A} manifold with a smooth
boundary}, 
Comm. Math. Phys. \textbf{167}, no.~1 (1995), 1--102. \MR{1316501 (96f:58184)}

\bibitem{KPW95} K. P. Wojciechowski,
{\em The additivity of the $\eta$-invariant. The case of a singular tangential operator},
Comm. Math. Phys. \textbf{169} (1995), 315--327. \MR{1329198 (96k:58210)}

\bibitem{WoK99} \bysame,
{\em The $\zeta$-determinant and the additivity of the $\eta$-invariant on
the smooth, self-adjoint {G}rassmannian}, 
Comm. Math. Phys. \textbf{201}, no.~2 (1999), 423--444. \MR{1682214
(2000f:58071)}

\end{thebibliography}

\end{document}