ADIABATIC DECOMPOSITION OF THE ζ-DETERMINANT AND SCATTERING THEORY.

JINSUNG PARK AND KRZYSZTOF P. WOJCIECHOWSKI

Abstract. We discuss the decompositon of the ζ-determinant of the square of the Dirac operator into contributions coming from different parts of the manifold. The “easy” case was worked out in paper [27]. Due to the assumptions made on the operators in [27], we were able to avoid the presence of the “small eigenvalues” which provide the large time contribution to the determinant. In the present work we analyze the general case. We offer a detailed analysis of the contribution made by the small eigenvalues. The idea is based on the method used by Werner Müller in his paper [23], where he studied the η-invariant on a manifold with cylindrical end. The analysis of the ζ-determinant, however, is more difficult. The difference is illustrated nicely in the proof of the Lesch-Wojciechowski formula for the ratio of the ζ-determinants of the Dirac operator subject to the two different boundary conditions provided in the final sections of the paper. The main results of the paper were announced in [28]

Introduction and Statement of the Results

The study of the behaviour of the geometric invariants of the Dirac operators with respect to the decomposition of the underlying manifold is a fascinating subject with a long and interesting history. The history of the subject and the most important developements were described in many different sources. In particular, we refer to a beautiful review paper [22] for the account of the recent developements in this area of mathematics.

In this paper we concentrate on the decomposition of the ζ-determinants of the Dirac Laplacian. We now proceed with the precise definitions of the operators and invariants involved in our story.

Let D : C^{∞}(M; S) → C^{∞}(M; S) be a compatible Dirac operator acting on sections of a bundle of Clifford modules S over a closed manifold M. The operator D is a self-adjoint operator, with discrete spectrum {λ_{k}}_{k∈ℤ} . Assume that we have a decomposition of M as M_{1} ∪ M_{2} , where M_{1} and

Date: April 16, 2002.

M_{2} are compact manifolds with boundaries such that

M = M_{1} ∪ M_{2} , M_{1} ∩ M_{2} = Y = ∂M_{1} = ∂M_{2} . (0.1)

The ζ-determinant of the operator D is given by the formula

det_{ζ}D = e^{\frac{iπ}{2} (ζ_{D^{2}}(0)−η_{D}(0))}·e^{− \frac{1}{2} ζ′_{D^{2}}(0)} , (0.2)

(see [32], see also the Introduction of [31]). The functions ζ_{D^{2}}(s) , and η_{D}(s) are defined as follows

ζ_{D^{2}}(s) = ∑_{λ_{k}≠0}(λ_{k}^{2})^{−s} = \frac{1}{Γ(s)} ∫_{0}^{∞} t^{s−1}[T r(e^{−tD^{2}}) − dim ker D] dt (0.3)

and

η_{D}(s) = ∑_{λ_{k}≠0} sign(λ_{k})|λ_{k}|^{−s} = \frac{1}{Γ(\frac{s+1}{2})} ∫_{0}^{∞} t^{\frac{s−1}{2}} T r De^{−tD^{2}}dt .

They are holomorphic functions of s for Re(s) large and they both have meromorphic extensions to the complex plane with simple poles only. In both cases s = 0 is a regular point.

In this paper we are going to study the decomposition of the ζ-determinant of D into the contributions coming from the submanifolds M_{1} and M_{2} . More precisely, we concentrate on

e^{−ζ′_{D^{2}}(0)} ,

the determinant of D^{2} , which provides the modulus of the ζ-determinant of D . The decomposition of the phase of the determinant is now a well-known story. We are going, however, to discuss this topic to a certain extent. The reason is that the adiabatic method we use here was originally developed in order to study the most important part of the phase, the η-invariant.

We make two further assumptions. First, we only discuss the odd-dimensional case

n = dim M = 2k + 1 .

Second, we assume that M and the operator D have product structures in a neighborhood of the boundary Y . This means that there is a bicollar neighborhood N ≅ [−1, 1]×Y of Y in M such that the Riemannian structure on M and the Hermitian structure on S are products when restricted to N . This implies that D has the following form when restricted to the submanifold N

D = G(∂_{u} + B) . (0.4)

DECOMPOSITION OF THE ζ-DETERMINANT AND SCATTERING THEORY 3

Here u denotes the normal variable, G : S|_{Y} → S|_{Y} is a bundle automorphism, and B is a corresponding Dirac operator on Y. Moreover, G and B do not depend on u and they satisfy

G^{∗} = −G , G^{2} = −Id , B = B^{∗} and GB = −BG . (0.5)

The first assumption is not of great importance from the technical point of view, and we discuss some technicalities and the minor differences between odd and even dimensional case, in the other paper. The second assumption has much more serious analytical consequences. The analysis of the nonproduct situation is much more complicated and we will not attempt to discuss it here.

Both the η_{D}(0) and ζ′_{D^{2}}(0) are non-local invariants and their values depend on global geometric information. Therefore, at first sight, it seems rather naive to expect any decomposition formula for the invariants from which the ζ-determinant is constructed. However, there is a method for extracting the useful informtion. The second author got the idea after studying the short, but extremely influential paper by Singer (see [33]). In this beautiful research note Singer studied, among the other things, the boundary contribution to the the η-invariant of the boundary value problem. It follows from the formula (0.3) that the η-invariant can be expressed via the trace of the corresponding heat operator, or the integral over the manifold from the kernel of the corresponding operator. Singer obtained the aforementioned boundary contribution to the η-invariant by stretching the collar neighborhood of the boundary. The trace he studied was divided into the interior contribution, the cylinder contribution and the error term. The key here was the fact that the error term disappeared as the length of the collar approached infinity and that the cylinder contribution could be explicitly computed.

The second author got acquainted with [33] in the Fall of 1987 and decided to apply this method to the study of the decomposition of the η-invariant. In short, the essence of the what we call here the Adiabatic Decomposition of the invariants of the Dirac operator is the stretching of M in the normal direction to Y .

To formulate the decomposition formula for the ζ-determinant we have to describe the invariants on a manifold with boundary which enter the picture at this point. The operator B has a discrete spectrum with infinitely many positive and infinitely many negative eigenvalues. Let Π_{>} denote the spectral projection onto the subspace spanned by the eigensections of B corresponding to the positive eigenvalues. Then Π_{>} is an elliptic boundary condition for D_{2} = D|_{M_{2}} (see [1]; see [3] for an exposition of the theory of elliptic boundary problems for Dirac operators). In fact, any orthogonal projection satisfying

(0.6) −GP G = Id − P and P − Π_{>} is a smoothing operator

is a self-adjoint elliptic boundary condition for the operator D_{2}. This means that the associated operator

(D_{2})_{P} : dom (D_{2})_{P} → L^{2}(M_{2}; S|_{M_{2}})

with dom (D_{2})_{P} = {s ∈ H^{1}(M_{2}; S|_{M_{2}}) | P (s|_{Y}) = 0} is a self-adjoint Fredholm operator with ker((D_{2})_{P}) ⊂ C^{∞}(M_{2}; S|_{M_{2}}) and discrete spectrum (see [37]). We denote by Gr_{∞}^{∗}(D_{2}) the space of P satisfying (0.6). The existence of the meromorphic extensions of the functions η_{(D_{2})_{P}} (s), ζ_{(D_{2})_{P}^{2}} (s) to the whole complex plane and their nice behaviour in a neighborhood of s = 0 were established in [37]. It follows that the ζ-determinant of the operator D_{P} , with P ∈ Gr_{∞}^{∗}(D_{2}) is well-defined. Let us observe that Id − P ∈ Gr_{∞}^{∗}(D_{1}) , if P is an element of Gr_{∞}^{∗}(D_{2}). We denote by η_{(D;P_{1},P_{2})}(s) the η-function of the operator G(∂_{u} + B) = D|_{[0,1]×Y} , subject to the boundary condition P_{2} at u = 0 and Id − P_{1} at u = 1 . We have the following pasting formula proved in [37]

η_{D}(0) = η_{(D_{1})_{Id−P_{1}}}(0) + η_{(D_{2})_{P_{2}}}(0) + η_{(D;P_{1},P_{2})}(0) mod ℤ . (0.7)

The special case of this formula with the boundary condition P_{1} = P_{2} = Π_{>} , under the assumption ker B = {0} , (otherwise (D_{2})_{Π_{>}} and (D_{1})_{Id−Π_{>}} would not be self-adjoint) was discussed by Ron Douglas and the second author in the Spring of 1988 and most of this discussion was included in [13]. The result is

η_{D}(0) = η_{(D_{1})_{Id−Π>}}(0) + η_{(D_{2})_{Π>}}(0) mod ℤ . (0.8)

The proof of (0.8) outlined by Douglas and the second author uses the fact that η_{D}(0), as well as the η-invariants of the boundary problems can be represented by the integral on the right side of the second equality in (0.2), where we put s = 0. This gives

η_{D}(0) = \frac{1}{\sqrt{π}} ∫_{0}^{∞} \frac{1}{\sqrt{t}}T r De^{−tD^{2}}dt (0.9)

= \frac{1}{\sqrt{π}} ∫_{0}^{∞} \frac{dt}{\sqrt{t}} ∫_{M} tr E(t; x, x)dx ,

where E(t; x, x) denotes the kernel of the operator De^{−tD^{2}} .

The first step in the proof is to replace the bicollar N by N_{R} = [−R, R] × Y. We now use Duhamel’s principle to split ∫_{M} tr E(t; x, x)dx into contributions coming from each side, plus the cylinder contribution and the error term. Explicit representation of the operators De^{−tD^{2}} , (D_{1})_{Id−Π_{>}}e^{−t(D_{1})_{Id−Π>}^{2}} and (D_{2})_{Π_{>}}e^{−t(D_{2})_{Π>}^{2}} on the corresponding cylinders show that in each case the cylinder contribution is trivial. It also easily follows that the error term disappears as R → ∞.

Second, though η_{D}(0) is not local, its variation (for instance with respect to the parameter R) is local and therefore the value of the contributions does not vary with R. This is enough to make explicit calculations of the formula (0.8).

One has to stress that the invertibility of the tangential operator B implies significant analytical simplifications. It allows us to ignore the large time contribution to the η-invariant given by the integral

∫_{R^{ε}}^{∞} t^{\frac{s−1}{2}} T r De^{−tD^{2}}dt ,

for any ε > 0 . This is due to the fact that the invertibility of B implies that all but finitely many eigenvalues of the operators involved stay away from 0 . There is finitely many small eigenvalues of the size e^{−cR} , and they produce only an integer contribution.

The situation is different in the case of the non-trivial kernel of the operator B . First of all we have to modify the spectral boundary conditions to obtain self-adjoint operators on manifolds M_{1} and M_{2} . Second, we have to deal with infinitely many eigenvalues approaching 0 as R → ∞ . Jeff Cheeger in his works on manifolds with conical singularities (see [9], [10]) introduced the appriopriate, finite-dimensional generalization of the Atiyah-Patodi-Singer conditions. He called them “ideal boundary conditions”. In this paper we use the “ideal boundary conditions” to define the operators on the manifolds with boundary. The Grassmannian Gr_{∞}^{∗}(D_{2}) of the selfadjoint boundary conditions of Atiyah-Patodi-Singer type discussed above was introduced by Bernhelm Boss-Bavnbek and second author (see [2], [13], [37]). The boundary correction term to the η-invariant, which appears in this situation was studied by Lesch and the second author in [20]. Lesch and Wojciechowski found a very nice expression for the variation of the η-invariant under the change of the ideal boundary condition. This was enough to formulate the decomposition formula (0.7) in the case P_{1} and P_{2} being ideal boundary conditions. Details of the proof appeared in [35] and [36]. The result was extended to the case of P_{1}, P_{2} ∈ Gr_{∞}^{∗}(D_{2}) in [37].

We have to point out that, due to the fact that the variation of the ηinvariant is local and that we have extra symmetries on the cylinder, there is no contribution to the gluing formula for the η-invariant made by small eigenvalues (or equivalently the large time contribution is trivial). On the contrary, the analysis of the contributions made by the small eigenvalues to the gluing formula for the ζ-determinant of the Dirac Laplacian is one of the most important part of the present exposition.

The decomposition of the η-invariant was also studied independently by many others mathematicians, who came up with several different variants of the gluing formula. Melrose and his collaborators used the process of blowing up the metric in the normal direction in the case where the initial structure was not product. This deformation and the necessary analytical apparaturs was carefully described in a series of papers (see [15], [21] and especially [16]). They were able to analyze the integer contribution to the gluing formula, at least in the case of the signature operator. As a result they came up with the analytic description of the Wall’s non-additivity formula for the signature of the manifold with corners (see [16]). The Wall’s nonadditivity was also discussed by Bunke (see [6]), who also gave his proof of the gluing formula for the η-invariant. This formula was also discussed by Dai and Freed (see [12]), Brüning and Lesch (see [5]) and Werner Müller (see [23] and [24]).

Let us also mention the recent work of Kirk and Lesch [18]. They present the gluing formula for the η-invariant in which the integer contribution is expressed as a triple index involving the boundary condition and the Calderon projectors of D_{1} and D_{2} . The key ingredient in the proof is the Scott-Wojciechowski Theorem for the determinant of the Dirac operator on a manifold with boundary (see [31]) and Brüning and Lesch approach to the mod ℤ gluing formula. To this day [18] contains the most complete and beautiful approach to the gluing of the η-invariant.

From the author’s point of view the work of Werner Müller was especially stimulating and useful in their approach to the decomposition of the ζ-determinant. He considered the case of the non-invertible tangential operator B , and showed that the adiabatic limit of the η-invariant of the boundary problem was well-defined and equal to the corresponding invariant on a manifold with a cylindrical end. In order to prove this statement he had to make a systematic study of the behaviour of the small eigenvalues. He showed that their behaviour is ruled by scattering matrix on the manifold with cylindrical end attached. Though we can avoid the scattering theory in the study of the decomposition of the η-invariant, this is not so in the case of the ζ-determinant in the case of the non-invertible operator B .

The origin of present work may be traced to the paper [19]. The focus of the study by Klimek and the second author was on the η-invariants of the local boundary problems introduced by Singer in [33]. The analysis of the analytic torsion was obtained as a byproduct of the method used in the study of the η-functions, and was treated by the authors more as an exercise, than the beginning of a serious study. We also have to point out that the decomposition of the analytic torsion was studied by Hassell in [17]. This paper is close in spirit to our paper and the companion work [27], which deals with the easy case of the invertible operator B . Hassell studied the analytic torsion using the machinery developed by himself, Mazzeo and Melrose. Some further remarks on the subject can be found in the exposition [29].

To finish with the history of our subject we would also like to mention the work of Burghelea, Friedlander and Kappeler (see [7]). They were first to offer a decomposition formula for det_{ζ}D^{2} . Their formula involved the local elliptic boundary problems and the correction term, which, up to constant, was equal to the ζ-determinant of the elliptic operator on Y . Though the aforementioned operator lives on the hypersurface Y it depends on the global geometric information on the whole M . We discuss the relation of our work with the results of [7] in another place (see [30]).

Let us now start with a discussion of our approach to the decomposition of the ζ-determinant of the Dirac Laplacian. The logarithm of this determinant is equal to “minus” derivative of the ζ-function at s = 0 and we have the following formula

\frac{d}{ds}ζ_{D^{2}}(s)|_{s=0} = ∫_{0}^{∞} \frac{1}{t}T r e^{−tD^{2}}dt , (0.10)

under the assumption that D is an invertible operator. The right side of (0.10) is “a priori” only a formal formula, but as explained, it makes sense in our context. We refer to a companion paper [27] for the discussion of this point.

We follow the original Douglas-Wojciechowski proof of the decomposition formula (0.8) for the η-invariant. However, we face two new problems, not present in the case of the η-invariant. First, \frac{d}{ds}ζ_{D^{2}}(s)|_{s=0} is a much more “non-local” invariant, than the η-invariant. This results, for instance, in the fact that the value of det_{ζ}D^{2} varies with the length of the cylinder. Second, the cylinder contribution to the determinant is now non-trivial.

We still follow the idea of paper [13] and we stretch our manifold M to separate M_{1} and M_{2} . Then, we split ζ′_{D_{R}^{2}}(0) into contributions coming from different submanifolds plus cylinder contributions and the error terms. Here D_{R} denotes the operator D on the manifold M_{R} equal to the manifold M with N replaced by N_{R} ≅ [−R, R] × Y . We introduce a manifold with boundary

M_{1,R} = M_{1} ∪ [−R, 0] × Y ,

where we identify the “old” collar neighborhood of the boundary Y on M_{1} with [−R − 1, −R] × Y . Similarly we introduce the manifold M_{2,R}. The ζ-determinant is going to blow up as R → ∞ , hence we study the ratio

\frac{det_{ζ}D_{R}^{2}}{det_{ζ}(D_{1,R})_{P_{1}}^{2}·det_{ζ}(D_{2,R})_{P_{2}}^{2}} . (0.11)

In (0.11) D_{i,R} denotes the operator D_{R} restricted to M_{i,R} and P_{i} denotes the boundary condition, a finite dimensional modification of Atiyah-Patodi-Singer condition, defined below. We select

σ_{1}, σ_{2} : ker B → ker B ,

two involutions on kernel of B which satisfy

Gσ_{i} = −σ_{i}G ,

and define π_{i} = \frac{Id−σ_{i}}{2} the orthogonal projections onto −1 eigenspaces of σ_{i} . Then, we define

P_{1} = Π_{<} + π_{1} , P_{2} = Π_{>} + π_{2} . (0.12)

The projections P_{i} are the “ideal boundary conditions” mentioned before. They were introduced by Cheeger in [9] (see also [10]).

If we assume that all three operators in (0.11) are invertible, with eigenvalues bounded away from 0 (for any R large enough), then the quotient in (0.11) is bounded and we obtain a finite limit as R → ∞ . This situation was discussed in [27] (see also [26]). The assumptions made in [27] allow us to discard the large time contributions. In the general case, however, the presence of small eigenvalues makes the “large time” part extremely important. The analysis of the contributions made by the small eigenvalues of the operators D_{R}^{2}, (D_{i,R})_{P_{i}}^{2} is the main achievement of the present paper.

The eigenvalues of D_{R} fall into three different categories. We have infinitely many large eigenvalues (l − values) bounded away from 0 . Secondly, there is an infinite family of small eigenvalues (s − values) of the size O(\frac{1}{R}) . Last, we have a finite amount of eigenvalues which decay exponentially with R (e − values). There exists R_{o} such that, for any R > R_{o} the number h_{M} of e − values is independent of R . The following formula gives h_{M}

h_{M} = dim ker_{L^{2}}D_{1,∞} + dim ker_{L^{2}}D_{2,∞} + dim L_{1} ∩ L_{2} , (0.13)

where the operator D_{i,∞} is the operator D_{i} extended in a natural way to the manifold M_{i,∞} , which is equal to M_{i} with the infinite semicylinder [0, ∞) × Y (or (−∞, 0] × Y) attached. More precisely, the operator D_{i} extends in a natural way to the manifold M_{i,∞} . It has a unique closed selfadjoint extension in L^{2}(M_{i,∞}; S) , which we denote by D_{i,∞} . The subspaces L_{i} ⊂ ker B are the spaces of limiting values of extended L^{2}-solutions of D_{i,∞} (see [1], see also [3]).

Remark 0.1. The decomposition of the eigenvalues of the operator D_{R} into different classes was discussed by Cappell, Lee and Miller (see [8]; see also [35]). The corresponding analysis for the operator (D_{i,R})_{P_{i}} was provided by Werner Müller (see [23]).

The spectrum of the operators (D_{i,R})_{P_{i}} splits in the same way as the spectrum of D_{R} . The only difference is that the operators (D_{i,R})_{P_{i}} do not have non-trivial e − values and the dimension of the space of the solutions of (D_{i,R})_{P_{i}} is equal to

h_{i} = dim ker (D_{i,R})_{P_{i}} = dim ker_{L^{2}}D_{i,∞} + dim L_{i} ∩ ker(σ_{i} − 1) . (0.14)

It was already mentioned that, in the adiabatic limit, the different types of eigenvalues make their contribution at different time intervals. The contribution made by l − values comes from the time interval 0 < t < R^{ε} , where ε is a positive number. More precisely, it is not difficult to show that the l − values contribution to the adiabatic limit of (0.11) from the time interval [R^{ε}, ∞] disappears as R → ∞ (see Section 1). The s − values contribution comes from the interval [R^{2−ε}, R^{2+ε}] , for ε small enough.

The contribution made by l − values was discussed in [27]. To be more precise in [27] (see also [26]) we discussed the case of the operator D , such that D_{i,∞} and the tangential operator B have trivial kernels. This condition implies the vanishing of the e − values and s − values . This allows us to reduce the computation of the quotient in (0.11) to the corresponding quotient on the cylinder. The result is that the limit of (0.11) as R → ∞ is equal to 2^{−ζ_{B^{2}}(0)} . Actually, even in the presence of e−values and s−values, we are able to show that in the adiabatic limit the contribution from the time interval [0, R^{2−ε}] comes from the cylinder only. This can be stated in the following way:

as R → ∞ the small time contribution forgets about the geometry involved and the operators in the quotient (0.11) can be replaced by their counterparts living on the cylinders ((−∞, 0] ∪ [0, ∞)) × Y and (−∞, +∞) × Y .

The precise meaning of the aforementioned statement can be found in Section 1. The method we use to prove this statement combines Duhamel’s Principle and finite propagation speed property of the wave operators.

The computation of the s−values contribution is the main achievement of the paper. We follow Werner Müller (see [23]) and use the Scattering T heory to get a description of the s−values. The operators D_{i,∞} on M_{i,∞} determine scattering matrices C_{i}(λ) . It appears that the matrix C_{12} = C_{1}(0)C_{2}(0) determines the s − values of the operator D_{R} for R large enough. More precisely we define a one dimensional Dirac operator D(C_{12}) equal to −\frac{i}{2} \frac{d}{du} acting on sections of a flat vector bundle over S^{1} determined by the unitary matrix \overline{C}_{12} (complex conjugate of C_{12}). It turns out that, up to a negligible error, the s−values of D_{R} are determined by the eigenvalues of the operator D(C_{12}) . The exact correspondence is stated in Section 3. The operators (D_{i,R})_{P_{i}} define corresponding matrices S_{σ_{i}} . The finite-dimensional unitary operator S_{σ_{i}} is determined by the scattering matrix C_{i}(0) and the involution σ_{i} . Those matrices provide us with the Dirac operators D(S_{σ_{i}}) on S^{1} . The eigenvalues of D(S_{σ_{i}}) , determine the s − values of the operators (D_{i,R})_{P_{i}} . Now, we can formulate the main result of the paper

Theorem 0.2. The following formula holds:

lim_{R→∞} \frac{R^{−2h}·det_{ζ}D_{R}^{2}}{det_{ζ}(D_{1,R})_{P_{1}}^{2}·det_{ζ}(D_{2,R})_{P_{2}}^{2}} = 2^{−ζ_{B^{2}}(0)}·\frac{det_{ζ} \frac{1}{4}D(C_{12})^{2}}{det_{ζ}D(S_{σ_{1}})^{2}·det_{ζ}D(S_{σ_{2}})^{2}} , (0.15)

where h = h_{M} − h_{1} − h_{2} = dim(L_{1} ∩ L_{2}) − dim(L_{1} ∩ ker(σ_{1} − 1)) − dim(L_{2} ∩ ker(σ_{2} − 1)) .

Remark 0.3. Let us observe that we can choose involutions σ_{i} in such a way that h = 0.

We study the relative ζ-function and its derivative to prove Theorem 0.2. We define

ζ_{rel,R}(s) := \frac{1}{Γ(s)} ∫_{0}^{∞} t^{s−1}[T r(e^{−tD_{R}^{2}} − e^{−t(D_{1,R})_{P_{1}}^{2}} − e^{−t(D_{2,R})_{P_{2}}^{2}}) − h] dt . (0.16)

We decompose ζ_{rel,R}(s) into two parts

ζ_{s}^{R}(s) = \frac{1}{Γ(s)} ∫_{0}^{R^{2−ε}}(·) dt , ζ_{l}^{R}(s) = \frac{1}{Γ(s)} ∫_{R^{2−ε}}^{∞}(·) dt .

The derivatives of ζ_{s}^{R}(s), ζ_{l}^{R}(s) at s = 0 give the small and large time contribution.

In Section 1 we deal with the small time contribution and prove the following proposition

Proposition 0.4.

lim_{R→∞}(\frac{d}{ds}ζ_{s}^{R}(s)|_{s=0} − h(γ − (2 − ε)·ln R)) = ln 2·ζ_{B^{2}}(0) . (0.17)

This gives the “small time contribution” and, as a result, the first factor in the right side of (0.15).

In Section 2 we obtain the description of the small eigenvalues. We follow [23] and use scattering theory in order to get a description of the s − values of the D_{R} and the boundary problems (D_{i,R})_{P_{i}} , which allows us to make a comparison of the eigenvalues of those operators with the eigenvalues of the corresponding operators over S^{1} . This is the central part of this paper.

In Section 3 we use the results of Section 2 to show that, in the adiabatic limit, the large time contribution to the quotient (0.11) is equal to

\frac{det_{ζ} \frac{1}{4}D(C_{12})^{2}}{det_{ζ}D(S_{σ_{1}})^{2}·det_{ζ}D(S_{σ_{2}})^{2}} .

This is the second factor on the right side of (0.15).

The e − values make their presence via the factor R^{−2h} on the left side of (0.15).

The last two Sections deal with the Lesch-Wojciechowski formulas. In Section 4 we review the formula for the η-invariant and offer a new proof based on the method used in order to prove Theorem 0.2. This proof is much more complicated than the original presented in [20]. However, it is a nice illustration of the differences we encounter when we deal with ζ-determinant instead of the η-invariant. In the last Section we prove the corresponding formula for the ζ-determinant of the Dirac Laplacian. The nice feature of the formula is that the ratio of the ζ-determinants is expressed as the ratio of the determinants of the finite-dimensional matrices. We refer to Section 5 for details.

Acknowledgements. The authors want to express their gratitude to Werner Müller for his helpful comments on the content of this paper and for the more general remarks on the theory of the ζ-determinant. It is fair to say that the present paper would not exist without the fundamental results of his work [23]. Part of this work was done during the first author’s stay at ICTP. He wishes to express his thanks to ICTP for the financial support and hospitality.

1. Small Time Contribution

In this Section we compute

lim_{R→∞} \frac{d}{ds}ζ_{s}^{R}(s)|_{s=0} .

This is done in two steps. First, we use Duhamel’s Principle and Finite Propagation Speed Property of the Wave Operator to show that we can reduce the problem to computations on the cylinder. Then, we perform the explicit calculations on the cylinder. Both parts are fairly standard. The cylinder contribution have been recently computed in [27]. Therefore, we only discuss the details of the reduction scheme and refer to [27] for the explicit computation on the cylinder.

Let E_{R}(t; x, y) denote the kernel of the operator e^{−tD_{R}^{2}} . We introduce the specific parametrix for E_{R}(t; x, y) , which fits our main purpose to “localize” the contribution coming from the cylinder [−R, R]×Y and the interior of M_{R} . In fact, the interesting point here is that we use E_{R}(t; x, y) to construct this parametrix. Let E_{cyl}(t; x, y) denote the kernel of the operator e^{−t(−∂_{u}^{2}+B^{2})} on the infinite cylinder (−∞, +∞) × Y . We introduce a smooth, increasing function ρ(a, b) : [0, ∞) → [0, 1] equal to 0 for 0 ≤ u ≤ a and equal to 1 for b ≤ u . We use ρ(a, b)(u) to define

ϕ_{1} = 1 − ρ(\frac{5}{7}R, \frac{6}{7}R) , ψ_{1} = 1 − ψ_{2}

and

ϕ_{2} = ρ(\frac{1}{7}R, \frac{2}{7}R) , ψ_{2} = ρ(\frac{3}{7}R, \frac{4}{7}R).

We extend these functions to symmetric functions on the whole real line. These functions are constant outside the interval [−R, R] and we use them to define the corresponding functions on a manifold M_{R} . Now, we define Q_{R}(t; x, y) a “parametrix” for the kernel E_{R}(t; x, y) using the following formula

Q_{R}(t; x, y) = ϕ_{1}(x)E_{cyl}(t; x, y)ψ_{1}(y) + ϕ_{2}(x)E_{R}(t; x, y)ψ_{2}(y) . (1.1)

It follows from Duhamel’s Principle that

E_{R}(t; x, y) = Q_{R}(t; x, y) + (E_{R} ∗ C_{R})(t; x, y) , (1.2)

where E_{R} ∗ C_{R} is a convolution given by

(E_{R} ∗ C_{R})(t; x, y) = ∫_{0}^{t} ds ∫_{M_{R}} dz{E_{R}(s; x, z)C_{R}(t − s; z, y)} ,

and the correction term C_{R}(t; x, y) is given by the formula

C_{R}(t; x, y) = −\frac{∂^{2}ϕ_{1}}{∂u^{2}} (x)E_{cyl}(t; x, y)ψ_{1}(y) − \frac{∂ϕ_{1}}{∂u} (x)\frac{∂E_{cyl}}{∂u} (t; x, y)ψ_{1}(y)

−\frac{∂^{2}ϕ_{2}}{∂u^{2}} (x)E_{R}(t; x, y)ψ_{2}(y) − \frac{∂ϕ_{2}}{∂u} (x)\frac{∂E_{R}}{∂u} (t; x, y)ψ_{2}(y).

The following elementary Lemma follows from the construction of Q_{R}(t; x, y) .

Lemma 1.1. The error term C_{R}(t; x, y) is equal to 0 outside of the cylinder [−\frac{6}{7}R, \frac{6}{7}R] × Y . Moreover it is equal to 0 if the distance between x and y is smaller than R/7 .

Now, following Cheeger, Gromov and Taylor (see [11]; see also Section 3 of [10]), we use the Finite Propagation Speed Property for the Wave Operator. The technique introduced in [11] allows us to compare the heat kernels determined by the operators D_{R}^{2} , (D_{1})_{P_{1}}^{2} and (D_{2})_{P_{2}}^{2} with the heat kernel of the operator −∂_{u}^{2} + B^{2} on the cylinder (−∞, +∞) × Y . We describe the case we need in our work. Let N_{1} and N_{2} be Riemannian manifolds of dimension n and S_{i} be spinors bundle with Dirac operators D_{i} . Assume that there exists a decomposition N_{i} = K_{i} ∪ U_{i} , where U_{i} is an open subset of N_{i} . Moreover, we assume that there exists an isometry h : U_{1} → U_{2} covered by the unitary bundle isomorphism Φ_{h} : S_{1}|_{U_{1}} → S_{2}|_{U_{2}} , which intertwines Dirac operators D_{1}|_{U_{1}} and D_{2}|_{U_{2}} . We identify

U ≅ U_{1} ≅ U_{2} ,

so that M_{1} and M_{2} have a common open subset U . Let E_{i}(t; x, y) denote the kernel of the operator e^{−tD_{i}^{2}} . Then we have the following estimate on the difference of the heat kernels on U Theorem 1.2. (see [11].) Let x, y ∈ U and let r = min(d(x, K_{1}), d(y, K_{1}), d(x, K_{2}), d(y, K_{2})) . Then there exist positive constants c_{1}, c_{2} , such that

∥E_{1}(t; x, y) − E_{2}(t; x, y)∥ ≤ c_{1}e^{−c_{2} \frac{r^{2}}{t}} . (1.3)

Remark 1.3. The same technique also gives estimate (1.3) for the derivatives of the kernels in the normal direction.

In our situation N_{1} = M_{R} (or M_{i,R}) and N_{2} is the cylinder (−∞, +∞)×Y . Note that E_{cyl}(t; x, y) , the kernel of the heat operator defined by operator D_{N_{2}}^{2} = −∂_{u}^{2} + B^{2} , satisfies the standard estimate. More precisely, for small time we have

∥E_{cyl}(t; (u, w), (v, z))∥ ≤ c_{1}t^{− \frac{n}{2}} e^{−c_{3} \frac{(u−v)^{2}}{t}} , (1.4)

where u, v ∈ R and w, z ∈ Y . For large t , the operator e^{−tB^{2}} approaches π_{ker B} , the orthogonal projection onto ker B . It follows from the Sobolev imbedding theorem that π_{ker B}(x, y) , the kernel of this projection, is uniformly bounded on Y (see for instance (1.5) in [13]). Hence, for large time,

the kernel of e^{−tB^{2}} is bounded by a constant, not by the factor c_{1}·e^{−c_{2} \frac{d^{2}(w,z)}{t}}.

We are going to use (1.3) and (1.4) to estimate the error term (E_{R} ∗ C)(t; x, y) . Lemma 1.1 and Theorem 1.2 gives the following estimate

Proposition 1.4. There exist positive constants c_{1} , c_{2} such that the following estimate holds for any ((u, w), (v, z)) ∈ supp C_{R}(t; ·, ·)

∥C_{R}(t; (u, w), (v, z))∥ ≤ c_{1}e^{−c_{2} \frac{R^{2}}{t}} , ∥E_{R}(t; (u, w), (v, z))∥ ≤ c_{1}e^{−c_{2} \frac{R^{2}}{t}} . (1.5)

Proof. We have

∥E_{R}(t; (u, w), (v, z))∥ ≤ ∥E_{cyl}(t; (u, w), (v, z))∥
+∥E_{R}(t; (u, w), (v, z)) − E_{cyl}(t; (u, w), (v, z))∥ .

It is obvious that in the case (u, w), (v, z) ∈ supp C_{R}(t; ·, ·) , both summands on the right side satisfy estimate (1.5) for some positive constants c_{1} and c_{2} . There is no problem with getting the same estimate for the normal derivatives of the kernels in the normal direction (see (1.3) and Remark 1.3), which takes care of the estimates for the kernel C_{R}(t; (u, w), (v, z)) .

Now, we are ready to prove the main technical result of this section

Theorem 1.5.

lim_{R→∞} ∫_{0}^{R^{2−ε}} \frac{dt}{t} ∫_{M_{R}} tr (E_{R} ∗ C_{R})(t; x, x)dx = 0 . (1.6)

Proof. The Theorem follows from estimate (1.5), which implies

|tr (E_{R} ∗ C_{R})(t; x, x)| ≤ ∥(E_{R} ∗ C_{R})(t; x, x)∥ ≤

∫_{0}^{t} ds ∫_{M_{R}} ∥E_{R}(s; x, z)C_{R}(t − s; z, x)∥dz =

∫_{0}^{t} ds ∫_{[− \frac{6}{7} R, \frac{6}{7} R]×Y} ∥E_{R}(s; x, z)C_{R}(t − s; z, x)∥dz =

c_{1}^{2}· ∫_{0}^{t} ds ∫_{[− \frac{6}{7} R, \frac{6}{7} R]×Y} e^{−c_{2} \frac{R^{2}}{s}} e^{−c_{2} \frac{R^{2}}{t−s}} dz ≤

c_{3}R· ∫_{0}^{t} e^{−c_{2} \frac{tR^{2}}{s(t−s)}} ds ≤ c_{3}R· ∫0^{\frac{t}{2}} e^{−2c_{2} \frac{R^{2}}{s}} ds ≤ c_{3}R·\frac{t}{2}e^{−4c_{2} \frac{R^{2}}{t}} ,

where the last estimate is a consequence of the elementary inequality

∫_{0}^{t} e^{− \frac{c}{s}} ds ≤ te^{− \frac{c}{t}} .

We have proved

|tr (E_{R} ∗ C_{R})(t; x, x)| ≤ c_{4}t·e^{−c_{5} \frac{R^{2}}{t}} , (1.7)

which allows us to estimate as follows

| ∫_{0}^{R^{2−ε}} \frac{dt}{t} ∫_{M_{R}} tr (E_{R} ∗ C_{R})(t; x, x)dx| ≤

c_{6}R· ∫_{0}^{R^{2−ε}} e^{−c_{5} \frac{R^{2}}{t}} ≤ c_{6}R^{3−ε}e^{−c_{5}R^{ε}} ,

and (1.6) follows.

The proof of (1.6) for the operator (D_{i,R})_{P_{i}} on a manifold M_{i,R} is carried out in exactly the same manner and we obtain the main result of this Section

Theorem 1.6. We have the following “adiabatic limit” formula

lim_{R→∞} ∫_{0}^{R^{2−ε}} \frac{1}{t}·(T r e^{−tD_{R}^{2}} − T r e^{−t(D_{1,R})_{P_{1}}^{2}} − T r e^{−t(D_{2,R})_{P_{2}}^{2}})dt = ln 2·ζ_{B^{2}}(0) . (1.8)

Proof. Estimate (1.6) gives

lim_{R→∞} ∫_{0}^{R^{2−ε}} \frac{1}{t}·(T r e^{−tD_{R}^{2}} − T r e^{−t(D_{1,R})_{P_{1}}^{2}} − T r e^{−t(D_{2,R})_{P_{2}}^{2}})dt =

lim_{R→∞} ∫_{0}^{R^{2−ε}} \frac{dt}{t}·[∫_{M_{1,R}}(tr Q_{R}(t; x, x) − tr Q_{1,R}(t; x, x))dx+

∫_{M_{2,R}}(tr Q_{R}(t; x, x) − tr Q_{2,R}(t; x, x))dx] .

Now, the interior contributions to the different parametrices, all determined by the kernel E_{R}(t; x, y) , cancel out and we are left with the cylinder contribution only. An explicit computation of this contribution was presented in [27].

We need the next result to finish the analysis of the “small time contribution”.

Proposition 1.7. We have the following equality

(ζ_{s}^{R})′(0) = ∫_{0}^{R^{2−ε}} t^{−1}T r(e^{−tD_{R}^{2}}−e^{−t(D_{1,R})_{P_{1}}^{2}}−e^{−t(D_{2,R})_{P_{2}}^{2}})dt+h(γ−(2−ε)ln R) .

Proof. First, we separate ζ_{s}^{R}(0) into two parts

\frac{1}{Γ(s)} ∫_{0}^{R^{2−ε}} t^{s−1}T r(e^{−tD_{R}^{2}} − e^{−t(D_{1,R})_{P_{1}}^{2}} − e^{−t(D_{2,R})_{P_{2}}^{2}}) dt − \frac{h}{Γ(s)} ∫_{0}^{R^{2−ε}} t^{s−1}dt .

We have the elementary identity

\frac{d}{ds}\{\frac{h}{Γ(s)} ∫_{0}^{R^{2−ε}} t^{s−1}dt\}|_{s=0} = −hγ + h(2 − ε) log R .

The derivative

\frac{d}{ds}\{\frac{1}{Γ(s)} ∫_{0}^{R^{2−ε}} t^{s−1}T r(e^{−tD_{R}^{2}} − e^{−t(D_{1,R})_{P_{1}}^{2}} − e^{−t(D_{2,R})_{P_{2}}^{2}})dt\}|_{s=0}

is obtained in the standard way following computation of the ζ′_{D^{2}}(0) (see [32], or the introduction to [27]). There is no problem with the pole at s = 0 , because the constant term in the asymptotic expansion of the

T r(e^{−tD_{R}^{2}} − e^{−t(D_{1,R})_{P_{1}}^{2}} − e^{−t(D_{2,R})_{P_{2}}^{2}})

disappears. Indeed, it is a linear combination of the corresponding terms in the expansion of T r e^{−tD_{R}^{2}} , T r e^{−t(D_{1,R})_{P_{1}}^{2}} and T r e^{−t(D_{2,R})_{P_{2}}^{2}} , but all of them are equal to 0 . The vanishing of this coefficient for the Dirac operator on a closed odd-dimensional manifold is a classical result (see for instance [14]). The result for the boundary problems considered in this paper has been recently established in the Appendix to the paper [27]. This ends the proof of the Proposition.

The proof of Proposition 0.4 is complete.

2. small eigenvalues and scattering matrices

In this section we investigate the relation between the s − values of the operators D_{R} , (D_{1,R})_{P_{1}} and (D_{2,R})_{P_{2}} and the scattering matrices C_{1}(λ), C_{2}(λ) determined by the operators D_{i,∞} on M_{i,∞} . We refer to [23] (section 4 and Section 8) for a more detailed exposition of the elements of Scattering Theory that we use in this paper.

Let μ_{1} denote the lowest positive eigenvalue of the tangential operator B . For any f ∈ ker(B) and λ ∈ ℂ − {(−∞, −μ_{1}] ∪ [μ_{1}, +∞)} we define E(f, λ) a generalized eigensection of D_{1,∞} over M_{1,∞} determined by the couple {f, λ} (see [23] Section 4). It is a smooth section of the bundle S over M_{1,∞} , which satisfies

DE(f, λ) = λE(f, λ) .

The section E(f, λ) has the following form over [0, ∞) × Y,

E(f, λ) = e^{−iλu}(f − iGf) + e^{iλu}C(λ)(f − iGf) + θ(f, λ)

where θ is a square integrable section such that, for each u , θ(f, λ, (u, ·)) is orthogonal to ker(B). The operator C(λ) : ker B → ker B is regular and unitary and equals the “Scattering Matrix” for |λ| < μ_{1} . Moreover, we have

C(λ)C(−λ) = Id , C(λ)G = −GC(λ) .

In particular

C(0)^{2} = Id , C(0)G = −GC(0) .

Therefore C(0) gives a distinguished unitary involution of ker(B). In fact, the space L_{1} ⊂ ker B , which appeared in the Introduction is equal to the +1 eigenspace of C(0)

L_{1} = ker(C(0) − 1) .

Now we study the s − values of the operator D_{R} over M_{R} . We introduce a manifold

M̄_{R} = M_{1,R} ⊔ M_{2,R} .

The boundary of M̄_{R} is equal to two copies of Y . We consider D_{R} the operator on a closed manifold M_{R} , as the Dirac operator on M̄_{R} , which satisfies the transmission boundary condition. In particular if λ = λ(R) is an s − value of D_{R} and ϕ denotes the corresponding eigensection, then ϕ satisfies the “transmission boundary condition”

ϕ|_{∂M_{1,R}} = ϕ|_{∂M_{2,R}} .

We refer to [5] for a more detailed discussion of the transmission problem (see also [26]). We want to warn the reader that, although we discuss transmission boundary condition so it would be natural to consider M_{1,R} as M_{1,R} = M_{1} ∪ [−R, 0] × Y , we still parametrize the cylindrical parts as

M_{1,R} = M_{1} ∪ [0, R] × Y , M_{2,R} = M_{2} ∪ [−R, 0] × Y .

In the following we assume that

|λ| < R^{−κ} , ∥ϕ∥ = 1 , (2.1)

where 0 < κ < 1 . In particular we have λ ≤ \frac{μ_{1}}{2} . The section ϕ can be represented in the following way on [0, R] × Y ⊂ M_{1,R}

ϕ = e^{−iλu}ψ_{1} + e^{iλu}ψ_{2} + ϕ_{1}

where ψ_{1} ∈ ker(G − i), ψ_{2} ∈ ker(G + i) and ϕ_{1} has the following form

ϕ_{1}(u, y) = ∑_{k} a_{k}e^{−\sqrt{μ_{k}^{2}−λ^{2}}u}\{(μ_{k} + \sqrt{μ_{k}^{2} − λ^{2}})ϕ_{k}(y) + λGϕ_{k}(y)\}

+ ∑_{k} b_{k}e^{\sqrt{μ_{k}^{2}−λ^{2}}u}\{(λϕ_{k}(y) + (μ_{k} + \sqrt{μ_{k}^{2} − λ^{2}})Gϕ_{k}(y)\}

where {μ_{k}}_{k∈ℕ} is the set of positive eigenavlues of B and ϕ_{k} denotes the eigensection of B corresponding to μ_{k} . We will use the following estimate on the L^{2}-norm of ϕ_{1} in the y-direction on the cylinder (see Lemma 2.1 in [35])

Lemma 2.1. There exist positive constants c_{1}, c_{2} such that

∥ϕ_{1}|_{{R}×Y} ∥ ≤ c_{1}e^{−c_{2}R} .

We also have the following result (see [23] Proposition 8.14, see also Theorem 2.2 in [35]).

Proposition 2.2. If the eigenvalue λ(R) is not bounded away from 0 , then the zero−eigenmode, e^{−iλu}ψ_{1}+e^{iλu}ψ_{2} , of ϕ is a non-trivial section of S|Y.

Assume that ψ_{1} is non-zero. Choose ψ ∈ L_{1} such that ψ_{1} = ψ − iGψ . Then the generalized eigensection E(ψ, λ) associated to ψ over M_{1,∞} has the following expression

E(ψ, λ) = e^{−iλu}(ψ − iGψ) + e^{iλu}C_{1}(λ)(ψ − iGϕ) + θ(ψ, λ)

over [0, ∞) × Y . Following [23], we introduce F = ϕ|_{M_{1,R}} − E(ψ, λ)|_{M_{1,R}}. Green’s Theorem gives

0 = ⟨DF, F ⟩_{M_{1,R}} − ⟨F, DF ⟩_{M_{1,R}} = ∫_{Y} ⟨GF, F ⟩dy.

On the other hand, Lemma 2.1 shows that

∫_{Y} ⟨GF, F ⟩dy = −i∥ C_{1}(λ)ψ_{1} − ψ_{2}∥^{2} + O(e^{−c_{3}R})

for some positive constant c_{3}. This produces the estimate

∥ C_{1}(λ)ψ_{1} − ψ_{2}∥^{2} < e^{−c_{3}R} (2.2)

(compare [23]). It follows from (2.2) that ψ_{2} is non-trivial section if and only if ψ_{1} is non-trivial.

We repeat the argument for the eigensection ϕ of D_{R} over [−R, 0] × Y ⊂ M_{2,R} . We have

ϕ = e^{iλu}ψ_{3} + e^{−iλu}ψ_{4} + ϕ_{2}

where ψ_{3} ∈ ker(G + i), ψ_{4} ∈ ker(G − i) and ϕ_{2} is orthogonal to ker(B). Choose ψ ∈ L_{2} such that ψ_{3} = ψ + iGψ . Then the generalized eigensection E(ϕ, λ) associated to ϕ over M_{2,∞} can be written as

E(ψ, λ) = e^{iλu}(ψ + iGψ) + e^{−iλu}C_{2}(λ)(ψ + iGψ) + θ(ψ, λ) .

We obtain the expected estimate

∥ C_{2}(λ)ψ_{3} − ψ_{4}∥^{2} < e^{−c_{3}R} . (2.3)

Transmission boundary condition over Y ⊔ Y = ∂M_{1,R} ⊔ ∂M_{2,R} implies the equality

e^{−iλR}ψ_{1} + e^{iλR}ψ_{2} = e^{−iλR}ψ_{3} + e^{iλR}ψ_{4}

so that

ψ_{1} = e^{2iλR}ψ_{4}, ψ_{3} = e^{2iλR}ψ_{2} . (2.4)

By (2.2), (2.3), (2.4), we have

∥e^{2iλR}C_{1}(λ)ψ_{4} − ψ_{2}∥^{2} < e^{−c_{3}R} , ∥e^{2iλR}C_{2}(λ)ψ_{2} − ψ_{4}∥^{2} < e^{−c_{3}R} .

We combine those inequalities and obtain

∥e^{4iλR}C_{1}(λ) ∘ C_{2}(λ)ψ_{2} − ψ_{2}∥^{2} < e^{−c_{4}R} , (2.5)

∥e^{4iλR}C_{2}(λ) ∘ C_{1}(λ)ψ_{4} − ψ_{4}∥^{2} < e^{−c_{4}R} (2.6)

for some positive constant c_{4}.

There exist a positive number δ such that the unitary operators C_{1}(λ), C_{2}(λ) are the analytic functions of λ for λ ∈ (−δ, δ). We define operator C_{12}(λ) as

C_{12}(λ) = C_{1}(λ) ∘ C_{2}(λ) : ker(G + i) → ker(G + i) .

The operator C_{12}(λ) is a unitary operator and it is an analytic function of λ for λ ∈ (−δ, δ). Therefore, there exist real analytic functions α_{j}(λ) of λ ∈ (−δ, δ) such that exp(iα_{j}(λ)) are the corresponding eigenvalues of C_{12}(λ) for λ ∈ (−δ, δ). We fix the constant δ_{1}, 0 < δ_{1} < δ, and define

r_{j} = max_{λ∈(−δ_{1},δ_{1})}|α′_{j}(λ)| .

The function f(λ) = 4Rλ + α_{j}(λ) is strictly increasing on (−δ_{1}, δ_{1}) , if we

assume that R ≥ r_{j} . Choose R_{0} ≥ max(r_{j}, δ_{1}^{− \frac{1}{κ}}). For R ≥ R_{0} and k ∈ ℤ, there exists at most one solution of the equation

4Rλ + α_{j}(λ) = 2πk, |λ| < R^{−κ} . (2.7)

Let k_{j,max} = k_{j,max}(R) be the maximal k for which (2.7) has a solution. Then

|k_{j,max}| ≤ R^{1−κ} (2.8) for R ≥ R_{0}. We now use (2.5) to show that λ(R) satisfies

4Rλ(R) + α_{j}(λ(R)) = 2πk + O(e^{−cR}) , |λ(R)| < R^{−κ} . (2.9)

We have proved the following result

Proposition 2.3. There exists R_{0} such that for R > R_{0} the s−value λ(R) of D_{R} satisfies

4Rλ(R) + α_{j}(λ(R)) = 2πk + O(e^{−cR}) (2.10)

for an integer k with |k| < R^{1−κ} , where exp(iα_{j}(λ)) is an eigenvalue of the unitary operator C_{1}(λ) ∘ C_{2}(λ) : ker(G + i) → ker(G + i) .

The map C_{12}(0) : ker(G+i) → ker(G+i) is not the identity map. However, we have the following proposition

Proposition 2.4. The map C_{12}(0) is equal to the identity when restricted to the subspace I_{+}(L_{1} ∩ L_{2}) where I_{+} = G − i : ker(B) → ker(G + i).

Proof. The subspace L_{i} is the +1-eigenspace of the operator C_{i}(0) . Hence it follows from the definition, that the restriction of the map C_{1}(0) ∘ C_{2}(0) to L_{1} ∩ L_{2} is the identity map. The claim follows easily from the following diagram

L_{1} ∩ L_{2} ⟶^{I_{+}} ker(G + i)
C_{1}(0)∘C_{2}(0)↓  ↓^{C_{1}(0)∘C_{2}(0)}
L_{1} ∩ L_{2} ⟶^{I_{+}} ker(G + i) .

It follows from Proposition 2.4 that the multiplicity of the eigenvalue +1 of the operator C_{12}(0) is equal to dim(L_{1} ∩ L_{2}) .

Moreover, (2.10) implies that the corresponding eigenvalues of the operator D_{R} are, in fact, e − values. Here we mean eigenvalues λ(R) , which corresponds to α_{j}(0) = 0 and k = 0 in the formula (2.10) in the Proposition 2.3. This provides another proof of the result, originally shown in [8] Theorem 2.5. The dimension of the space of eigensections corresponding to e − values which are not determined by ker_{L^{2}}(D_{i,∞}) for i = 1, 2 is equal to the dimension of the space L_{1} ∩ L_{2}.

The corresponding analysis for the operators (D_{i,R})_{P_{i}} for i = 1, 2 follows in the same manner. The case of the boundary condition determined by C_{1}(0) was studied in [23], but it is not difficult to show, that everything works well for any involution σ on ker(B). Let ϕ be an eigensection of (D_{1,R})_{P_{1}} , which corresponds to the s − value λ = λ(R) . That is,

D_{1,R}ϕ = λϕ with |λ| < R^{−κ}

for some fixed κ with 0 < κ < 1 and

P_{1}(ϕ|_{{R}×Y}) = 0. (2.11)

The section ϕ can be represented in the following way on [0, R] × Y ⊂ M_{1,R}

ϕ = e^{−iλu}ψ_{1} + e^{iλu}ψ_{2} + ϕ_{1}

where ψ_{1} ∈ ker(G − i), ψ_{2} ∈ ker(G + i) and ϕ_{1} is smooth L^{2} section orthogonal to ker(B). By (2.11), we have

e^{−2iλR}P_{1}(ψ_{1}) = −P_{1}(ψ_{2}) . (2.12)

The next result corresponds to Proposition 2.2.

Proposition 2.6. If the eigenvalue λ(R) is not bounded away from 0 , then the zero − eigenmode, e^{−iλu}ψ_{1} + e^{iλu}ψ_{2} , of ϕ is a non-trivial section on the cylinder. Moreover, the following formula holds

∥C_{1}(λ)ψ_{1} − ψ_{2}∥^{2} < e^{−c_{3}R} . (2.13)

We define

I_{−} = G + i : ker(B) → ker(G − i)

I_{σ_{1}} = I_{−}|_{ker(σ_{1}+1)} : ker(σ_{1} + 1) → ker(G − i) ,

P_{σ_{1}} = \frac{1}{2}(σ_{1} − 1) : ker(B) → ker(σ_{1} + 1)

and

S_{σ_{1}}(λ) = −P_{σ_{1}} ∘ C_{1}(λ) ∘ I_{σ_{1}} : ker(σ_{1} + 1) → ker(σ_{1} + 1) .

We know that there exists a unique ϕ ∈ ker(σ_{1} +1) such that ψ_{1} = ϕ−iGϕ. Using the equation (2.12) and (2.13), we have

∥e^{2iλR}S_{σ_{1}}(λ)ϕ − ϕ∥^{2} < e^{−c_{3}R}

for ϕ ∈ ker(σ_{1}+1). Similarly we define S_{σ_{2}} using C_{2}(0), I_{+} instead of C_{1}(0), I_{−} by

S_{σ_{2}}(λ) := −P_{σ_{2}} ∘ C_{2}(λ) ∘ I_{σ_{2}} : ker(σ_{2} + 1) → ker(σ_{2} + 1) ,

where

I_{σ_{2}} := I_{+}|_{ker(σ_{2}+1)} : ker(σ_{2} + 1) → ker(G + i) .

We also have

∥e^{2iλR}S_{σ_{2}}(λ)ϕ − ϕ∥^{2} < e^{−c_{3}R}

for ϕ ∈ ker(σ_{2} + 1) . We repeat the argument we used to prove the Proposition 2.3 and obtain

Proposition 2.7. There exists R_{0} such that for R > R_{0} the s − value λ = λ(R) of (D_{i,R})_{P_{i}} satisfies

2Rλ(R) + β_{j}(λ(R)) = 2πk + O(e^{−cR}) (2.14)

for an integer k with |k| < R^{1−κ} , where exp(iβ_{j}(λ)) denotes an eigenvalue of the unitary operator S_{σ_{i}}(λ) = −P_{σ_{i}} ∘C_{i}(λ)∘I_{σ_{i}} : ker(σ_{i}+1) → ker(σ_{i}+1) .

The restriction of S_{σ_{i}}(0) to ker(σ_{i} + 1) ∩ ker(C_{i}(0) + 1) is equal to Id , so that the number of β_{j}(0)’s with β_{j}(0) = 0 is the same as

dim(ker(σ_{i} + 1) ∩ ker(C_{i}(0) + 1)) = dim(ker(σ_{i} − 1) ∩ ker(C_{i}(0) − 1)) .

We have the following proposition Proposition 2.8. There is a natural isomorphism

ker((D_{i,R})_{P_{i}}) ≅ ker_{L^{2}}(D_{i,∞}) ⊕ ker(σ_{i} − 1) ∩ L_{i}

for i = 1, 2.

Proof. Let ϕ ∈ ker(D_{1,R})_{P_{1}}. Section ϕ satisfies G(\frac{d}{du} + B)ϕ = 0 on the cylinder [0, R] × Y, and it has the following representation when restricted to this cylinder

ϕ = ϕ_{0} + ∑_{μ_{j}>0} c_{j}e^{−μ_{j}u}ϕ_{j}

where (σ_{i}−1)(ϕ_{0}) = 0 . We follow [23] (see also section 6 of [13]) and use this expansion to extend ϕ to a smooth section ϕ̃ on M_{1,∞} satisfying D_{1,∞}ϕ̃ = 0. This means that ϕ̃ belongs to the extended L^{2}-kernel of D_{1,∞}. Hence ϕ_{0} is an element of L_{1} . Let E(ϕ_{0}, λ) be the generalized eigensection attached to ϕ_{0}. Then ϕ̃ − \frac{1}{2}E(ϕ_{0}, 0) is square integrable and D_{1,∞}(ϕ̃ − \frac{1}{2}E(ϕ_{0}, 0)) = 0 , and the map

ϕ ⟶ (ϕ̃ − \frac{1}{2}E(ϕ_{0}, 0), ϕ_{0})

gives the expected isomorphism.

It follows from Proposition 2.8 that the number of elements in the set {β_{j}(0) : eigenvalue of S_{σ_{i}}(0) | β_{j}(0) = 0 } is equal to the dimension of the subspace of ker((D_{i,R})_{P_{i}}), which is not determined by ker_{L^{2}}(D_{i,∞}) for i = 1, 2 .

Now we define our model operator. Let C : W → W denote a unitary operator acting on d-dimensional vector space W with the eigenvalues e^{iα_{j}} for j = 1, ⋯, d. We introduce the operator D(C),

D(C) := −i\frac{1}{2} \frac{d}{du} : C^{∞}(S^{1}, E_{C}) → C^{∞}(S^{1}, E_{C})

where E_{C} is the flat vector bundle over S^{1} = ℝ/ℤ defined by the holonomy \overline{C} , the complex conjugate of C . The spectrum of D(C) is equal to

\{ πk − \frac{1}{2}α_{j} | k ∈ ℤ, j = 1, ⋯, d \} . (2.15)

The operators C_{12} = C_{1}(0) ∘ C_{2}(0) , S_{σ_{1}} := S_{σ_{1}}(0) and S_{σ_{2}} := S_{σ_{2}}(0) are the unitary operators acting on the finite dimensional vector spaces. Hence we can define self-adjoint, elliptic operators D(C_{12}), D(S_{σ_{1}}), D(S_{σ_{2}}) on S^{1}.

Proposition 2.9. There exists R_{0} such that for R > R_{0}, the s − value λ(R) of D_{R} (or (D_{i,R})_{P_{i}}) satisfies

2Rλ(R) = λ_{k} + O(\frac{1}{R^{κ}}) (or Rλ(R) = λ_{k} + O(\frac{1}{R^{κ}})) , (2.16)

where λ_{k} is an eigenvalue with |λ_{k}| ≤ R^{1−κ} of D(C_{12}) (or D(S_{σ_{i}})).

Proof. We know that α_{j}(λ) is an analytic function of λ for λ ∈ (−δ, δ) , which has the following expansion

α_{j}(λ) = α_{j0} + α_{j1}λ + α_{j2}λ^{2} + · · · .

It follows from (2.15) that the eigenvalues of D(C_{12}) are of the form πk−\frac{1}{2}α_{j0} . Equation (2.10) implies

2Rλ(R) = (πk − \frac{1}{2}α_{j0}) − \frac{1}{2}λ(R)(α_{j1} + α_{j2}λ(R) + ⋯) + O(e^{−cR})

which gives the (2.16) since |λ(R)| ≤ R^{−κ}.

3. Large time contribution

In this section, we prove the following proposition

Proposition 3.1.

lim_{R→∞}{∫_{R^{2−ε}}^{∞} t^{−1}[T r(e^{−tD_{R}^{2}} − e^{−t(D_{1,R})_{P_{1}}^{2}} − e^{−t(D_{2,R})_{P_{2}}^{2}}) − h] dt + h(γ + ε·ln R) (3.1)

−\frac{d}{ds}\{\frac{1}{Γ(s)} ∫_{0}^{∞} t^{s−1}[T r(e^{−t \frac{1}{4} D(C_{12})^{2}}−e^{−tD(S_{σ_{1}})^{2}}−e^{−tD(S_{σ_{2}})^{2}})−h]dt\} |_{s=0}\} = 0

where h = dim(L_{1} ∩ L_{2}) − dim(L_{1} ∩ ker(σ_{1} − 1)) − dim(L_{2} ∩ ker(σ_{2} − 1)) .

Proposition 3.1 shows that the large time contribution to the adiabatic decomposition formula for the ζ-determinant is equal to

\frac{det_{ζ} \frac{1}{4}D(C_{12})^{2}}{det_{ζ}D(S_{σ_{1}})^{2}·det_{ζ}D(S_{σ_{2}})^{2}} .

We start with the following result which corresponds to the Proposition 1.7 in our study of the “small time contribution”

Proposition 3.2. We have the following formula for the small time contribution defined by the 1-dimensional operators

lim_{R→∞}(\frac{d}{ds}\{\frac{1}{Γ(s)} ∫_{0}^{R^{−ε}} t^{s−1}[T r(e^{−t \frac{1}{4} D(C_{12})^{2}} −e^{−tD(S_{σ_{1}})^{2}} −e^{−tD(S_{σ_{2}})^{2}})−h]dt\} |_{s=0}

−h(γ + ε·ln R)) = 0 .

Proof. First, note that there exists a constant c_{1} such that

|T r(e^{−t \frac{1}{4} D(C_{12})^{2}} − e^{−tD(S_{σ_{1}})^{2}} − e^{−tD(S_{σ_{2}})^{2}})| < c_{1}\sqrt{t} . (3.2)

This is due to the fact that D(C_{12})^{2} , D(S_{σ_{1}})^{2} and D(S_{σ_{1}})^{2} are Dirac Laplacians on S^{1} , so the expansion of the trace of the heat operator for each of those Laplacians is of the form

\sqrt{\frac{π}{t}} ·\frac{dim ker B}{2} + O(\sqrt{t}) .

Now a factor of \frac{1}{4} in front of D(C_{12})^{2} provides a factor of 2 in front of \sqrt{\frac{π}{t}} in the expansion for the operator \frac{1}{4}D(C_{12})^{2} . This allows us to estimate

| ∫_{0}^{R^{−ε}} t^{−1}T r(e^{−t \frac{1}{4} D(C_{12})^{2}} − e^{−tD(S_{σ_{1}})^{2}} − e^{−tD(S_{σ_{2}})^{2}}) dt| ≤

≤ c_{2}· ∫_{0}^{R^{−ε}} \frac{dt}{\sqrt{t}} = 2c_{2}·R^{− \frac{ε}{2}} ,

We also have

\frac{d}{ds}\{\frac{h}{Γ(s)} ∫_{0}^{R^{−ε}} t^{s−1}dt\} |_{s=0}= −h(γ + ε · ln R) .

This completes the proof.

It follows from the Proposition 3.2 that equality (3.1) is equivalent to the following equation

lim_{R→∞} ∫_{R^{2−ε}}^{∞} t^{−1}[T r(e^{−tD_{R}^{2}} − e^{−t(D_{1,R})_{P_{1}}^{2}} − e^{−t(D_{2,R})_{P_{2}}^{2}}) − h] dt

= lim_{R→∞} \frac{d}{ds}\{\frac{1}{Γ(s)} ∫_{R^{−ε}}^{∞} t^{s−1}[T r(e^{−t \frac{1}{4} D(C_{12})^{2}}−e^{−tD(S_{σ_{1}})^{2}}−e^{−tD(S_{σ_{2}})^{2}})−h] dt\} |_{s=0}

= lim_{R→∞} ∫_{R^{−ε}}^{∞} t^{−1}[T r(e^{−t \frac{1}{4} D(C_{12})^{2}} − e^{−tD(S_{σ_{1}})^{2}} − e^{−tD(S_{σ_{2}})^{2}}) − h] dt .

We change variables and obtain

∫_{R^{2−ε}}^{∞} t^{−1}[T r(e^{−tD_{R}^{2}} − e^{−t(D_{1,R})_{P_{1}}^{2}} − e^{−t(D_{2,R})_{P_{2}}^{2}}) − h]dt
= ∫_{R^{−ε}}^{∞} t^{−1}[T r(e^{−tR^{2}D_{R}^{2}} − e^{−tR^{2}(D_{1,R})_{P_{1}}^{2}} − e^{−tR^{2}(D_{2,R})_{P_{2}}^{2}}) − h] dt .

Now (3.1) is equivalent to

lim_{R→∞} ∫_{R^{−ε}}^{∞} t^{−1}[T r(e^{−tR^{2}D_{R}^{2}} − e^{−tR^{2}(D_{1,R})_{P_{1}}^{2}} − e^{−tR^{2}(D_{2,R})_{P_{2}}^{2}}) − h]dt (3.3)

= lim_{R→∞} ∫_{R^{−ε}}^{∞} t^{−1}[T r(e^{−t \frac{1}{4} D(C_{12})^{2}} − e^{−tD(S_{σ_{1}})^{2}} − e^{−tD(S_{σ_{2}})^{2}}) − h] dt .

We split

T r(e^{−tR^{2}D_{R}^{2}} − e^{−tR^{2}(D_{1,R})_{P_{1}}^{2}} − e^{−tR^{2}(D_{2,R})_{P_{2}}^{2}})

into two parts T r_{I,R}(e^{−tR^{2}D_{R}^{2}} − e^{−tR^{2}(D_{1,R})_{P_{1}}^{2}} − e^{−tR^{2}(D_{2,R})_{P_{2}}^{2}}) ,

T r_{II,R}(e^{−tR^{2}D_{R}^{2}} − e^{−tR^{2}(D_{1,R})_{P_{1}}^{2}} − e^{−tR^{2}(D_{2,R})_{P_{2}}^{2}})

where T r_{I,R}(·) (T r_{II,R}(·)) is taken over the eigenvalues of R^{2}D_{R}^{2}, R^{2}(D_{1,R})_{P_{1}}^{2} , R^{2}(D_{2,R})_{P_{2}}^{2} which are larger (smaller or equal to) than R^{\frac{1}{2}} . The next Proposition shows that T r_{I,R}(·) can be neglected as R → ∞

DECOMPOSITION OF THE ζ-DETERMINANT AND SCATTERING THEORY 29

Proposition 3.3. We have the following estimate

∫_{R^{−ε}}^{∞} t^{−1}T r_{I,R}(e^{−tR^{2}D_{R}^{2}} − e^{−tR^{2}(D_{1,R})_{P_{1}}^{2}} − e^{−tR^{2}(D_{2,R})_{P_{2}}^{2}})dt ≤ b_{3}e^{−b_{4}R^{\frac{1}{2} −ε}}

for some positive constants b_{3}, b_{4}.

Proof. Let λ_{k_{0}}(R) denote the smallest “large” eigenvalue of D_{R}^{2}, that is, we have

λ_{k_{0}}(R) > \frac{1}{R\sqrt{R}} .

We estimate T r_{I,R}e^{−tR^{2}D_{R}^{2}} , which is the sum of the “large” eigenvalues

T r_{I,R}e^{−tR^{2}D_{R}^{2}} = ∑_{λ_{k}≥ \frac{1}{R\sqrt{R}}} e^{−tR^{2}λ_{k}} = ∑_{λ_{k}≥ \frac{1}{R\sqrt{R}}} e^{−(tR^{2}−1)λ_{k}}e^{−λ_{k}} <

e^{−(tR^{2}−1)λ_{k_{0}}} ∑_{λ_{k}≥ \frac{1}{R\sqrt{R}}} e^{−λ_{k}} < e^{−(tR^{2}−1)λ_{k_{0}}}T r e^{−D_{R}^{2}} <

b_{1}Re^{−(tR^{2}−1) \frac{1}{R\sqrt{R}}} < b_{1}Re^{−b_{2}t\sqrt{R}} ,

for some positive constants b_{1}, b_{2} . Now we have

∫_{R^{−ε}}^{∞} \frac{1}{t}T r_{I,R}e^{−tR^{2}D_{R}^{2}}dt < ∫_{R^{−ε}}^{∞} \frac{1}{t}b_{1}Re^{−b_{2}t\sqrt{R}}dt <

\frac{b_{1}}{b_{2}} R^{\frac{1}{2} +ε} ∫_{b_{2}R^{\frac{1}{2} −ε}}^{∞} e^{−v}dv < b_{3}e^{−b_{4}R^{\frac{1}{2} −ε}} .

The trace T r_{I,R}e^{−tR^{2}(D_{i,R})_{P_{i}}^{2}} for i = 1, 2 is estimated in the same way.

We also split T r(e^{−t \frac{1}{4} D(C_{12})^{2}} − e^{−tD(S_{σ_{1}})^{2}} − e^{−tD(S_{σ_{2}})^{2}}) into two parts T r_{I,R}(e^{−t \frac{1}{4} D(C_{12})^{2}} − e^{−tD(S_{σ_{1}})^{2}} − e^{−tD(S_{σ_{2}})^{2}}) ,

T r_{II,R}(e^{−t \frac{1}{4} D(C_{12})^{2}} − e^{−tD(S_{σ_{1}})^{2}} − e^{−tD(S_{σ_{2}})^{2}}) .

where T r_{I,R}(·) (T r_{II,R}(·)) is taken over the eigenvalues of \frac{1}{4}D(C_{12})^{2}, D(S_{σ_{1}})^{2}, D(S_{σ_{2}})^{2} which are larger (smaller or equal to) than R^{\frac{1}{2}} . The following Proposition corresponds to the Proposition 3.3 and its proof is essentially the same as the proof of Proposition 3.3

Proposition 3.4. There exist positive constants c_{1}, c_{2} such that

∫_{R^{−ε}}^{∞} t^{−1}T r_{I,R}(e^{−t \frac{1}{4} D(C_{12})^{2}} − e^{−tD(S_{σ_{1}})^{2}} − e^{−tD(S_{σ_{2}})^{2}}) ≤ c_{1}e^{−c_{2}R^{\frac{1}{2} −ε}} .

Now we can see that the equality (3.3) is equivalent to

lim_{R→∞}(∫_{R^{−ε}}^{∞} t^{−1}[T r_{II,R}(e^{−tR^{2}D_{R}^{2}} − e^{−tR^{2}(D_{1,R})_{P_{1}}^{2}} − e^{−tR^{2}(D_{2,R})_{P_{2}}^{2}}) − h]dt (3.4)

− ∫_{R^{−ε}}^{∞} t^{−1}[T r_{II,R}(e^{−t \frac{1}{4} D(C_{12})^{2}} − e^{−tD(S_{σ_{1}})^{2}} − e^{−tD(S_{σ_{2}})^{2}}) − h] dt) = 0 .

Equation (3.4) is a consequence of the next result

Proposition 3.5. For sufficiently large R , there exist positive constants c_{6}, c_{7} independent of R and t , such that

|T r_{II,R}(e^{−tR^{2}D_{R}^{2}}) − T r_{II,R}(e^{−t \frac{1}{4} D(C_{12})^{2}})| < c_{6} \frac{t}{R^{\frac{1}{4}}} e^{−c_{7}t} ,

|T r_{II,R}(e^{−tR^{2}(D_{i,R})_{P_{i}}^{2}}) − T r_{II,R}(e^{−tD(S_{σ_{i}})^{2}})| < c_{6} \frac{t}{R^{\frac{1}{4}}} e^{−c_{7}t}

for any positive t.

Proof. We know that the dimension of the kernel of \frac{1}{4}D(C_{12})^{2} is h_{M} , and it follows that the contributions of the e − values and zero eigenvalues cancel out in the difference of the traces

T r_{II,R}(e^{−tR^{2}D_{R}^{2}}) − T r_{II,R}(e^{−t \frac{1}{4} D(C_{12})^{2}}) .

Hence, in the following we neglect those eigenvalues. Now, we use the analysis of s−values developed in Section 2. We fix κ = \frac{3}{4} . It follows from Proposition 2.9 that, for any eigenvalue λ(R) of D_{R} with |λ(R)| < R^{− \frac{3}{4}} there exists an analytic function α(λ) such that

Rλ(R) = λ_{j} + \frac{1}{4}λ(R)α(λ(R)) + O(e^{−cR})

where λ_{j} is an eigenvalue of \frac{1}{2}D(C_{12}) with |λ_{j}| < R^{\frac{1}{4}} . Therefore, there exists a function c_{3}(R) such that

R^{2}λ(R)^{2} = λ_{j}^{2} + c_{3}(R)\frac{λ_{j}}{R^{\frac{3}{4}}} + \frac{c_{3}(R)^{2}}{R^{\frac{3}{2}}} ,

|c_{3}(R)| ≤ c_{4} ,

for any sufficiently large R . We use the elementary inequality |e^{−λ} − 1| ≤ |λ|e^{|λ|} to get

|e^{−tR^{2}λ(R)^{2}} − e^{−tλ_{j}^{2}}| = |e^{−tλ_{j}^{2}}(e^{−t[R^{2}λ(R)^{2}−λ_{j}^{2}]} − 1)|

≤ (\frac{c_{3}(R)|λ_{j}|}{R^{\frac{3}{4}}} + \frac{c_{3}(R)^{2}}{R^{\frac{3}{2}}} )te^{−(λ_{j}^{2}−_{\frac{c_{3}(R)|λ_{j} |}{R \frac{3}{4}}} − \frac{c_{3}(R)^{2}}{R \frac{3}{2}} )t} ≤ \frac{max(c_{4}, c_{4}^{2})t}{R^{\frac{1}{2}}}e^{− \frac{1}{2} λ_{j}^{2}t} .

In the last inequality we used the fact that |λ_{j}| < R^{\frac{1}{4}} . Let us fix a sufficiently large R and a positive t. We take the sum over finitely many eigenvalues λ_{j}^{2} of \frac{1}{4}D(C_{12})^{2} with λ_{j}^{2} < R^{\frac{1}{2}} for fixed R, and obtain

|T r_{r,II,R}(e^{−tR^{2}D_{R}^{2}}) − T r_{II,R}(e^{−t \frac{1}{4} D(C_{12})^{2}})| ≤ c_{5} \frac{t}{R^{\frac{1}{2}}} ∑_{λ_{j}^{2}<R^{\frac{1}{2}}} e^{− \frac{1}{2} λ_{j}^{2}t} .

for some constant c_{5} . The operator D(C_{12})^{2} is a Laplace type operator over S^{1} , hence the number of eigenvalues λ_{j}^{2} with λ_{j}^{2} < R^{\frac{1}{2}} can be estimated by R^{\frac{1}{4}} . Therefore, we have

c_{5} \frac{t}{R^{\frac{1}{2}}} ∑_{λ_{j}^{2}<R^{\frac{1}{2}}} e^{− \frac{1}{2} λ_{j}^{2}t} ≤ c_{6} \frac{t}{R^{\frac{1}{2}}} R^{\frac{1}{4}} e^{−λ_{1}^{2}t}

where λ_{1} denotes the first non-zero eigenvalue of \frac{1}{4}D(C_{12})^{2} . Note that c_{6} and λ_{1} are independent of R and t. This proves the first claim. The proof of the second claim goes the same way.

The proof of Proposition 3.1 is now complete. We have shown that

lim_{R→∞}(ζ_{s}^{R}′(0) − h(γ − (2 − ε)·ln R) + ζ_{l}^{R}′(0) + h(γ + ε·ln R)) = (3.5)

ln 2·ζ_{B^{2}}(0)+\frac{d}{ds}\{\frac{1}{Γ(s)} ∫_{0}^{∞} t^{s−1}[T r(e^{−t \frac{1}{4} D(C_{12})^{2}}−e^{−tD(S_{σ_{1}})^{2}}−e^{−tD(S_{σ_{2}})^{2}})−h]dt\} |_{s=0}\} .

which is equal to the formula (0.15) from Theorem 0.2.

4. Relations with the decomposition of the η-invariant

In this Section we discuss the relation of our results to the decomposition formulas for the η-invariant. The formula (0.7) has been proved by several authors and the proof we discuss in this Section is not the simplest one. Still we believe that it is worthy to present the “scattering approach” to the decomposition of the η-invariant. The key in our proof is to show that the scattering data provide you with the Lesch-Wojciechowski formula which gives the cylinder contribution in the decomposition formula for the η-invariant (see [20]).

Let us remind the reader that the η-function of a Dirac operator on a closed manifold, introduced in [1], is defined as

η_{D}(s) = ∑_{λ≠0} sign(λ)|λ|^{−s} ,

where the sum is taken over all non-zero eigenvalues of D . The η-function is well-defined for Re(s) large and it has a meromorphic extension to the whole complex plane and s = 0 is a regular point, hence η_{D}(0) is welldefined. If we consider {D_{r}} , a smooth family of the Dirac operators, then the function r → η_{D_{r}}(0) is a well-defined smooth function modulo integer contribution. The reason is that every time the eigenvalue changes sign the value of the η_{D_{r}}(0) jumps. Following [1] we introduce the η-invariant of D as

η(D) = \frac{1}{2}·(η_{D}(0) + dim ker D) . (4.1)

The η-invariant is a smooth function of the Dirac operator modulo integer contribution. The η-function of the boundary problem, defined by the boundary condtion of the type discussed earlier in this paper is well-defined and it has the same properties as the η-function of the Dirac operator on a closed manifold. Now, let us assume that we have a given Dirac operator D on a closed odd-dimensional manifold M , which decomposes in the way described in the Introduction. The following result was proved by several authors and we refer to [36] and [37] for the proof and bibliographical remarks.

Theorem 4.1. We have the following formula

η(D_{M}) = η((D_{1})_{P_{1}}) + η((D_{2})_{P_{2}}) + η(D; σ_{1}, σ_{2}) mod ℤ , (4.2)

where η(D; σ_{1}, σ_{2}) denotes the η-invariant of the operator D restricted to the collar N ≅ [−1, 1] × Y (hence equal to G(∂_{u} + B)) , subject to the boundary condition P_{2} at u = −1 and P_{1} at u = 1 (see [36]).

Though the formula holds for any generalized self-adjoint boundary conditions of Atiyah-Patodi-Singer type (see [37]) we restrict our attention here to the “ideal boundary conditions” defined in the Introduction, hence

P_{1} = Π_{<} + π_{1} , P_{2} = Π_{>} + π_{2} ,

where π_{i} is an orthogonal projection onto the −1-eigenspace of an involution σ_{i} : ker B → ker B , which anticommutes with G . Let us observe that

U = σ_{1}σ_{2} : ker B → ker B

is the unitary operator, such that

UG = GU , det U = 1 , U^{∗} = σ_{1}Uσ_{1} . (4.3)

It follows that the spectrum of U is invariant under complex conjugation. Moreover, let G_{±} denote the ±i eigenspace of the operator G : ker B → ker B , then the maps U_{±} = U|_{G_{±}} : G_{±} → G_{±} are well-defined.

Let T (U_{±}) : G_{±} → G_{±} denote a self-adjoint operator such that

U_{±} = e^{iT (U_{±})} . (4.4)

The following result proved in Section 2 of [20] was the key ingredient in the proof of Theorem 4.1.

Theorem 4.2. (see [20]). We have the following formulas

η_{(D;σ_{1},σ_{2})}(0) = −\frac{T r T (U_{+})}{π} = \frac{T r T (U_{−})}{π} mod ℤ , (4.5)

and

η(D; σ_{1}, σ_{2}) = \frac{T r T (U_{−})}{2π} mod ℤ .

Remark 4.3. Let us also notice that the formula for the η-invariant of the operator D(V) on S^{1} defined by V ∈ U(N)

η(D(V)) = −\frac{T r T (V)}{2π} mod ℤ . (4.6)

(see for instance [4]).

One way to prove the decomposition formula (4.2) is to use the adiabatic analysis we developed in the proof of Theorem 0.2. This analysis easily gives us the following Theorem

Theorem 4.4. The following decomposition formula holds for the η-invariant

η(D) − η(D_{σ_{1}}) − η(D_{σ_{2}}) (4.7)

= η(D(C_{12})) − η(D(S_{σ_{1}})) − η(D(S_{σ_{2}})) mod ℤ

Proof. In fact the analysis for the η-invariant is simpler than for the ζdeterminant of the Dirac Laplacian. It provides us with the expected formula

lim_{R→∞}{η(D_{R}) − η((D_{1,R})_{σ_{1}}) − η((D_{2,R})_{σ_{2}})}

= η(D(C_{12})) − η(D(S_{σ_{1}})) − η(D(S_{σ_{2}})) mod ℤ

Now, we use the fact that the η-invariant on the manifolds M_{R}, M_{1,R} and M_{2,R} is independent of R (modulo integer contributions). This was carefully explained in [23] (see Proposition 2.16).

It seems that we have a problem here. The cylinder contribution in formula (4.2) in Theorem 4.1, contains only boundary data, and

η(D(C_{12})) − η(D(S_{σ_{1}})) − η(D(S_{σ_{2}}))

the cylinder contribution in the formula (4.7) in Theorem 4.4 seems to contain global information from M encoded into scattering matrices. However, there is a cancellation mechanism which leads to the equality of the η(D; σ_{1}, σ_{2}) and the right side of (4.7).

The scattering matrix C_{i} = C_{i}(0) can be represented in the following way, with respect to the decomposition ker B = G_{+} ⊕ G_{−}

C_{i} = (\begin{matrix}0 & C(i)_{−} \\ C(i)_{+} & 0 \end{matrix}) where C(i)_{±}C(i)_{∓} = Id .

Similar formulas hold for the involutions σ_{i} . We see that

C_{12} = C(1)_{+}C(2)_{−} : G_{−} → G_{−} .

We also need the corresponding formula for the operator S_{σ_{1}}

S_{σ_{1}} = −P_{σ_{1}} ∘ C_{1} ∘ I_{σ_{1}} = −\frac{1}{2} (\begin{matrix}−Id & σ(1)_{−} \\ σ(1)_{+} & −Id \end{matrix}) (\begin{matrix}0 & C(1)_{−} \\ C(1)_{+} & 0 \end{matrix}) (\begin{matrix}2i & 0 \\ 0 & 0 \end{matrix})

= (\begin{matrix}−iσ(1)_{−}C(1)_{+} & 0 \\ iC(1)_{+} & 0 \end{matrix}) .

The subspace ker(σ_{1} + Id) is the graph of the operator

(\begin{matrix}Id & 0 \\ 0 & −σ(1)_{+} \end{matrix}) ,

which means that each element of G_{i} is represented in the form (\begin{matrix}f \\ −σ(1)_{+}f \end{matrix})

, for some f ∈ G_{+} . This allows us to represent the map S_{σ_{1}} = −P_{σ_{1}} ∘ C_{1} ∘ I_{σ_{1}} : ker(σ_{1} + Id) → ker(σ_{1} + Id) as

(4.8)  S_{σ_{1}} (\begin{matrix}f \\ −σ(1)_{+}f \end{matrix}) = (\begin{matrix}−iσ(1)_{−}C(1)_{+} & 0 \\ iC(1)_{+} & 0 \end{matrix}) (\begin{matrix}f \\ −σ(1)_{+}f \end{matrix})

= (\begin{matrix}−iσ(1)_{−}C(1)_{+}f \\ σ(1)_{+}(iσ(1)_{−}C(1)_{+}f) \end{matrix}) .

It turns out that, from the spectral point of view, the operator S_{σ_{1}} is equal to the operator

−iσ(1)_{−}C(1)_{+} : G_{+} → G_{+} ,

or equivalently to the operator

−iC(1)_{+}σ(1)_{−} : G_{−} → G_{−} .

The corresponding analysis for the operator S_{σ_{2}} implies that S_{σ_{2}} is equivalent to

iC(2)_{−}σ(2)_{+} : G_{+} → G_{+} or iσ(2)_{+}C(2)_{−} : G_{−} → G_{−} .

As a result, when we study the quantity \frac{det(C_{12})}{det(S_{σ_{1}})det(S_{σ_{2}})} , we obtain

\frac{det(C_{12})}{det(S_{σ_{1}})det(S_{σ_{2}})} = det (C_{12}S_{σ_{2}}^{−1}S_{σ_{1}}^{−1})

= det C(1)_{+}σ(2)_{−}σ(1)_{+}C(1)_{−} = det σ(2)_{−}σ(1)_{+} .

In the same manner we analyze the trace of the logarithm of the corresponding operator which, of course, leads us to the η-invariant. Let us remind (see (4.4)) that we introduced a self-adjoint matrix T (U_{−}) , such that U_{−} = exp(iT (U_{−})) , where

U_{−} = σ(1)_{+}σ(2)_{−} .

If we combine our computations with formula (4.6), we have

η(D(C_{12})) − η(D(S_{σ_{1}})) − η(D(S_{σ_{2}})) = −\frac{T r T (\overline{σ(2)_{−}σ(1)_{+}})}{2π}

\frac{T r T (σ(2)_{−}σ(1)_{+})}{2π} = \frac{T r T (σ(1)_{+}σ(2)_{−})}{2π} = \frac{T r T (U_{−})}{2π} mod ℤ .

This ends the proof of the following Theorem

Theorem 4.5. The cylinder contributions in the formulas (4.2) and (4.7) are equal to each other

η(D; σ_{1}, σ_{2}) = η(D(C_{12})) − η(D(S_{σ_{1}})) − η(D(S_{σ_{2}})) mod ℤ . (4.9)

5. Lesch-Wojciechowski formula for the ζ-determinant of the Dirac Laplacians

Let us now assume that we have a given Dirac operator D on a compact manifold N with boundary Y and that we specified two different boundary conditions determined by involutions σ_{1} , σ_{2} of the kernel of the tangential operator B . The formula for the difference η(D_{P_{2}})−η(D_{P_{1}}) can be obtained by studying the variation of the η-invariant. This is how the original proof in [20] went. We obtained the equality

η(D_{P_{2}}) − η(D_{P_{1}}) = η(D; σ_{1}, σ_{2}) mod ℤ . (5.1)

and we used this equality to study the decomposition formula for the ηinvariant (see [35], [36], [37]).

You can actually reverse the argument. We just proved the decomposition formula for the η-invariant using the adiabatic argument and scattering matrices in the previous Section. Now, the formula (5.1) follows if we apply the decomposition formula in the case of the manifold M = Ñ = (−N) ∪ N where M_{1} = (−N) is the manifold N with the reverse orientation. The corresponding Dirac operator is constructed via the well-known doubling construction introduced by the second author in [34]. We define a bundle of Clifford modules on Ñ as S^{G} = S ∪_{G} S . The set of smooth sections of S^{G} can be identified with the set of couples (s_{1}, s_{2}) , where

s_{1} ∈ C^{∞}((−N) ∪ [0, 1) × Y; S) and s_{2} ∈ C^{∞}((−1, 0] × Y ∪ N; S) ,

and we identify

s_{2}(u, y) ≅ G(y)s_{1}(u, y)

over (−1, 1)×Y . The operator D̃ on C^{∞}(M; S^{G}) is defined via the formula

D̃(s_{1}, s_{2}) = (−Ds_{1}, Ds_{2}) . (5.2)

This definition works due to the fact that −D on (−1, 1) × Y is represented as

−D|_{(−1,1)×Y} = −G(−∂_{u} + B)

while D|_{(−1,1)×Y} = G(∂_{u} + B) . We can now check

G((−D)s_{1})|_{(−1,1)×Y}) = (−∂_{u} + B)s_{1} = −(−∂_{u} + B)G^{2}s_{1}
= −(−G∂_{u} − GB)Gs_{1} = Ds_{2}|_{(−1,1)×Y} .

A similar computation shows that the scattering matrix C_{1} is equal to −C_{2} , where C_{2} = C_{2}(0) is the scattering matrix determined by the operator D_{2,∞} on N_{2,∞} ≅ ((−∞, 0] × Y) ∪ N . It follows from the formula (4.6) that η(D(C_{12})) = η(D(−Id)) = 0 and now formula (4.9) from Theorem 4.5 gives (5.1), which can be also reformulated as

Proposition 5.1. The following equality holds for the η-invariants of the operators D_{P_{1}} and D_{P_{2}}

η(D_{P_{2}}) − η(D_{P_{1}}) = −η(D(S_{σ_{1}})) − η(D(S_{σ_{2}})) mod ℤ . (5.3)

The interesting feature of formula (5.3) is that, according to (4.6), the number on the right side of (5.3) is the trace of a finite dimensional matrix. In this final Section of our paper we prove the corresponding result for the ζ-determinant of the Dirac Laplacian.

We consider only the case of invertible operators D_{P_{1}} and D_{P_{2}} . It follows that D(S_{σ_{i}}) are invertible operators. We also have ker_{L^{2}}D_{∞} equal to {0} , and C_{12} = −Id , hence the operator D(C_{12}) is invertible as well. Theorem 0.2 provides the formulas

lim_{R→∞} \frac{det_{ζ}D_{R}^{2}}{det_{ζ}(D_{1,R})2 ; P̃_{1}·det_{ζ}(D_{2,R})_{P_{2}}^{2}} = \frac{2^{−ζ_{B^{2}}(0)}·det_{ζ} \frac{1}{4}D(C_{12})^{2}}{det_{ζ}D(S_{σ_{1}})^{2}·det_{ζ}D(S_{σ_{2}})^{2}} ,

and

lim_{R→∞} \frac{det_{ζ}D_{R}^{2}}{det_{ζ}(D_{1,R})2 ; P̃_{1}·det_{ζ}(D_{2,R})_{P_{1}}^{2}} = \frac{2^{−ζ_{B^{2}}(0)}·det_{ζ} \frac{1}{4}D(C_{12})^{2}}{det_{ζ}D(S_{σ_{1}})^{2}·det_{ζ}D(S_{σ_{1}})^{2}} ,

where P̃_{1} = Π_{<} + π_{1} , as the original P_{1} projection is equal to Π_{>} + π_{1} . The result is the following equality

lim_{R→∞} \frac{det_{ζ}(D_{2,R})_{P_{2}}^{2}}{det_{ζ}(D_{2,R})_{P_{1}}^{2}} = \frac{det_{ζ}D(S_{σ_{2}})^{2}}{det_{ζ}D(S_{σ_{1}})^{2}} . (5.4)

Theorem 0.2 gives us only the adiabatic equality in (5.4). However, we observed in [29] that the quotient on the left side of (5.4) is independent of R . The proof relies on the results of the Scott-Wojciechowski work on the determinants of the boundary problems (see [31]) and on the Nicolaescu study of the dynamics of the Calderon projections (see [25]). We have the folowing Proposition

Proposition 5.2. Under the assumptions we have made in this section the quotient

\frac{det_{ζ}(D_{2,R})_{P_{2}}^{2}}{det_{ζ}(D_{2,R})_{P_{1}}^{2}}

does not depend on R.

Proof. The proof of this proposition is an application of the following Scott– Wojciechowski formula, Proposition 4.1 in [31],

(5.5) \frac{det_{ζ}(D_{R})_{P_{1}}^{2}}{det_{ζ}(D_{R})_{P_{2}}^{2}} = |det_{F r}U_{P_{2}}(U_{P_{1}})^{−1}S_{R}(P_{1})S_{R}(P_{2})^{−1}|^{2}

where det_{F r} is the Fredholm determinant, and

U_{P_{2}}(U_{P_{1}})^{−1} : Range(P_{1}) → Range(P_{2})

is a unitary map which depends only on P_{1}, P_{2}. The operator S_{R}(P_{i}) is defined by

S_{R}(P_{i}) := P_{i}P (D_{R}) : H(D_{R}) → Ran P_{i} ,

where P (D_{R}) is the Calderon projection defined by D_{R} and H(D_{R}) is the Cauchy data space (see for instance [3]) . The definitions of the boundary condition P_{i} implies the following decomposition of S_{R}(P_{i})

Π_{>}S_{R}(P_{i}) = Π_{>}P (D_{R}) and σ_{i}S_{R}(P_{i}) = σ_{i}P (D_{R}) .

We also have the corresponding decomposition of S_{R}(P_{i})^{−1} into its restrictions to the images of Π_{>} and σ_{i} which implies the obvious identities

σ_{i}P (D_{R})S_{R}(P_{i})^{−1} = σ_{i} and Π_{>}P (D_{R})S_{R}(P_{i})^{−1} = Π_{>} .

Hence, the operator S_{R}(P_{1})S_{R}(P_{2})^{−1} has the following form

S_{R}(P_{1})S_{R}(P_{2})^{−1} = (\begin{matrix}Π_{>}S_{R}(P_{1})S_{R}(P_{2})^{−1}Π_{>} & Π_{>}S_{R}(P_{1})S_{R}(P_{2})^{−1}σ_{2} \\ σ_{1}S_{R}(P_{1})S_{R}(P_{2})^{−1}Π_{>} & σ_{1}S_{R}(P_{1})S_{R}(P_{2})^{−1}σ_{2} \end{matrix}) (5.6)

= (\begin{matrix}Π_{>} & 0 \\ 0 & σ_{1}S_{R}(P_{1})S_{R}(P_{2})^{−1}σ_{2} \end{matrix}) .

Nicolaescu proved that the dynamics of the Cauchy data is determined by the operator e^{−RB} on L^{2}(Y, S|_{Y}) (see [25]). However, this operator is equal to the identity when restricted to the subspace ker(B) . It follows that σ_{1}S_{R}(P_{1})S_{R}(P_{2})^{−1}σ_{2} is independent of R. Now the Proposition follows from (5.5) and (5.6).

We combine (5.4), Proposition 5.2, and the formula for the one dimensional det_{ζ} to express the quotient

\frac{det_{ζ}(D_{2,R})_{P_{2}}^{2}}{det_{ζ}(D_{2,R})_{P_{1}}^{2}}

as a finite-dimensional determinant. Let us remind the reader of the formula for the ζ-determinant of the operator D(U)^{2} . Let us start with the operator D_{a} = −i\frac{d}{dx} + a acting on functions on S^{1} = ℝ mod 2πℤ . In this case there is the well-known formula (see for instance [4])

det_{ζ}D_{a}^{2} = 4· sin^{2} πa . (5.7)

Now, if we have an operator −i\frac{d}{dx} on S^{1} = ℝ mod ℤ acting on sections of the bundle determined by the holonomy U ∈ U(N) , this translates into a direct sum of the operators −i\frac{d}{dx} + \frac{a_{j}}{2π} , where e^{ia_{j}} is an eigenvalue of U . Hence the corresponding formula is

det_{ζ}D(U)^{2} = 4^{N} ∏_{j=1}^{N} sin^{2} \frac{a_{j}}{2} . (5.8)

We can use (5.8) to rewrite the right side of (5.4) in terms of the suitable sin − matrices. Another possibility is to use equality

det_{ζ}D(U) = 4^{N}det \frac{2 − U − U^{−1}}{4} , (5.9)

which is equivalent to (5.8). This leads to the Lesch-Wojciechowski formula for the ζ-determinant of the Dirac Laplacian

Theorem 5.3. The following formula holds for the quotient of the ζ-determ-inants of the square of D subject to the boundary conditions defined by involutions σ_{1} and σ_{2}

\frac{det_{ζ}D_{P_{2}}^{2}}{det_{ζ}D_{P_{1}}^{2}} = det((Id − \frac{S_{σ_{2}} + S_{σ_{2}}^{−1}}{2} )(Id − \frac{S_{σ_{1}} + S_{σ_{1}}^{−1}}{2} )^{−1}) . (5.10)

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Max Planck Institute of Mathematics, Vivatgasse 7, D-53111 Bonn, Germany

  E-mail address: jspark@mpim-bonn.mpg.de

Department of Mathematics, IUPUI (Indiana/Purdue), Indianapolis IN 46202–3216, U.S.A.

  E-mail address: kwojciechowski@math.iupui.edu