Scattering at low energies on manifolds with cylindrical ends and stable systoles
SSCATTERING AT LOW ENERGIES ON MANIFOLDS WITHCYLINDRICAL ENDS AND STABLE SYSTOLES
WERNER M ¨ULLER AND ALEXANDER STROHMAIER
Abstract.
Scattering theory for p -forms on manifolds with cylindrical ends has a directinterpretation in terms of cohomology. Using the Hodge isomorphism, the scatteringmatrix at low energy may be regarded as operator on the cohomology of the boundary.Its value at zero describes the image of the absolute cohomology in the cohomology of theboundary. We show that the so-called scattering length, the Eisenbud-Wigner time delayat zero energy, has a cohomological interpretation as well. Namely, it relates the normof a cohomology class on the boundary to the norm of its image under the connectinghomomorphism in the long exact sequence in cohomology. An interesting consequence ofthis is that one can estimate the scattering lengths in terms of geometric data like thevolumes of certain homological systoles. Introduction and main results
Scattering theory for manifolds with cylindrical ends deals with the following geometricsituation. Let M be an oriented, connected, compact Riemannian manifold with boundary Y = ∂M such that the metric is a product near the boundary, i.e., there is a tubularneighborhood of Y which is isometric to ( − (cid:15), × Y, equipped with the product metric du + h , where h is a Riemannian metric on Y . The non-compact elongation X of M isthen obtained from M by attaching the half-cylinder Z = R + × Y over the boundary (seefigure 1):(1) X = Z ∪ Y M. X ZM Y
Figure 1.
Elongation X of M .The Riemannian metric on M is extended to one on X in the obvious way so that g | R + × Y = du + h. (2) The second author was supported by the Leverhulm trust and the MPI Bonn. a r X i v : . [ m a t h . A P ] N ov W. M ¨ULLER AND A. STROHMAIER
Such a manifold is called a manifold with cylindrical ends .Scattering theory on X investigates how wave packets coming in from infinity are scat-tered in M . The scattering of p -forms on X is described by the scattering matrix C p ( λ ) ∈ End (cid:32)(cid:77) µ ≤ λ Eig µ (∆ (cid:48) p ) ⊕ Eig µ (∆ (cid:48) p − ) (cid:33) , (3)where ∆ (cid:48) p is the Laplace-Beltrami operator acting on p -forms on Y and Eig µ (∆ (cid:48) p ) theeigenspace of ∆ (cid:48) p with eigenvalue µ . In particular for small values of the spectral parameter λ we have C p ( λ ) ∈ End (cid:0) H p ( Y ) ⊕ H p − ( Y ) (cid:1) , (4)where H p ( Y ) = ker ∆ (cid:48) p is the space of harmonic p -forms on Y . For the purposes of thisarticle one can think of the scattering matrix for small values of the spectral parameter asbeing defined by the statement of Theorem 2.1. In this case it can be shown that C ( λ )leaves the direct summands invariant and is of the form C p ( λ ) = (cid:18) S p ( λ ) 00 − S p − ( λ ) (cid:19) , (5)where S p ( λ ) ∈ End( H p ( Y )) is the scattering matrix describing the scattering of coclosedforms. Again S p can be defined by the statement of Theorem 2.6 as the matrix relatingthe incoming and outgoing waves. The first observation is that the total scattering matrixfor coclosed p -forms at energy 0, S (0) = ⊕ p S p (0) ∈ End( H ∗ ( Y )) , is a self-adjoint involution which anti-commutes with the Hodge star operator. The +1eigenspace of S (0) coincides with the space of harmonic forms that represent cohomologyclasses in Im( r : H ∗ ( X, R ) → H ∗ ( Y, R )).The Eisenbud-Wigner time-delay operator T p ( λ ) describes the time-delay a p -form-waveundergoes when being scattered in M (see Appendix A). It can be calculated from theEisenbud-Wigner formula (see Appendix A), and for small λ it is given by T p ( λ ) = − i C p ( λ ) ∗ ddλ C p ( λ ) . Of course T p ( λ ) = (cid:18) T p ( λ ) 00 T p − ( λ ) (cid:19) , (6)where T p ( λ ) is the time delay operator for coclosed forms defined by T p ( λ ) = − i S p ( λ ) − S (cid:48) p ( λ ) . (7)Its value T p (0) at zero energy is of particular interest and we call it the scattering length .The physical interpretation of the scattering length is as follows. If a coclosed wave packethas very low energy then, by the uncertainty relation, it will be far spread out. In particularit will not be able to “feel” details of the geometry of M . The effect of the manifold M CATTERING AT LOW ENERGIES 3 in the scattering process for a wave is then close to that of a cylindrical obstacle of lengthgiven by the scattering length. It is therefore an interesting question to determine whatgeometric properties of M have an effect on the scattering length. Since T (0) commuteswith the Hodge star operator it is enough to know its restriction to the − S (0). Denote by (cid:107) · (cid:107) st the stable norm of a homology class, and by (cid:107) · (cid:107) ∞ the comass normon the cohomology groups (see [Gro99]). Let ν > (cid:48) p . Let us define Vol ∗ ( M ) = Vol( M ) + 1 √ ν Vol( Y ) . Furthermore, in section 5 we introduce for each n ∈ N and 0 ≤ p ≤ n constants C ( n, p ) > p R n . They are equal to 1 for p = 0 and p = 1. One of our main results relates the scattering length to certains norms inhomology (Theorem 4.7) and gives rise to the following estimation of the scattering lengthin terms of geometric data. Theorem 1.1.
Let ≤ p ≤ n . For every φ in the − -eigenspace of S p (0) we have C ( n, p + 1) − Vol ∗ ( M ) − (cid:107) [ M ] ∩ ∂φ (cid:107) st ≤ (cid:104) φ, T (0) − φ (cid:105) ≤ C ( n, p + 1)Vol( M ) (cid:107) ∂φ (cid:107) ∞ , where ∂ : H p ( Y ) → H p +1 ( M, Y ) is the connecting homomorphism in the long exact se-quence in cohomology and [ M ] ∩ ∂φ is the Poincare dual of the class ∂φ . As an example we treat the case when Y has two connected components Y and Y and p = 0. In this case there is a canonical basis in H ( Y, R ) with respect to which T (0) hasthe form T (0) = (cid:18) t t (cid:19) , (8)so that t = 2 Vol( M )Vol( Y ) , (9) C ≤ t ≤ C , (10)and the constants C and C are given by C = 2Vol ∗ ( M ) Vol( Y )Vol( Y ) (cid:107) ι ∗ ([ Y ]) (cid:107) st (Vol( Y ) + Vol( Y )) , (11) C = 2Vol( M ) − dist( Y , Y ) Vol( Y )Vol( Y )Vol( Y ) + Vol( Y ) . (12)The map ι is the inclusion of Y into M . So we get an estimate for the scattering length bypurely geometric quantities. The physical interpretation of this is as follows. For a waveof low energy the reflection coefficient r and the transmission coefficient r for a wave W. M ¨ULLER AND A. STROHMAIER coming in at the boundary component Y are approximated by their values at zero, namely(see section 7.3) r = Vol( Y ) − Vol( Y )Vol( Y ) + Vol( Y ) , r = 2Vol( Y )Vol( Y ) + Vol( Y ) . (13)The time-delay is then determined by t and t . For example in the case where Vol( Y ) =Vol( Y ) the reflection coefficient at zero energy is zero and the time delay of the transmittedwave is equal to ( t + t ) (see section 7.3). Another example, the full-torus, is treated insection 7.4.Given a >
0, let M a be the manifold that is obtained from M by attaching the cylinder[0 , a ] × Y to M . We also investigate how the L -norm of a class in H p ( M a , R ) and the L -norm of its image in H p +1 ( M a , R ) under the connecting homomorphism are related forthe manifold with boundary M a . There is an operator q a that relates the L -norm of classesin the complement of the kernel of the connecting homomorphism to the L -norm of theimage of this class under the connecting homomorphism. So q a measures to what extentthe connecting homomorphism is a partial isometry. Theorem 3.3 shows that the operator q a has an expansion of the form q a = a + 12 T (0) + O ( e − ca ) , (14)as a → ∞ . This shows that one can calculate the scattering length by approximating X by the compact manifolds M a and consider the constant term in the above expansion. Theexponential decay of the remainder term may also be useful for numerical computations.The paper is organized as follows. In sections 2 we review stationary scattering theory for p -forms on manifolds with cylindrical ends. Section 3 and section 4 deal with cohomologyof M and X , and their relation to scattering theory and the continuous spectrum of theLaplacian of X . We also derive a cohomological formula for the scattering length. Insections 5 and 6 it is shown that the L -scalar products on the cohomology groups of X can be estimated in terms of geometric quantities and that these estimates imply estimateson the scattering length. Section 7 treats the case of functions in the case of two boundarycomponents and the case of a full-torus. In this section we demonstrate how our mainresult can be used to obtain estimates of the scattering length in terms of geometric data.In appendix A we discuss the relation between the stationary and the dynamical approachto scattering theory and we establish the Eisenbud-Wigner formula for manifolds withcylindrical ends.Whereas for the sake of notational simplicity we restricted ourselves in this paper to man-ifolds with cylindrical ends most of our analysis carries over in a straightforward mannerto waveguides if Neumann boundary conditions are imposed. Acknowledgements.
Much of the work on this paper has been done during the visitof the second author to the MPI in Bonn and he would like to thank the MPI for thekind support. Both authors would also like to thank the MSRI in Berkeley for hospitalityduring the program “Analysis on Singular Spaces”. We are grateful to Werner Ballmann,
CATTERING AT LOW ENERGIES 5
Alexej Bolsinov, Peter Perry, and Sasha Pushnitski for useful discussions and comments.The authors would like to thank the referee for his very useful comments and suggestions.2.
Stationary scattering theory for manifolds with cylindrical ends
In this section we review stationary scattering theory for differential forms on manifoldswith cylindrical ends and establish some elementary relations which we will use in subse-quent sections. For functions this can be found e.g in [Chr95, Chr02, Mel93] or in [IKL10]which also covers the asymptotically cylindrical case. Scattering theory for Dirac typeoperators that are of product type on the cylinder is also well covered in the literature (seee.g. [Mu94])As before let M be an oriented, connected, compact Riemannian manifold with boundary Y = ∂M such that the metric is a product near the boundary. Let X be the elongation of M . For any Riemannian manifold W we denote by Λ p ( W ) (resp. Λ pc ( W ), L Λ p ( W )) thespace of smooth p -forms (resp. smooth p -forms with compact supports, L -forms) on W .Let ∆ p be the Laplace operator on Λ p ( W ). Throughout this paper a harmonic p -form willmean a p -form φ ∈ Λ p ( W ) with ∆ p φ = 0.Since X is complete, the Laplace-Beltrami operator ∆ p on p -forms is essentially self-adjoint when regarded as operator in L Λ p ( X ) with domain Λ pc ( X ) ([Che73]). We continueto denote its self-adjoint extension by ∆ p . In this section we recall some facts concerningthe generalized eigenforms of ∆ p and derive some properties of the scattering matrix forlow energy.Let ∆ (cid:48) p denote the Laplacian on p -Forms of Y . Let 0 ≤ ν < ν < · · · be the distincteigenvalues of of ∆ (cid:48) p ⊕ ∆ (cid:48) p − . Let Σ → C the minimal Riemann surface on which (cid:112) λ − ν j is a single-valued function for all j ∈ N . As proved by Melrose[Mel93] the resolvent(∆ p − λ ) − , regarded as operator Λ pc ( X ) → L Λ p ( X ), admits a meromorphic extensionfrom the half-plane Im( λ ) > µ > (cid:48) p ⊕ ∆ (cid:48) p − . Then it follows in particular that (∆ p − λ ) − extends to a meromorphicfunction on the disc { λ : | λ | < µ } . As a consequence one can define analytic families ofgeneralized eigenforms.For a ≥ Y a the hypersurface ( a, Y ) ⊂ R +0 × Y ⊂ X . Note that therestriction of the bundle Λ p T ∗ X to Y a is canonically isomorphic to the direct sum Λ p T ∗ Y ⊕ Λ p − T ∗ Y since each vector f ∈ Λ p T ∗ X at some point ( u, x ) can be uniquely decomposedas f = f + du ∧ f with f ∈ Λ p T ∗ X and f ∈ Λ p − T ∗ X . Accordingly, the restriction ofany p -form ω ∈ Λ p ( X ) to the cylinder R + × Y is of the form ω = ω + du ∧ ω , (15)where ω and ω are sections of the pulled back bundles π ∗ Λ p T ∗ Y and π ∗ Λ p − T ∗ Y , respec-tively, and π : R + × Y → Y is the canonical projection. We think of ω ( u ) and ω ( u ) asforms on Y that depend smoothly on the additional parameter u . The map j p : π ∗ Λ p T ∗ Y ⊕ π ∗ Λ p − T ∗ Y → Λ p T ∗ Z, ( ω , ω ) (cid:55)→ ω + du ∧ ω (16) W. M ¨ULLER AND A. STROHMAIER is an isomorphism of vector bundles. The exterior differential of such a form ω is thengiven by dω = d (cid:48) ω + du ∧ ∂ u ω − du ∧ d (cid:48) ω , (17)where d (cid:48) denotes the exterior differential on Y . In matrix notation this means j − ◦ d ◦ j = (cid:18) d (cid:48) ∂ u − d (cid:48) (cid:19) , (18)where we denote j = ⊕ j p . Since the metric has product structure the decomposition isorthogonal and the formal adjoint δ of d is therefore j − ◦ δ ◦ j = (cid:18) δ (cid:48) − ∂ u − δ (cid:48) (cid:19) . (19)Here again we use the notation δ (cid:48) for the codifferential on Y . We therefore have j − ◦ ( d + δ ) ◦ j = (cid:18) − (cid:19) ( d (cid:48) + δ (cid:48) ) + (cid:18) −
11 0 (cid:19) ∂ u . (20)and j − p ◦ ∆ p ◦ j p = (cid:18) − ∂ u + ∆ (cid:48) p − ∂ u + ∆ (cid:48) p − (cid:19) for all p . This form of an operator allows for a separation of variables on the cylinder R + × Y . Suppose that ( ψ i ) is an orthonormal sequence of eigenforms of (cid:18) ∆ (cid:48) p
00 ∆ (cid:48) p − (cid:19) (21)with eigenvalues µ ψ i . Then for | λ | < µ , any solution F of the equation (∆ p − λ ) F = 0has an expansion of the form(22) F ( u ) = ∞ (cid:88) i =0 (cid:18) a i e − i (cid:113) λ − µ ψi u + b i e i (cid:113) λ − µ ψi u (cid:19) j p ( ψ i ) + (cid:40)(cid:80) µ ψi =0 c i uj p ( ψ i ) , λ = 0;0 , λ (cid:54) = 0 . The series converges in the C ∞ -topology. The square roots are chosen throughout thearticle as √ re i ϕ = √ re i2 ϕ if r > ≤ ϕ < π . From the analytic continuation of theresolvent one gets the following result. Theorem 2.1.
For each ψ in ker(∆ (cid:48) p ⊕ ∆ (cid:48) p − ) there exists a p -form (cid:101) F ( ψ, λ ) which ismeromorphic in λ ∈ { z : | z | < µ } such that the following conditions hold (i) (cid:101) F ( ψ, λ ) is holomorphic in λ for Im( λ ) > . (ii) ∆ (cid:101) F ( ψ, λ ) = λ (cid:101) F ( ψ, λ ) . (iii) There exists (cid:101) R p ( ψ, λ ) ∈ L Λ p ( Z ) such that on Z we have (cid:101) F ( ψ, λ ) = e − i λu j p ( ψ ) + e i λu j p ( C p ( λ ) ψ ) + (cid:101) R p ( ψ, λ ) . (iv) C p ( λ ) : ker(∆ (cid:48) p ⊕ ∆ (cid:48) p − ) → ker(∆ (cid:48) p ⊕ ∆ (cid:48) p − ) is a linear operator, and C p ( λ ) and (cid:101) R p ( ψ, λ ) are meromorphic functions of λ . CATTERING AT LOW ENERGIES 7
Moreover, C p ( λ ) , (cid:101) R p ( ψ, λ ) and (cid:101) F ( ψ, λ ) are uniquely determined by these properties.Proof. This follows from[Gui89], [Mel93]. For the convenience of the reader we include thedetails. Let Σ → C be the minimal Riemann surface to which all the functions ( λ − µ ψ i ) / extend to be holomorphic. By [Gui89, Th´eor`em 0.2], [Mel95, Theorem7.1] the resolvent(∆ p − λ ) − , regarded as operator Λ pc ( X ) → L Λ p ( X ), extends to a meromorphic functionof λ ∈ Σ. Especially (∆ p − λ ) − extends to a meromorphic function of λ ∈ { z ∈ C : | z | <µ } as an operator Λ pc ( X ) → L Λ p ( X ). Let χ be a smooth function with support in Z which is equal to 1 outside a compact set. Put (cid:101) F ( ψ, λ ) := χe − i λu j p ( ψ ) − (∆ p − λ ) − (cid:8) (∆ p − λ )( χe − i λu j p ( ψ )) (cid:9) . (23)Then (cid:101) F ( ψ, λ ) is a smooth p -form on X which depends meromorphically on λ ∈ Σ. More-over, it satisfies (∆ p − λ ) (cid:101) F ( ψ, λ ) = 0 . (24)Since (cid:101) F ( ψ, λ ) − χe − i λu j p ( ψ ) is square integrable for Im( λ ) >
0, the expansion (22) has theform (cid:101) F ( ψ, λ ) = e − i λu j p ( ψ ) + e i λu j p ( C p ( λ ) ψ ) + (cid:101) R p ( ψ, λ )where C p ( λ ) ψ ∈ ker(∆ (cid:48) P ⊕ ∆ (cid:48) p − ) and (cid:101) R p ( ψ, λ, u ) = (cid:88) µ ψi (cid:54) =0 (cid:18) a i ( λ ) e − i (cid:113) λ − µ ψi u j p ( ψ i ) + b i ( λ ) e +i (cid:113) λ − µ ψi u j p ( ψ i ) (cid:19) . (25)Moreover, (cid:101) R p ( ψ, λ ) is square integrable for Im( λ ) >
0. This implies that a i ( λ ) = 0 forIm( λ ) >
0. Since a i ( λ ) is meromorphic, it follows that a i ( λ ) = 0 for | λ | < µ . Thuswe conclude that R p ( ψ, λ ) is a meromorphic function in the disc | λ | < µ with values in L Λ p ( X ).The uniqueness is an immediate consequence of the self-adjointness of ∆ p . Namely, if (cid:101) F ( ψ, λ ) and (cid:101) F ( ψ, λ ) both have the above properties then their difference G ( ψ, λ ) is squareintegrable for Im( λ ) > p − λ ). Using that ∆ p isself-adjoint, we get G ( ψ, λ ) = 0 for Im( λ ) >
0. Since it is meromorphic in λ we concludethat G ( ψ, λ ) = 0. (cid:3) The requirement that (cid:101) R p ( ψ, λ ) is in L Λ p ( Z ) in fact implies a much faster decay atinfinity. Lemma 2.2. If (cid:101) R ( ψ, λ ) is regular at λ , then all derivatives of (cid:101) R ( ψ, λ ) in u and x areexponentially decaying as u → ∞ . More precisely | ∂ ku (∆ (cid:48) ) l (cid:101) R ( ψ, λ ) | ≤ C (cid:48) k,l,λ e − C λ u , for all k, l ≥ and some positive constants C (cid:48) k,l,λ and C λ . W. M ¨ULLER AND A. STROHMAIER
Proof.
The proof is already implicitly contained in the proof of the previous theorem.Namely, (cid:101) R ( ψ, λ ) is smooth by elliptic regularity and thus the expansion (25) converges inΛ p ( Z ). Since a i = 0, we get the decay for | λ | < µ with C λ = Re (cid:16)(cid:112) µ − λ (cid:17) . (cid:3) Let ψ ∈ ker ∆ (cid:48) p . Put(26) F ( ψ, λ ) := (cid:101) F (( ψ, , λ ) , F ( du ∧ ψ, λ ) := (cid:101) F ((0 , ψ ) , λ ) . Then the expansion (iii) of Theorem 2.1 takes the form F ( ψ, λ ) = e − iλu ψ + e iλu C p ( λ ) ψ + e iλu du ∧ C p ( λ ) ψ + R p ( ψ, λ ); F ( du ∧ ψ, λ ) = e − iλu du ∧ ψ + e iλu du ∧ C p ( λ ) ψ + e iλu C p ( λ ) ψ + R p ( du ∧ ψ, λ ) , (27)where(28) C p ( λ ) = C p ( λ ) C p ( λ ) C p ( λ ) C p ( λ ) as endomorphism of ker∆ (cid:48) p ⊕ ker∆ (cid:48) p − . A priory there is no reason why C p ( λ ) should leavethe summands invariant. Nevertheless this is guaranteed by a continuous version of theHodge decomposition as the proof of the following proposition shows. Proposition 2.3. C p ( λ ) leaves the spaces ker∆ (cid:48) p and ker∆ (cid:48) p − invariant, i.e., C p ( λ ) = 0 and C p ( λ ) = 0 .Proof. Let ψ ∈ ∆ (cid:48) p . Using the expansion of 27, it follows that on R + × Y we have δdF ( ψ, λ ) = λ e − i λu ψ + λ e i λu C p ( λ ) ψ + δdR p ( ψ, λ ) ,dδF ( du ∧ ψ, λ ) = λ e − i λu du ∧ ψ + λ e i λu du ∧ C p ( λ ) ψ + dδR p ( du ∧ ψ, λ ) . (29)Comparing the leading terms, it follows that λ − δdF ( ψ, λ ) = F ( ψ, λ ) , λ − dδF ( du ∧ ψ, λ ) = F ( du ∧ ψ, λ ) . Since all derivatives of R p ( ψ, λ ) and R p ( du ∧ ψ, λ ) are exponentially decaying, the unique-ness statement in theorem 2.1 implies immediately C p ( λ ) = 0 and C p ( λ ) = 0. (cid:3) Note that the splitting Λ p T ∗ Z = π ∗ (Λ p T ∗ Y ) ⊕ π ∗ (Λ p − T ∗ Y ) indeed corresponds to theHodge decomposition. Let P and P the projection on the first and second summand,respectively. Then we have Proposition 2.4.
We have P ψ = 0 , i.e. ψ ∈ ker∆ (cid:48) p if and only if δ (cid:101) F ( ψ, λ ) = 0 . Similarly P ψ = 0 , i.e. ψ ∈ ker∆ (cid:48) p − if and only if d (cid:101) F ( ψ, λ ) = 0 .Proof. Let ψ = ( ψ , ψ ) ∈ ker ∆ (cid:48) p ⊕ ker ∆ (cid:48) p − . Suppose that δ (cid:101) F ( ψ, λ ) = 0. Then, inparticular, we have δF ( du ∧ ψ , λ ) = 0. Applying δ to the second equation of (27) andusing Proposition 2.3, it follows that ψ = 0. For the other direction, observe that by (27), δF ( ψ , λ ) = δR ( ψ , λ ) on Z . Hence, δF ( ψ , λ ) is exponentially decaying. In particular itis square integrable for Im( λ ) >
0. Since ∆ p is self-adjoint and (∆ p − λ ) δF ( ψ , λ ) = 0 we CATTERING AT LOW ENERGIES 9 get δF ( ψ , λ ) = 0. Thus, if ψ = 0, it follows that δ (cid:101) F ( ψ, λ ) = 0. The proof of the othercase is analogous. (cid:3) Proposition 2.5.
The following relation holds for ≤ p < n . C p ( λ ) | ker∆ (cid:48) p = − C p +1 ( λ ) | ker∆ (cid:48) p , (30) Proof.
Let ψ ∈ ker∆ (cid:48) p . Using (27), we geti λ − dF ( ψ, λ ) | Z = e − i λu du ∧ ψ − e i λu du ∧ C p ( λ ) ψ + i λ − dR p ( ψ, λ )(31)Comparing the leading terms, it follows from Theorem 2.1 thati λ − dF ( ψ, λ ) = F ( du ∧ ψ, λ ) . Using (27), we get C p ( λ ) = − C p +1 ( λ ), which is equivalent to the statement of the Propo-sition. (cid:3) Let us use the notation S p ( λ ) for the restriction of C p ( λ ) to ker∆ (cid:48) p . Then the aboveproposition shows that C p ( λ ) = (cid:18) S p ( λ ) 00 − S p − ( λ ) (cid:19) . (32)In the following we will suppress the index p and write S ( λ ), meaning that S ( λ ) is actingon the space of harmonic forms, leaving the space of p -forms invariant. It is the scatteringmatrix at low energy for the scattering problem for coclosed forms. Summarizing we haveestablished the following theorem. Theorem 2.6.
For each harmonic p -form ψ ∈ ker(∆ (cid:48) p ) there exists a p -form F ( ψ, λ ) on X , which is meromorphic in λ in the disc | λ | < µ such that (i) δF ( ψ, λ ) = 0(ii) F ( ψ, λ ) is holomorphic in λ for Im( λ ) > . (iii) (∆ p − λ ) F ( ψ, λ ) = 0 . (iv) F ( ψ, λ ) = e − i λu ψ + e i λu S ( λ ) ψ + R ( ψ, λ ) on R + × Y , (v) S ( λ ) ∈ End(ker(∆ (cid:48) p )) and R ( ψ, λ ) ∈ L Λ p ( Z ) are meromorphic functions of λ .Moreover, S ( λ ) , R ( ψ, λ ) and F ( ψ, λ ) are uniquely determined by these properties. The scattering matrix has the following properties
Theorem 2.7.
The function S ( λ ) satisfies the following equations (i) S ( λ ) ∗ S ( λ ) = 1 . (ii) S ( λ ) S ( − λ ) = 1 . (iii) S ( λ ) ∗ = − ∗ S ( λ ) ,where ∗ is the Hodge star operator on Y .Proof. For a > X a be the manifold ([0 , a ) × Y ) ∪ Y X with boundary Y a = { a } × Y .Let ω X be the volume form of X . By Theorem 2.6, (iii), we have (cid:90) X a (cid:104) F ( ψ, λ ) , ∆ p F ( ψ, λ ) (cid:105) ω X − (cid:90) X a (cid:104) ∆ p F ( ψ, λ ) , F ( ψ, λ ) (cid:105) ω X = 0 . Using Green’s formula, we obtain (cid:90) Y a (cid:104) F ( ψ, λ ) , − ∂ u F ( ψ, λ ) (cid:105) ω Y + (cid:90) Y a (cid:104) ∂ u F ( ψ, λ ) , F ( ψ, λ ) (cid:105) ω Y = 0 . (33)In the limit a → ∞ this expression can be evaluated using Theorem 2.6. We obtainlim a →∞ (cid:90) Y a ( (cid:104) F ( ψ, λ ) , − ∂ u F ( ψ, λ ) (cid:105) + (cid:104) ∂ u F ( ψ, λ ) , F ( ψ, λ ) (cid:105) ) ω Y == − λ ( (cid:107) ψ (cid:107) − (cid:104) S ( λ ) ψ, S ( λ ) ψ (cid:105) ) = 0 , (34)which proves the first statement. The second statement follows from the functional equa-tion F ( C ( λ ) ψ, − λ ) = F ( ψ, λ ) , which is a simple consequence of the uniqueness statement in theorem 2.1. To show that C ( λ ) anticommutes with the Hodge star operator on Y we note that that Hodge staroperator ∗ X on X commutes with the Laplace operator ∆, i.e. ∗ X ∆ p = ∆ n − p ∗ X . Applyingthis to F ( ψ, λ ) and using the uniqueness statement we obtain immediately ∗ X C p = C n − p ∗ X . For ψ ∈ ker∆ (cid:48) p we get ∗ X ψ = ( − p du ∧ ∗ ψ and consequently ∗ S ( λ ) = − S ( λ ) ∗ , where weused that du ∧ anticommutes with C ( λ ). (cid:3) As an application we obtain the following well known result about the signature sign( Y )of a closed manifold Y . Corollary 2.8.
Let Y be a closed oriented manifold. Assume that Y is the boundary of acompact manifold. Then sign( Y ) = 0 .Proof. We may assume that dim Y = 4 k . Otherwise the signature is zero. Pick a Rie-mannian metric on Y . Let H k ± ( Y ) be the ± ∗ acting in H k ( Y ). Then thesignature sign( Y ) of Y is given bysign( Y ) = dim H k + ( Y ) − dim H k − ( Y ) . If Y is the boundary of a compact Riemannian manifold M , it follows from Theorem 2.7,that S ( λ ) is regular at λ = 0, S (0) = 1 I and S (0) intertwines H k + ( Y ) and H k − ( Y ). Hencewe get sign( Y ) = 0. (cid:3) Remark 2.9.
The same proof works equally well for Dirac type operators. It implies thecobordism invariance of the index of Dirac operators.
Proposition 2.10. S ( λ ) , F ( ψ, λ ) , and R ( ψ, λ ) are regular for real λ . If ψ = − S (0) ψ then F ( ψ,
0) = 0 .Proof.
For real λ it follows from Theorem 2.7 that S ( λ ) S ∗ ( λ ) = 1 I and therefore (cid:107) S ( λ ) (cid:107) = 1.In particular, S is bounded on the real line and can not have a pole there. It remains to CATTERING AT LOW ENERGIES 11 show that F is regular for real λ . Suppose that φ is a square integrable eigensection of ∆with real eigenvalue λ (cid:48) . Then the expansion (22) of φ on R + × Y takes the form φ ( u, y ) = (cid:88) µ ψi >λ (cid:48) a i e − (cid:113) µ ψi − λ (cid:48) u j p ( ψ i ) . This implies that φ is exponentially decaying. Since for real λ (cid:104) (∆ − λ ) F ( ψ, λ ) , φ (cid:105) = − ( λ − λ (cid:48) ) (cid:104) F ( ψ, λ ) , φ (cid:105) , (35)and (cid:104) F ( ψ, λ ) , φ (cid:105) is a meromorphic function we get (cid:104) F ( ψ, λ ) , φ (cid:105) = 0. Suppose now that F ( ψ, λ ) has a pole of order k at λ (cid:48) . Then G := lim λ → λ (cid:48) ( λ − λ (cid:48) ) k F ( ψ, λ ) is an eigenformwith eigenvalue λ (cid:48) and it also is square integrable since S is regular at λ (cid:48) . By the above (cid:104) G, G (cid:105) = 0. It follows that G = 0 and therefore F ( ψ, λ ) is regular at λ (cid:48) . If ψ = − S (0) ψ then F ( ψ,
0) is square integrable and harmonic and by the same argument F ( ψ,
0) = 0. (cid:3)
By the above F ( ψ,
0) and F (cid:48) ( ψ,
0) := ∂∂λ F ( ψ, λ ) | λ =0 are well defined. From the proof ofProposition 2.10 we obtain the following corollary. Corollary 2.11. F ( ψ, and F (cid:48) ( ψ, are orthogonal to the space H p (2) ( X ) of square inte-grable harmonic forms. The scattering matrix S ( λ ) is also regular at 0 and it follows from Theorem 2.7 that S (0)is a self-adjoint involution. Hence, ker(∆ (cid:48) p ) decomposes into +1 and − ψ ∈ ker(∆ (cid:48) p ) with S (0) ψ = ψ we get that F ( ψ,
0) is a smooth coclosed harmonic p -formwhose restriction to Z equals F ( ψ,
0) = 2 ψ + R ( ψ, , where R ( ψ,
0) and its derivatives are exponentially decaying. That is, ψ is a limiting valueof F ( ψ,
0) in the sense of Atiyah, Patodi, and Singer ([APS75]). It turns out that theconverse is also true.
Proposition 2.12.
The +1 eigenspace for S (0) is the set of limiting values of coclosedharmonic forms on X , i.e, it equals { ψ ∈ ker∆ (cid:48) p | ∃ G ∈ Λ p ( X ) : G | Z − ψ ∈ L Λ p ( Z ) , ∆ p G = 0 , δG = 0 } . Furthermore for each ψ ∈ ker ∆ (cid:48) p , F ( ψ, satisfies dF ( ψ,
0) = 0 and δF ( ψ,
0) = 0 .Proof.
Suppose that F ∈ Λ p ( X ) and G ∈ Λ n − − p ( X ) are both coclosed harmonic forms on X with limiting values ψ and φ , respectively. Since ψ − G and φ − F are both exponentiallydecaying and ψ and φ are closed and coclosed on Y the forms dG and dF are exponentiallydecaying. Using Green’s formula, we get0 = (cid:104) ∆ G, G (cid:105) = (cid:104) δdG, G (cid:105) = (cid:104) dG, dG (cid:105) . Thus dG = 0. Similarly we get dF = 0. Using Stokes formula, it follows that0 = (cid:90) X a dG ∧ F = ± (cid:90) Y ψ ∧ φ + O ( e − cu ) . Thus, (cid:104) ψ, ∗ φ (cid:105) = 0. Now suppose that φ is a limiting value which is in the − S (0). Since ∗ anticommutes with S (0), it follows that ∗ φ is in the +1 eigenspace. Itis therefore a limiting value. Since ∗ φ and φ are both limiting values it follows from theabove that (cid:107) φ (cid:107) = ±(cid:104) φ, ∗ ∗ φ (cid:105) = 0 and therefore, φ = 0. Since the set of limiting vectorscontains the +1 eigenspace it has to coincide with the +1 eigenspace. (cid:3) Finally we derive some formulas concerning F (cid:48) ( ψ,
0) which we are going to use in thenext section. Note that the restriction of F (cid:48) ( ψ, λ ) to the cylinder Z has the form F (cid:48) ( ψ, λ ) | Z = − i u ( e − i λu ψ − e +i λu S ( λ ) ψ ) + e i λu S (cid:48) ( λ ) ψ + R (cid:48) ( ψ, λ ) , (36)and for λ = 0:(37) F (cid:48) ( ψ, | Z = − i u (1 − S (0)) ψ + S (cid:48) (0) ψ + R (cid:48) ( ψ, . Differentiating the equation (∆ − λ ) F ( ψ, λ ) = 0(38)it follows that ∆ F (cid:48) ( ψ,
0) = 0 . (39)Hence, dF (cid:48) ( ψ,
0) is in the kernel of ∆ and its restriction to the cylinder has the form dF (cid:48) ( ψ, | Z = − i du ∧ (1 − S (0)) ψ + dR (cid:48) ( ψ, . (40)By Theorem 2.7, (iii), we get(41) ∗ X dF (cid:48) ( ψ, | Z = − i(1 + S (0)) ∗ ψ + ∗ X dR (cid:48) ( ψ, . Thus, ∗ X dF (cid:48) ( ψ,
0) is an extended harmonic form with limiting value − i(1 + S (0)) ∗ ψ .Since ∗ X dF (cid:48) ( ψ,
0) is coexact and bounded it is orthogonal to the space H ∗ (2) ( X ) of squareintegrable harmonic forms. The proof of Prop. 2.10 shows that F ( − i(1 + S (0)) ∗ ψ, H ∗ (2) ( X ). Their difference is therefore square integrable,harmonic and orthogonal to H ∗ (2) ( X ). Thus, it vanishes and we have the following niceformula ∗ X dF (cid:48) ( ψ,
0) = − i2 F ((1 + S (0)) ∗ ψ, . (42)In particular if ψ is in the − S (0) we have ∗ X dF (cid:48) ( ψ,
0) = − i F ( ∗ ψ, , (43)or equivalently(44) dF (cid:48) ( ψ,
0) = − i F ( du ∧ ψ, . Remark 2.13.
Note that whereas F ( ψ, λ ) is regular at λ = 0 the meromorphic continu-ation of the resolvent (∆ p − λ ) − is in general not regular at zero but has a pole of theform − K λ − + ( K + K ) λ − , where K , K and K are finite rank operators. K is theorthogonal projection onto the space of L -harmonic p -forms and comes from the discretepart of the spectrum. K corresponds to the part of the continuous spectrum spanned byco-closed generalized eigenforms. Using a deformation of the contour of integration in the CATTERING AT LOW ENERGIES 13 spectral representation of the resolvent one can show (see e.g. Prop. 4.2 in [Str05] ) thatthe integral kernel k ( x, y ) of K is given by k ( x, y ) = − i4 (cid:88) i F ( ψ i , x ) F ( ψ i , y ) , where ψ i is an orthonormal basis in H p ( Y ) . The part K comes from the part of thecontinuous spectrum spanned by the closed generalized eigenforms. In the same way as forco-closed generalized eigenforms one obtains k ( x, y ) = − i4 (cid:88) i F ( du ∧ ψ i , x ) F ( du ∧ ψ i , y ) , where ψ i is an orthonormal basis in H p − ( Y ) . Cohomology and Hodge theory on M As before let M be a compact manifold with boundary Y and X = ( R + × Y ) ∪ Y M theassociated manifold with a cylindrical end. We consider the long exact sequence(45) . . . ∂ (cid:47) (cid:47) H k ( M, Y, R ) e (cid:47) (cid:47) H k ( M, R ) r (cid:47) (cid:47) H k ( Y, R ) ∂ (cid:47) (cid:47) H k +1 ( M, Y, R ) e (cid:47) (cid:47) . . . , in de Rham cohomology. Here e is the canonical embedding and r is the restriction ho-momorphism. There are three cochain complexes which compute the relative de Rhamcohomology. Let Λ p ( M, Y ) := { ω ∈ Λ p ( M ) : i ∗ ω = 0 } , where i : Y → M is the inclusion. Since d commutes with i ∗ , we get a complex Λ ∗ ( M, Y ).Its cohomology is denoted by H ∗ ( M, Y, R ). There is an exact sequence of complexes0 (cid:47) (cid:47) Λ ∗ ( M, Y ) j (cid:47) (cid:47) Λ ∗ ( M ) i ∗ (cid:47) (cid:47) Λ ∗ ( Y ) (cid:47) (cid:47) , where j is the inclusion map. It gives rise to the long exact sequence (45). The connectinghomomorphism ∂ is defined as follows. Let [ φ ] ∈ H k ( Y, R ). Extend φ to a k -form ω on M such that ω = φ in a neighborhood of the boundary. Then ∂ [ φ ] = [ dω ] . (46)For the second description consider the cochain complex Λ ∗ rel ( M, Y ) of the mapping coneof i ∗ which is defined by Λ prel ( M, Y ) := Λ p ( M ) ⊕ Λ p − ( Y )with differential d given by d ( ω, θ ) = ( dω, i ∗ ω − dθ ) , ω ∈ Λ p ( M ) , θ ∈ Λ p − ( Y ) . Let α : Λ p − ( Y ) → Λ prel ( M, Y ) and β : Λ prel ( M, Y ) → Λ p ( M ) be defined by α ( θ ) = (0 , ( − p − θ ) and β ( ω, θ ) = ω , respectively. Then α and β are cochainmaps and we get a second exact sequence of cochain complexes0 (cid:47) (cid:47) Λ ∗− ( Y ) α (cid:47) (cid:47) Λ ∗ rel ( M, Y ) β (cid:47) (cid:47) Λ ∗ ( M ) (cid:47) (cid:47) γ : Λ ∗ ( M, Y ) → Λ ∗ rel ( M, Y ) , ω (cid:55)→ ( ω, . It follows from the corresponding commutative diagram of long exact sequences that γ induces an isomorphism γ : H ∗ ( M, Y, R ) ∼ = H ∗ rel ( M, Y, R ) . Finally H ∗ rel ( M, Y, R ) is also naturally isomorphic to the cohomology with compact supports H ∗ c ( X ). The isomorphism can be described as follows. Let p : Z = R + × Y → Y be thecanonical projection. Integration over the fibre R + of p induces a mapping p ∗ : Λ pc ( R + × Y ) → Λ p − ( Y ) . Define a map ξ : ω ∈ Λ pc ( X ) (cid:55)→ ( ω | M , − p ∗ ( ω | Z )) ∈ Λ prel ( M, Y )This is a chain map. If the support of ω is contained in M \ Y , then ξ ( ω ) = ( ω, H pc ( X ) has a representative of this form, it follows that ξ inducesan isomorphism ¯ ξ : H pc ( X ) → H prel ( M, Y ) . If we fix a metric on Y we may identify H k ( Y, R ) with the space of harmonic forms H k ( Y, R ). In fact, the image of e , i.e. the kernel of r can be read off from the scatteringmatrix at 0. Theorem 3.1.
The +1 eigenspace of the scattering matrix S (0) on H p ( Y, R ) coincideswith Im( H p ( M, R ) → H p ( Y, R )) .Proof. Let ψ ∈ H p ( Y, R ) with ψ = S (0) ψ . By Proposition 2.12, F ( ψ,
0) is closed andcoclosed. Therefore the restriction of F ( ψ,
0) to M defines a cohomology class in H p ( M ).Expand F ( ψ,
0) on Z in terms of an orthonormal basis of ker ∆ (cid:48) p ⊕ ker ∆ (cid:48) p − . Using that F ( ψ,
0) is closed and coclosed, it follows that its expansion on Z has the form F ( ψ,
0) = 2 ψ + (cid:88) µ φi > a i e − µ φi u ( d (cid:48) φ i − µ φ i du ∧ φ i ) , (47)where { φ i } i ∈ N is an orthonormal basis of δ (Λ p ( Y )) consisting of eigenforms of ∆ (cid:48) p − witheigenvalues µ φ i . In particular i ∗ Y F ( ψ,
0) = 2 ψ + d (cid:48) (cid:32)(cid:88) i a i φ i (cid:33) , (48)and therefore the image of the cohomology class [ F ( ψ, r is precisely [ ψ ]. There-fore we have shown that the +1 eigenspace of S (0) is contained in the image of H p ( M, R )in H p ( Y, R ). Now let φ be an element in the image of r , i.e. φ is a harmonic form that is CATTERING AT LOW ENERGIES 15 in the same cohomology class as the restriction of a closed form f on M . If ψ is in the − S (0) then by Theorem 2.7, ∗ ψ is in the +1 eigenspace and we have0 = (cid:90) M df ∧ F ( ∗ ψ,
0) = (cid:90) Y i ∗ Y ( f ) ∧ ∗ ψ = (cid:90) Y φ ∧ ∗ ψ = (cid:104) φ, ψ (cid:105) . (49)Hence, any element in Im( H p ( M, R ) → H p ( Y, R )) is in the orthogonal complement to the − S (0) which is exactly the +1 eigenspace. This shows that Im( H p ( M, R ) → H p ( Y, R )) is contained in the +1 eigenspace and this concludes the proof. (cid:3) Hence, the scattering matrix at 0 is determined completely by the metric on the bound-ary. Namely, it is equal to 1 on the kernel of ∂ and equal to − ∂ . Recall that by Proposition 2.12, F ( ψ,
0) is a closed andcoclosed p -form. Let ˆ F : H k ( Y, R ) → H k ( M, R ) be the map defined byˆ F ( ψ ) = [ 12 F ( ψ, | M ] . (50)Then, by construction, r ◦ ˆ F is the orthogonal projection onto the kernel of ∂ .Hodge theory for manifolds with boundary shows that absolute and relative cohomologyclasses have unique harmonic representatives that satisfy certain boundary conditions.We recall the definition of the relative and absolute boundary conditions for the Laplaceoperator. The operator ∆ rel is the closure of the Laplace operator with respect to therelative boundary conditions ω | Y = 0 , ( δω ) | Y = 0 . The operator ∆ abs is the closure of the Laplace operator with respect to the absoluteboundary conditions ( ∗ ω ) | Y = 0 , ( ∗ dω ) | Y = 0 . Both operators are self-adjoint and have compact resolvents. Their kernels are the space ofharmonic forms satisfying relative and absolute boundary conditions. Equivalently, theyare given by H prel ( M ) = { ω ∈ Λ p ( M ) : dω = δω = 0 , ω | Y = 0 } , H pabs ( M ) = { ω ∈ Λ p ( M ) : dω = δω = 0 , ( ∗ ω ) | Y = 0 } , Hodge theory for manifolds with boundary shows that the canonical maps H prel ( M ) → H p ( M, Y, R ) , H pabs ( M ) → H p ( M, R )are isomorphisms, that is, every absolute/relative cohomology class has a unique harmonicrepresentative satisfying absolute/relative boundary conditions (see e.g. [DS52]).The harmonic representative φ of the cohomology class [ φ ] ∈ H p ( M, Y, R ) is the uniqueminimizer of the functional ω (cid:55)→ (cid:104) ω, ω (cid:105) L ( M ) in [ φ ]. Similarly, any harmonic form satisfying absolute boundary conditions minimizesthe L -norm in its absolute cohomology class. Apart from these minimax principles thereis another interesting minimizing problem which is described in the following proposition. Theorem 3.2.
Let φ ∈ Λ p ( Y ) . Consider the functional F on { ω ∈ Λ p ( M ) : ω | Y = φ } which is defined by F ( ω ) = (cid:104) dω, dω (cid:105) L . Then there exists a unique coclosed harmonic form ω with ω | Y = φ such that ω isorthogonal to H prel ( M ) . The minimum of F is attained at ω and ω is the unique coclosedminimizer that is orthogonal to H prel ( M ) . If φ is closed dω is the harmonic representativein ∂ [ φ ] .Proof. We divide the proof into several steps.
Uniqueness:
If two forms ω and ω (cid:48) are harmonic, coclosed, and their restrictions to Y coincide, it follows that ω − ω (cid:48) is harmonic, coclosed and satisfies relative boundaryconditions. Therefore, ω − ω (cid:48) ∈ H prel ( M ). If both ω and ω (cid:48) are orthogonal to H prel ( M ) itfollows that ω = ω (cid:48) . Existence:
Choose any extension ˜ ψ of φ to M which in a neighborhood of Y is of theform φ + udu ∧ δφ. (51)Then δ ˜ ψ vanishes near Y . Next we claim that the form ∆ ˜ ψ is in the orthogonal complementof the kernel of ∆ rel . Indeed, if ξ ∈ H prel ( M ), then (cid:104) ξ, ∆ ˜ ψ (cid:105) = (cid:90) M dδ ˜ ψ ∧ ∗ ξ + (cid:90) M ξ ∧ ∗ δd ˜ ψ == (cid:90) Y δ ˜ ψ ∧ ∗ ξ − (cid:90) Y ξ ∧ ∗ d ˜ ψ = 0 , where the first integral vanishes because ˜ ψ is coclosed near Y and the second integral van-ishes because ξ satisfies relative boundary conditions. Let us denote by p the orthogonalprojection onto the kernel of ∆ rel . Since ∆ ˜ ψ is in the orthogonal complement of the kernelof ∆ rel the following form is well defined and harmonic ω = ˜ ψ − (cid:16) ∆ rel | H prel ( M ) ⊥ (cid:17) − (∆ ˜ ψ ) − p ˜ ψ. By construction it is also in the orthogonal complement of H prel ( M ) and satisfies ω | Y = φ .Moreover, ω is harmonic and since ∆ rel commutes with δ , we have δω = δ ˜ ψ − (cid:16) ∆ rel | H prel ( M ) ⊥ (cid:17) − ∆ δ ˜ ψ Since δ ˜ ψ vanishes near Y , it is in the domain of ∆ rel and therefore, the right hand sidevanishes. Hence δω = 0. CATTERING AT LOW ENERGIES 17
Minimizing property:
Suppose that ψ is a p -form with ψ | Y = 0, then (cid:104) d ( ω + ψ ) , d ( ω + ψ ) (cid:105) = (cid:104) dω , dω (cid:105) + 2 (cid:104) dω , dψ (cid:105) + (cid:104) dψ, dψ (cid:105) == (cid:104) dω , dω (cid:105) + (cid:104) dψ, dψ (cid:105) , because (cid:104) dω , dψ (cid:105) = (cid:104) δdω , ψ (cid:105) = 0. Thus, ω is a minimizer. One can see immediatelythat the Euler-Lagrange equations for the minimizer are the equations δdω = 0. Thus,any coclosed minimizer has to be harmonic. The uniqueness statement for the minimizerthus follows from the above uniqueness statement. (cid:3) Since restriction to the boundary commutes with the differential the map which sends φ to ω commutes with the differential. Therefore, if φ is exact or closed, so is ω .Now consider the manifold M a which is obtained from M by attaching the cylinder[0 , a ] × Y to M . Then M a is a manifold with boundary Y a . Let φ ∈ H p − ( Y ). We regardit as a harmonic form on Y a . By the Hodge theorem there is a unique harmonic form in H prel ( M a ) which represents ∂ [ φ ] ∈ H p ( M a , ∂M a ). Let us denote this form by ∂ a φ , wherethe notation ∂ a indicates that ∂ a maps H p − ( Y ) to different spaces depending on a . Foreach a we have the L -inner product on H prel ( M a ). It is a natural question to ask whether ∂ as a map from one Hilbert space to another one is a partial isometry. The scalar producton H prel ( M a ), however, depends on a . Theorem 3.3.
Let Q a be the sesquilinear form on ker( ∂ ) ⊥ which is defined by Q a ( ψ, φ ) = (cid:104) ∂ a ψ, ∂ a φ (cid:105) L ( M a ) and let q ( a ) be the unique linear operator ker( ∂ ) ⊥ → ker( ∂ ) ⊥ such that Q a ( ψ, φ ) = (cid:104) ψ, q ( a ) φ (cid:105) . Then, as a → ∞ : q ( a ) − = a I + i2 S (cid:48) (0) | ker( ∂ ) ⊥ + O ( ae − µ a ) . Proof.
Recall that ker( ∂ ) ⊥ = ker(I + S (0)). Let φ ∈ ker( ∂ ) ⊥ . Then by (37) the restrictionof F (cid:48) ( φ,
0) to the cylinder Z has the following form F (cid:48) ( φ, | Z = − uφ + S (cid:48) (0) φ + R (cid:48) ( φ, . (52)It follows that F (cid:48) ( φ,
0) is a coclosed harmonic form. Thus, by theorem 3.2 F (cid:48) ( φ,
0) is acoclosed minimizer of the functional η → (cid:104) dη, dη (cid:105) L ( M a ) with boundary condition η | Y a = − aφ + S (cid:48) (0) φ + R (cid:48) ( φ, | Y a . Let H be the minimizer with boundary conditions H | Y = aφ + i2 S (cid:48) (0) φ so that dH is the unique harmonic representative of ∂ a (cid:0) ( a I + i2 S (cid:48) (0)) φ (cid:1) .Again by theorem 3.2 G φ := i F (cid:48) ( φ, − H (53)minimizes the functional η → (cid:104) dη, dη (cid:105) L ( M a ) with boundary conditions η | Y a = R (cid:48) ( φ, | Y a . (54) Then for every ψ ∈ ker( ∂ ) ⊥ we have Q a ( ψ, ( a I + i2 S (cid:48) (0)) φ ) = (cid:104) ∂ a ψ, i dF (cid:48) ( φ, (cid:105) L ( M a ) − (cid:104) ∂ a ψ, dG φ (cid:105) L ( M a ) . (55)The second term on the right hand side can be estimated using the Cauchy-Schwarz in-equality |(cid:104) ∂ a ψ, dG φ (cid:105) L ( M a ) | ≤ (cid:107) ∂ a ψ (cid:107) L ( M a ) · (cid:107) dG φ (cid:107) L ( M a ) . (56)The first term in (55) can be explicitly calculated. Using (44) we get (cid:104) ∂ a ψ, i dF (cid:48) ( φ, (cid:105) L ( M a ) = 12 (cid:90) Y ψ ∧ ∗ M F ( du ∧ φ,
0) == (cid:90) Y ψ ∧ ∗ φ = (cid:104) ψ, φ (cid:105) , where we used that R ( du ∧ φ, | Y is orthogonal to ψ . Thus, | Q a ( ψ, ( a I + i2 S (cid:48) (0)) φ ) − (cid:104) ψ, φ (cid:105)| ≤ (cid:107) ∂ a ψ (cid:107) L ( M a ) · (cid:107) dG φ (cid:107) L ( M a ) . (57)The terms on the right hand side can be estimated as follows. First note that (cid:107) dG φ (cid:107) L ( M a ) minimizes (cid:107) dη (cid:107) L ( M a ) over all forms η which restrict to R (cid:48) ( φ, | Y a on Y a . Moreover, χ a := R (cid:48) ( φ, | Y a is exponentially decaying in a . Define the form η a by η a := ua χ a on the cylinderand 0 elsewhere. Then, (cid:107) dη a (cid:107) L ( M a ) = a (cid:107) dχ a (cid:107) L ( Y ) + 1 a (cid:107) χ a (cid:107) L ( Y ) . (58)By Lemma 2.2 we have (cid:107) dχ a (cid:107) L ( Y ) + (cid:107) χ a (cid:107) L ( Y ) ≤ C φ e − µ a , (59)which implies (cid:107) dG φ (cid:107) L ( M a ) ≤ ˜ C φ e − µ a . (60)To estimate (cid:107) ∂ a ψ (cid:107) L ( M a ) , recall that by Theorem 3.2, ∂ a ψ = dω , where ω minimizes thefunctional η (cid:55)→ (cid:107) dη (cid:107) L ( M a ) with boundary conditions η | Y a = ψ . Let f ∈ C ∞ ( R − ) such that f ( u ) = 1 for − / ≤ u ≤ f ( u ) = 0 for u ≤ − /
4. For a ≥ ψ a ∈ Λ p ( M a ) byˆ ψ a ( x ) = (cid:40) f ( u − a ) ψ ( y ) , if x = ( u, y ) ∈ [0 , a ] × Y ;0 , otherwise . Then it follows that there exists
C > (cid:107) ∂ a ψ (cid:107) L Λ p +1 ( M a ) ≤ (cid:107) d ˆ ψ a (cid:107) L Λ p +1 ( M a ) ≤ C (cid:107) ψ (cid:107) L Λ p ( Y ) . Thus we get | Q a ( ψ, ( a I + i2 S (cid:48) (0)) φ ) − (cid:104) ψ, φ (cid:105)| ≤ C (cid:48)(cid:48) φ (cid:107) ψ (cid:107) e − µ a . (61) CATTERING AT LOW ENERGIES 19
By compactness of the unit sphere in ker( ∂ ) ⊥ the constant can be chosen independent of φ if we use vectors of norm 1 only. Hencesup (cid:107) ψ (cid:107) , (cid:107) φ (cid:107) =1 (cid:28) ψ, ( q ( a )( a I + i2 S (cid:48) (0)) − I ) φ (cid:29) ≤ Ce − µ a , which implies(62) (cid:107) a I + i2 S (cid:48) (0) − q ( a ) − (cid:107) ≤ C (cid:107) q ( a ) − (cid:107) e − µ a . From this inequality we deduce that (cid:107) q ( a ) − (cid:107) ≤ C (1 + a ). Combined with (62) the state-ment of the theorem follows. (cid:3) Hodge theory on X and the scattering length In this section we give a description of the long exact cohomology sequence of X in termsof harmonic forms and we derive a cohomological formula for the scattering length.Let H ∗ c ( X ) denote the de Rham cohomology groups with compact supports. It is wellknown (see [Mel93], Sec. 6.4) that H ∗ ( X ) and H ∗ c ( X ) are canonically isomorphic to certainspaces of extended harmonic forms on X . We recall some details.The space of extended harmonic forms H pext ( X ) is defined to be the subspace of all (realvalued) ψ ∈ Λ p ( X ) satisfying 1) ∆ p ψ = 0 and 2) there exist φ ∈ ker ∆ (cid:48) p and φ ∈ ker ∆ (cid:48) p − such that ψ | Z − φ − du ∧ φ ∈ L Λ p ( Z ) . Note that for a given ψ ∈ H pext ( X ) the sections φ and φ are uniquely determined. Weregard φ (resp. φ ) as the tangential (resp. normal) boundary value of ψ at infinity andwe denote them by ψ t and ψ n , respectively. The spaces satisfying absolute and relativeboundary conditions at infinity are then defined as H pext,abs ( X ) := { ψ ∈ H pext ( X ) | ψ n = 0 } , H pext,rel ( X ) := { ψ ∈ H pext ( X ) | ψ t = 0 } . Since ψ ∈ H pext ( X ) is harmonic and the form ψ − ψ t − du ∧ ψ n is square integrable, itfollows from (22) that there exists c > ψ − ψ t − du ∧ ψ n )( u, y ) (cid:28) e − cu , ( u, y ) ∈ Z. Moreover dψ and δψ are also exponentially decaying. Applying Greens formula to M a , weget 0 = (cid:104) ∆ ψ, ψ (cid:105) M a = (cid:107) dψ (cid:107) M a + (cid:107) δψ (cid:107) M a + O ( e − ca ) , which implies that(64) dψ = 0 , δψ = 0 for all ψ ∈ H pext ( X ) . The intersection H pext,abs ( X ) ∩H pext,rel ( X ) is the space H p (2) ( X ) of square integrable harmonicforms. On H pext ( X ) we introduce an inner product as follows. For ψ, φ ∈ H pext ( X ) let(65) (cid:104) ψ, φ (cid:105) = (cid:90) M ψ ∧ ∗ φ + (cid:90) Z ( ψ − ψ t − du ∧ ψ n ) ∧ ∗ ( φ − φ t − du ∧ φ n ) . To verify that this is an inner product, we only need to show that (cid:107) φ (cid:107) = 0 implies φ = 0. So suppose that (cid:107) φ (cid:107) = 0. Then, in particular, we have φ | M = 0 and the uniquecontinuation property for harmonic forms (see e.g. [BB00, BW93]) implies φ = 0. We notethat the inner product can be also defined by the following formula:(66) (cid:104) ψ, φ (cid:105) = lim a →∞ (cid:18)(cid:90) M a ψ ∧ ∗ φ − a ( (cid:104) ψ t , φ t (cid:105) + (cid:104) ψ n , φ n (cid:105) ) (cid:19) . This inner product coincides on the subspace H p (2) ( X ) with the usual inner product on H p (2) ( X ). The orthogonal projections define canonical maps H pext,rel ( X ) → H p (2) ( X ) , H pext,abs ( X ) → H p (2) ( X ) . Moreover, we have the mapsˆ F : H p ( Y ) → H pext,abs ( X ) , φ (cid:55)→ F ( φ, , ˆ G : H p ( Y ) → H pext,rel ( X ) , φ (cid:55)→ i2 dF (cid:48) ( φ, . (67)Next we define maps into the de Rham cohomology. Let φ ∈ H pext,abs ( X ). By (64), φ isclosed and we get a canonical map(68) R : H pext,abs ( X ) → H p ( X, R ) . Now consider ψ ∈ H pext,rel ( X ). By (64) ψ is closed and on the cylinder ψ is of the form(69) ψ | Z = du ∧ ψ n + dθ, where θ is exponentially decaying. Let χ be a function with support on the cylinder Z which is equal to 1 outside a compact set. Following [Mel93] we can then define a map R c : H pext,rel ( X ) → H pc ( X, R ) , ψ (cid:55)→ [ ψ − d ( χ ( uψ n + θ ))] . (70)This map is well defined and independent of the choice of χ . Indeed changing χ on acompact subset changes ψ − d ( χ ( uψ n + θ ))by the differential of a compactly supported form. Let( · , · ) : H pc ( X ) × H n − p ( X ) → R be the canonical pairing defined by([ φ ] , [ ψ ]) = (cid:90) X φ ∧ ψ, [ φ ] ∈ H pc ( X ) , [ ψ ] ∈ H n − p ( X ) . Define a pairing ( · , · ) ext : H pext,rel ( X ) × H n − pext,abs ( X ) → R (71) CATTERING AT LOW ENERGIES 21 by taking the constant term in the asymptotic expansion of (cid:90) M a ψ ∧ φ as a → ∞ . Applying Green’s formula to M a , it follows that there exists c > (cid:90) M a ψ ∧ φ = a (cid:90) Y ψ n ∧ φ t + ([ ψ − d ( χ ( u · ψ n + θ ))] , [ φ ]) + O ( e − ca )as a → ∞ . This implies that the following diagram commutes (72) H pext,rel ( X ) × H n − pext,abs ( X ) R H pc ( X, R ) × H n − p ( X, R ) ........................................................................................................ .................................................................................................................................................................................................................................................................................... R c ...................................................................................................................................................... R , where the horizontal maps are given by the corresponding pairing.Let φ ∈ H pext ( X ) and ω ∈ Λ p − c ( X ). Since φ is co-closed (64), it follows that (cid:104) dω, φ (cid:105) = 0.Therefore, for φ ∈ H pext ( X ), [ ψ ] ∈ H pc ( X ), and ψ (cid:48) ∈ [ ψ ], the inner product (cid:104) ψ (cid:48) , φ (cid:105) is inde-pendent of the representative of the cohomology class [ ψ ] and will be denoted by (cid:104) [ ψ ] , φ (cid:105) .This leads to the following alternative description of the inner product in H pext,rel ( X ). Lemma 4.1.
For all φ, ψ ∈ H pext,rel ( X ) we have (cid:104) ψ, φ (cid:105) = (cid:104) R c ( ψ ) , φ (cid:105) . Proof.
Applying Stoke’s theorem and using that θ is rapidly decreasing, we get (cid:90) M a d ( χ ( uψ n + θ )) ∧ ∗ φ = a (cid:104) ψ n , φ n (cid:105) + O ( e − ca ) . By (66) we get (cid:104) ψ, φ (cid:105) = (cid:104) ψ − d ( χ ( uψ n + θ )) , φ (cid:105) = (cid:104) R c ( ψ ) , φ (cid:105) . (cid:3) Our next goal is to describe the connecting homomorphism ∂ : H p ( Y, R ) → H p +1 c ( X, R )on the level of harmonic forms. To this end we need some preparation. Let ψ be in the − S (0). Then by (37), we have on Z i2 F (cid:48) ( ψ, | Z = uψ + i2 S (cid:48) (0) ψ + θ, where θ is exponentially decaying. Let χ be a smooth function with support in Z which isequal to 1 outside a compact set. Theni2 F (cid:48) ( ψ, − χ ( uψ + θ ) is equal to i2 S (cid:48) (0) ψ outside a compact set and we conclude that d ( i2 F (cid:48) ( ψ, − χ ( uψ + θ ))represents ∂ [ i2 S (cid:48) (0) ψ ] in H p +1 c ( X, R ). Let κ : H p ( Y ) → H p ( Y ) be the canonical isomor-phism. Then we have shown that for each ψ ∈ H p ( Y ) we have(73) R c ( ˆ G ( ψ )) = ∂ (cid:20) κ ( i S (cid:48) (0) ψ ) (cid:21) , where ˆ G ( ψ ) is defined by (67). Lemma 4.2.
The operator S (cid:48) (0) in H ∗ ( Y ) is invertible.Proof. Differentiating equations (ii) and (iii) of Theorem 2.7, it follows that S (cid:48) (0) commuteswith S (0) and anti-commutes with ∗ . Therefore, it suffices to show that the restriction of S (cid:48) (0) to the − E − of S (0) is invertible. Let ψ ∈ E − . Then S (cid:48) (0) ψ ∈ E − . ByTheorem 3.1 we have E − = (ker ∂ ) ⊥ . Using (73), it follows that it suffices to show that R c ( ˆ G ( ψ )) (cid:54) = 0 whenever ψ (cid:54) = 0. By Lemma 4.1 we have (cid:104) ˆ G ( ψ ) , ˆ G ( ψ ) (cid:105) = (cid:104) R c ( ˆ G ( ψ )) , ˆ G ( ψ ) (cid:105) for all ψ ∈ H pext,rel ( X ). Recall that ˆ G ( ψ ) is a harmonic form, which is non-zero, if ψ (cid:54) = 0.Therefore, the left hand side of the above equality is non-zero, if ψ (cid:54) = 0. (cid:3) Now we define maps˜ e : H pext,rel ( X ) → H pext,abs ( X ) , ˜ r : H pext,abs ( X ) → H p ( Y ) , ˜ ∂ : H p ( Y ) → H p +1 ext,rel ( X )as follows. Let ˜ e be the composition of the orthogonal projection H pext,rel ( X ) → H p (2) ( X )and the inclusion H p (2) ( X ) → H pext,abs ( X ). ˜ r assigns to φ ∈ H pext,abs ( X ) its limiting value φ t . To define ˜ ∂ , we note that by Lemma 4.2, S (cid:48) (0) is an invertible operator. Put˜ ∂ = ˆ G ◦ (cid:18) i2 S (cid:48) (0) (cid:19) − . Proposition 4.3.
The sequence . . . H pext,rel ( X ) H pext,abs ( X ) H p ( Y ) H p +1 ext,rel ( X ) . . . ............................................................................................. ............ ˜ ∂ ............................................... ............ ˜ e ......................................................................... ............ ˜ r .......................................................................... ............ ˜ ∂ ............................................................................................. ............ ˜ e is exact.Proof. Let E ± = ker( S (0) ∓ Id). By Proposition 2.12 and (42) it follows that(74) Im(˜ r ) = E + = ker( ˆ G ) . Since S (cid:48) (0) preserves E ± , we get Im(˜ r ) = ker( ˜ ∂ ). By definition we have Im(˜ e ) = H p (2) ( X )and this is also equal to ker(˜ r ). Finally by Corollary 2.11 it follows that Im( ˜ ∂ ) = H p +1(2) ( X ) ⊥ .On the other hand, by definition we have ker(˜ e ) = H p +1(2) ( X ) ⊥ . Thus Im( ˜ ∂ ) = ker(˜ e ). (cid:3) CATTERING AT LOW ENERGIES 23
Using the definition of ˜ ∂ and (73), it follows that R c ◦ ˜ ∂ = ∂ ◦ κ. By [APS75] every element in the image of H ∗ c ( X, R ) in H ∗ ( X, R ) can be representedby a unique square integrable harmonic form. Using these facts we obtain the followingcommutative diagram. . . . H pext,rel ( X ) H p ( Y ) H p (2) ( X ) H pext,abs ( X ) H p ( Y )ker( S (0) − I ) H p ( Y ) H p +1 ext,rel ( X ) . . .. . . H pc ( X, R ) H p ( X, R ) H p ( Y, R ) H p +1 c ( X, R ) . . . ....................................................................................................... ............ ∂ ..................................................................... ............ e ....................................................................... ............ r ........................................................... ............ ∂ .......................................................................................... ............ e ............................................................................................. ............ ˜ ∂ .................................................................................................................................................................................................. ..................................................................................................................................................... ......................................................................... ............ ˜ r .......................................................................... ............ ˜ ∂ ............................................................................................. ............ ˜ e ...................................................................................................................................................... R c ...................................................................................................................................................... R ...................................................................................................................................................... κ ...................................................................................................................................................... R c .............................................................................................................................................................................................................................................................................................................................................................. ˆ F ............................................... ............ ˜ e ................................................................................................................................................................................................................... i2 S (cid:48) (0) ...................................................................................................................................................... ˆ G ............................................................................................................................................................................................................ i2 S (cid:48) (0) ...................................................................................................................................................... ˆ G Proposition 4.4.
The maps R : H pext,abs ( X ) → H p ( X, R ) , R c : H pext,rel ( X ) → H pc ( X, R ) are isomorphisms.Proof. We first consider R . Let H p ! ( X, R ) = Im( e ). By [APS75], R induces an isomorphismof H p (2) ( X ) onto H p ! ( X ). Let φ ∈ H pext,abs ( X ) and suppose that R ( φ ) = 0. Then it followsthat ˜ r ( φ ) = 0. Hence φ ∈ H p (2) ( X ). Since R is an isomorphism on φ ∈ H p (2) ( X ), we get φ =0. This proves injectivity. Let ψ ∈ H p ( X, R ). Using H p ( X, R ) ∼ = H p ( M, R ) and Theorem3.1, it follows that κ − ( r ( ψ )) ∈ ker( S (0) − Id). Thus by (74) there exists φ ∈ H pext,abs ( X )such that ˜ r ( φ ) = κ − ( r ( ψ )). Then r ( R ( φ ) − ψ ) = 0. Hence R ( φ ) − ψ ∈ H p ! ( X, R ). By theabove remark there is ω ∈ H p (2) ( X ) such that R ( ω ) = R ( φ ) − ψ . Thus R is surjective andhence an isomorphism. Applying the commutativity of the diagramm and the 5-Lemma,it follows that R c is an isomorphism too. (cid:3) This can also be proved by slightly different methods (e.g. [Mel95] and [Mel93], Sec.6.4).
Corollary 4.5.
In every class in H p ( X, R ) there is a unique representative in H pext,abs ( X ) . Corollary 4.6.
For every class [ ψ ] in H pc ( X, R ) there is a unique element ˆ ψ in H pext,rel ( X ) such that for any φ ∈ H pext,abs ( X ) : (cid:104) [ ψ ] , [ φ ] (cid:105) = (cid:104) ˆ ψ, φ (cid:105) . Moreover, the map H pc ( X, R ) → H pext,rel ( X ) , [ ψ ] (cid:55)→ ˆ ψ is an isomorphism. We can now consider the scattering length(75) T (0) := − i S (0) ∗ S (cid:48) (0) = − i S (0) S (cid:48) (0) . Let ∂ : H p ( Y, R ) → H p +1 c ( X, R ) be the connecting homomorphism. We identify H p ( Y, R )with H p ( Y ) and H p +1 c ( X, R ) with H p +1 ext,rel ( X ) via Corollary 4.6. Thus we may regard theconnecting homomorphism as a map ∂ : H p ( Y ) → H p +1 ext,rel ( X ) . Let (ker ∂ ) ⊥ be the orthogonal complement of ker ∂ . Theorem 4.7.
The scattering length T (0) is a positive, invertible operator in H p ( Y ) . Itis uniquely determined by the following conditions. ∀ φ, ψ ∈ (ker ∂ ) ⊥ : (cid:104) ∂φ, ∂ ( T (0) ψ ) (cid:105) = 2 (cid:104) φ, ψ (cid:105) , (76) T (0) ∗ = ∗ T (0) . (77) Proof.
By Theorem 3.1, Im( r ) = ker ∂ equals the +1 eigenspace of S (0). Therefore (ker ∂ ) ⊥ equals the − S (0). By Theorem 2.7, ∗ anti-commutes with S (0). Hence itinterchanges the ± ∗ also anti-commutes with S (cid:48) (0) and consequently thescattering length T (0) commutes with the Hodge star operator. It is therefore completelydetermined by its restriction to (ker ∂ ) ⊥ . Let φ, ψ ∈ (ker ∂ ) ⊥ . Using (73) and Lemma 4.1,we have (cid:104) ∂φ, ∂T (0) (cid:105) = (cid:104) ∂φ, − i∂ ( S (cid:48) (0) S (0) ψ ) (cid:105) = 2 (cid:104) ∂φ, ∂ (( i/ S (cid:48) (0) ψ ) (cid:105) = 2 (cid:104) ∂φ, R c ( ˆ G ( ψ )) (cid:105) = 2 (cid:104) ∂φ, ˆ G ( ψ ) (cid:105) . (78)Let ˜ φ be the pull-back of φ to Z and let χ ∈ C ∞ ( Z ) such that χ = 0 on [0 , × Y and χ = 1 outside a compact set. Then d ( χ ˜ φ ) ∈ Λ p +1 c ( X ) represents ∂ [ φ ] ∈ H p +1 c ( X, R ). Bythe same argument as in the proof of Lemma 4.1 we get (cid:104) ∂φ, ˆ G ( ψ ) (cid:105) = (cid:104) d ( χ ˜ φ ) , ˆ G ( ψ ) (cid:105) . By (41), ∗ ˆ G ( ψ ) = ∗ i dF (cid:48) ( ψ,
0) is an extended harmonic form with limiting value ∗ ψ . UsingStokes theorem, applied to M a , it follows that(79) (cid:104) d ( χ ˜ φ ) , ˆ G ( ψ ) (cid:105) = lim a →∞ (cid:90) ∂M a φ ∧ ∗ ˆ G ( ψ ) = (cid:90) Y φ ∧ ∗ ψ = (cid:104) φ, ψ (cid:105) . This concludes the proof of the theorem. (cid:3)
Corollary 4.8.
For the scattering length T (0) = − i S (0) ∗ S (cid:48) (0) we have the following for-mula T − (0) = 12 (cid:0) ∂ ∗ ∂ + ( ∗ ) − ∂ ∗ ∂ ∗ (cid:1) = 12 (cid:0) rr ∗ + ( ∗ ) − rr ∗ ∗ (cid:1) . This implies that r ∗ T (0) r is equal to the orthogonal projection onto the orthogonalcomplement of ker r . CATTERING AT LOW ENERGIES 25 Estimates on the norm of extended harmonic forms
By the results of the previous sections we have canonical isomorphisms η abs : H pext,abs ( X ) ∼ = H p ( X, R ) ∼ = H p ( M, R ) ,η rel : H pext,rel ( X ) ∼ = H pc ( X, R ) ∼ = H p ( M, Y, R ) . (80)In this section we establish relations between some norms on these spaces. On the coho-mology groups H p ( M, R ) and H p ( M, Y, R ) there is the so-called comass norm (see [Gro99,Ch. 4C]) which is defined as follows. If V is a finite dimensional inner product space, thenΛ p V ∗ has a natural inner product as well and we denote the norm that is induced by thisinner product by (cid:107) · (cid:107) . The comass norm (cid:107) · (cid:107) ∞ on Λ p V ∗ is defined by (cid:107) ω (cid:107) ∞ = sup { ω ( e , . . . , e p ) | e k ∈ V, (cid:107) e k (cid:107) = 1 } (81)Since the norms are equivalent there is a constant C such that (cid:107) ω (cid:107) ≤ C (cid:107) ω (cid:107) ∞ , (82)and we denote by C ( n, p ) the optimal such constant. Since all n -dimensional inner productspaces are unitarily equivalent the constant depends only on n and p . Of course (see also[Fed69]), C ( n,
0) = C ( n,
1) = 1 , (83) C ( n, p ) ≤ (cid:18) np (cid:19) . (84)Moreover, since the Hodge star operator leaves the space of primitive forms invariant, wehave C ( n, n − p ) = C ( n, p ) . (85)It is also known that C ( n,
2) = [ n B be a differentiable manifold. Let ω ∈ Λ p ( B ). The comass (cid:107) ω (cid:107) ∞ of ω is defined by (cid:107) ω (cid:107) ∞ = sup { ω x ( e , . . . , e p ) | x ∈ B, e i ∈ T x B, g ( e i , e i ) = 1 } =(87) = sup {(cid:107) ω x (cid:107) ∞ | x ∈ B } . For a compact manifold B with smooth boundary ∂B this induces a norm on H p ( B, ∂B, R )by (cid:107) φ (cid:107) ∞ = inf {(cid:107) ω (cid:107) ∞ | φ = [ ω ] , ω ∈ Λ p ( B, ∂B ) , dω = 0 } , (88)which we also refer to as the comass norm. To compare the norms on the various co-homology groups, we need some preparation. Let ψ ∈ Λ p ( M ). We define an extension ˆ ψ ∈ L ∞ Λ p ( X ) of ψ in the following way. The restriction ψ | Y can be expanded into eigen-sections of the Laplace-Beltrami operator on Y : ψ | Y = φ + ∞ (cid:88) i =1 a i φ i , where φ is harmonic and φ i is an orthonormal basis in the orthogonal complement of thespace of harmonic forms such that ∆ (cid:48) φ i = µ φ i φ i . Now defineˆ ψ ( x ) = (cid:26) ψ ( x ) for x ∈ M,φ ( y ) + (cid:80) ∞ i =1 a i e − µ φi u (cid:0) φ i ( y ) − µ − φ i du ∧ δ (cid:48) φ i ( y ) (cid:1) for x = ( u, y ) ∈ Z (89)As usually x = ( u, y ). The map ψ (cid:55)→ ˆ ψ is of course linear and maps into the space ofbounded sections. Note that in general ˆ ψ is not continuous. However, it satisfies i ∗ Y ( ˆ ψ | M ) = i ∗ Y ( ˆ ψ | Z ) . Therefore, using Green’s formula, it follows that the distributional derivative d ˆ ψ satisfies d ˆ ψ ( x ) = ∞ (cid:88) i =1 a i e − µ φi u (cid:0) d (cid:48) φ i − µ φ i du ∧ φ i + µ − φ i du ∧ d (cid:48) δ (cid:48) φ i (cid:1) =(90) = ∞ (cid:88) i =1 a i e − µ φi u (cid:0) d (cid:48) φ i − µ − φ i du ∧ δ (cid:48) d (cid:48) φ i (cid:1) = (cid:99) dψ. In particular d ˆ ψ is again bounded. If ψ is closed, then ˆ ψ is also closed. Note that theextension map ψ ∈ Λ p ( M ) (cid:55)→ ˆ ψ ∈ L ∞ Λ p ( X ) is chosen so that it inverts the restrictionoperator on the space H pext,abs ( X ). Namely, for F ∈ H pext,abs ( X ) it follows from (47) that F = (cid:100) F | M . (91)We can now establish the comparison results for the norms. Lemma 5.1.
Let ψ ∈ Λ p ( M ) be closed and F ∈ H pext,abs ( X ) a harmonic form such that F | M and ψ represent the same element in H p ( M, R ) . Let φ ∈ H p ( Y ) be the unique harmonicrepresentative of the cohomology class of ψ | Y . Then (cid:107) F (cid:107) ≤ (cid:107) ψ (cid:107) L Λ p ( M ) + 1 µ (cid:107) ψ | Y − φ (cid:107) L Λ p ( Y ) , (92) where µ is the smallest positive eigenvalue of ∆ Y . In particular (cid:107) F (cid:107) ≤ C ( n, p )Vol ∗ ( M ) (cid:107) [ F | M ] (cid:107) ∞ , (93) where we define the effective volume Vol ∗ ( M ) by Vol ∗ ( M ) = Vol( M ) + 1 µ Vol( Y ) . (94) CATTERING AT LOW ENERGIES 27
Proof.
Let F ∈ H pext,abs ( X ). Let ψ ∈ Λ p ( M ) be a closed form such that F | M − ψ = dh, (95)for some h ∈ Λ p − ( M ). Denote by φ the unique harmonic representative of the class of ψ | Y . If χ Z is the characteristic function of Z ⊂ X , then the norm of F is by definition the L -norm of F − φχ Z . Therefore, (cid:107) ˆ ψ − φχ Z (cid:107) = (cid:107) F − φχ Z + ( ˆ ψ − F ) (cid:107) =(96) = (cid:107) F − φχ Z (cid:107) + (cid:107) ( ˆ ψ − F ) (cid:107) + 2 (cid:104) F − φχ Z , ( ˆ ψ − F ) (cid:105) . (97)Now observe that by (95) and the definition of ˆ ψ , the expansion of ˆ ψ − F on Z containsno harmonic form. Hence by (63) and (89) the restriction of ˆ ψ − F to Z is exponentiallydecreasing. This implies (cid:104) F − φχ Z , ˆ ψ − F (cid:105) = (cid:104) F, ˆ ψ − F (cid:105) . By (95) and (90) we have F − ˆ ψ = (cid:99) dh = d ˆ h , where the latter means the distributionalderivative. Since F is closed and coclosed, it follows that (cid:104) F − φχ Z , F − ˆ ψ (cid:105) = (cid:104) F, (cid:99) dh (cid:105) = (cid:104) F, d ˆ h (cid:105) = (cid:104) δF, ˆ h (cid:105) = 0 . (98)Therefore we get (cid:107) ˆ ψ − φχ Z (cid:107) = (cid:107) F − φχ Z + ( ˆ ψ − F ) (cid:107) == (cid:107) F − φχ Z (cid:107) + (cid:107) ( ˆ ψ − F ) (cid:107) ≥ (cid:107) F (cid:107) . (99)On the other hand using an expansion of the form (47) with a basis ( φ i ) i ∈ N of co-exacteigenforms on the boundary we have (cid:107) ˆ ψ − φχ Z (cid:107) = (cid:107) ψ (cid:107) L Λ p ( M ) + (cid:90) ∞ ∞ (cid:88) i =1 | a i | e − µ φi u (cid:0) (cid:107) d (cid:48) φ i (cid:107) + µ φ i (cid:107) φ i (cid:107) (cid:1) du == (cid:107) ψ (cid:107) L Λ p ( M ) + ∞ (cid:88) i =1 µ − φ i | a i | (cid:107) d (cid:48) φ i (cid:107) = (cid:107) ψ (cid:107) L Λ p ( M ) + (cid:107) ∆ − Y ( ψ | Y − φ ) (cid:107) L Λ p ( Y ) ≤≤ (cid:107) ψ (cid:107) L Λ p ( M ) + 1 µ (cid:107) ψ | Y − φ (cid:107) . (100) (cid:3) Lemma 5.2.
Let φ ∈ H pext,rel ( X ) . Let ψ ∈ Λ p ( M, Y ) be a representative of the class [ ψ ] ∈ H p ( M, Y, R ) which corresponds to φ with respect to the isomorphism (80) . Then (cid:107) φ (cid:107) ≤ (cid:107) ψ (cid:107) L Λ p ( M ) , (101) and in particular (cid:107) φ (cid:107) ≤ C ( n, p )Vol( M ) (cid:107) [ ψ ] (cid:107) ∞ . (102) Proof.
Let φ ∈ H pext,rel ( X ). Recall the definition of R c by (70). Choose χ such that thesupport of φ − d ( χ ( uφ n + θ )) is contained in M . Then [ φ − d ( χ ( uφ n + θ ))] is the image of φ w.r.t. the isomorphism (80). Since ψ ∈ Λ p ( M, Y ) represents this cohomology class, thereis ω ∈ Λ p − ( M, Y ) such that φ − d ( χ ( uφ n + θ )) = ψ + dω. (103)Let ˜ ψ (resp. ˜ ω ) be the differential form on X which is equal to ψ (resp. ω ) on M and 0on Z . Then (cid:107) ψ (cid:107) = (cid:107) ˜ ψ − φ + χ Z du ∧ φ n + ( φ − χ Z du ∧ φ n ) (cid:107) = (cid:107) ˜ ψ − φ + χ Z du ∧ φ n (cid:107) + (cid:107) φ − χ Z du ∧ φ n (cid:107) + 2 (cid:104) ˜ ψ − φ + χ Z du ∧ φ n , φ − χ Z du ∧ φ n (cid:105) . (104)By the definition (65) of the norm in H pext,rel ( X ), we have (cid:107) φ (cid:107) = (cid:107) φ − χ Z du ∧ φ n (cid:107) L . By (103) we have (cid:104) ˜ ψ − φ + χ Z du ∧ φ n , φ − χ Z du ∧ φ n (cid:105) = (cid:104) ˜ ψ − φ, φ − χ Z du ∧ φ n (cid:105) = −(cid:104) d ˜ ω + d ( χ ( uφ n + θ )) , φ − χ Z du ∧ φ n (cid:105) . Since ω ∈ Λ p − ( M, Y ), it follows from Green’s formula that (cid:104) d ˜ ω, φ − χ Z du ∧ φ n (cid:105) = (cid:90) M dω ∧ ∗ φ = (cid:90) Y i ∗ Y ( ω ) ∧ i ∗ Y ( ∗ φ ) = 0 . Similarly, by Green’s formula and (69) we have (cid:104) d ( χ ( uφ n + θ )) , φ − χ Z du ∧ φ n (cid:105) = (cid:90) M d ( χ ( uφ n + θ )) ∧ ∗ φ + (cid:90) Z d ( χ ( uφ n + θ )) ∧ ∗ ( φ − χ Z du ∧ φ n )= (cid:90) Y θ ∧ ∗ ( du ∧ φ n + dθ ) − (cid:90) Y θ ∧ ∗ dθ = 0 . Thus (cid:104) ˜ ψ − φ + χ Z du ∧ φ n , φ − χ Z du ∧ φ n (cid:105) = 0and by (104) we get (cid:107) ψ (cid:107) = (cid:107) φ (cid:107) + (cid:107) ˜ ψ − φ + χ Z du ∧ φ n (cid:107) ≥ (cid:107) φ (cid:107) . (cid:3) CATTERING AT LOW ENERGIES 29 Estimates on the scattering matrix and stable systoles
We recall some notions from geometric measure theory. Suppose B is a compact orientedRiemannian manifold and let A be a closed submanifold. In our case A will be either theboundary of B or the empty set. For z ∈ H p ( B, A, Z ) let the minimal volume be definedas the infimum of the volumes of all its representatives, i.e.vol( z ) = inf { (cid:88) i | α i | Vol( c i ) | z = (cid:88) i α i [ c i ] , α i ∈ Z } . (105)where the infimum is over all Lipschitz continuous simplices c i . The stable norm (cid:107) z (cid:107) st ofan element z ∈ H p ( B, A, R ) is defined similarly by (cid:107) z (cid:107) st := inf { (cid:88) i | α i | Vol( c i ) | z = (cid:88) i α i [ c i ] , α i ∈ R } . (106)This defines indeed a norm (see [FF60, 9.6, 9.9], and [Fed75, § z ∈ H p ( B, A, Z ) is by definition thestable norm of its image in H p ( B, A, R ). Clearly, (cid:107) z (cid:107) st ≤ vol( z )(107)and equality does not hold in general. However, as shown by Federer (see [Fed75, § (cid:107) z (cid:107) st = lim k →∞ k vol( kz ) . (108)Moreover, for z ∈ H n − ( B, A, Z ) we have equality, i.e. (cid:107) z (cid:107) st = vol( z ).By a general result in geometric measure theory the stable norm and the comass aredual to each other, i.e., (cid:107) z (cid:107) st = sup {| φ ( z ) | | φ ∈ H p ( B, A, R ) , (cid:107) φ (cid:107) ∞ ≤ } (109)(see [Fed75, 4.10], and also [Gro99, 4.35], and [AuB06] for a sketch of the proof in the casewithout boundary).6.1. Estimate of the L -norm on H ∗ ( Y ) . Now we will apply this result to our problemin the case where B = Y and A = ∅ . We equip H p ( Y ) with the norm induced from H p ( Y )by the de Rham isomorphism. Suppose that ω is a p -form on Y . Then using (82) oneobtains (cid:107) ω (cid:107) ≤ C ( n − , p )Vol( Y ) (cid:107) ω (cid:107) ∞ , (110)Now let φ be a harmonic p -form representing an element [ φ ] ∈ H p ( Y, R ). Using (110), weget (cid:107) φ (cid:107) = sup (cid:26)(cid:90) φ ∧ ω | ω ∈ Λ n − p − ( Y ) , dω = 0 , (cid:107) ω (cid:107) ≤ (cid:27) ≥ C ( n − , p ) − / Vol( Y ) − / sup (cid:26)(cid:90) φ ∧ ω | ω ∈ Λ n − p − ( Y ) , dω = 0 , (cid:107) ω (cid:107) ∞ ≤ (cid:27) = C ( n − , p ) − / Vol( Y ) − / sup (cid:8) (cid:104) [ φ ] ∪ α, [ Y ] (cid:105) | α ∈ H n − p − ( Y, R ) , (cid:107) α (cid:107) ∞ ≤ (cid:9) Since the stable norm is dual to the comass norm, we finally get (cid:107) φ (cid:107) ≥ C ( n − , p ) − / Vol( Y ) − / (cid:107) [ Y ] ∩ φ (cid:107) st , (111)for any φ ∈ H p ( Y, R ).On the other hand the inequality (110) may also be used directly. Since (cid:107) φ (cid:107) is theinfimum of the L -norms of all representatives of the cohomology class [ φ ], we have Proposition 6.1.
The Hilbert space norm on H p ( Y, R ) , induced from the harmonic forms,satisfies C ( n − , p ) − / Vol( Y ) − / (cid:107) [ Y ] ∩ φ (cid:107) st ≤ (cid:107) φ (cid:107) ≤ C ( n − , p ) / Vol( Y ) / (cid:107) φ (cid:107) ∞ . (112)6.2. Estimate of the norms on H ext ( X ) . We can use Lemmas 5.1 and 5.2 in the sameway as above to get similar estimates of the norms of elements in H pext,rel ( X ). First notethat ∗ induces an isomorphism ∗ : H pext,rel ( X ) → H n − pext,abs ( X ) . Let ( · , · ) ext be the pairing (71). Then we have (cid:104) φ, ψ (cid:105) = ( φ, ∗ ψ ) ext , φ, ψ ∈ H pext,rel ( X ) . Let F ∈ H pext,rel ( X ). Then we get (cid:107) F (cid:107) = sup {(cid:104) F, ω (cid:105) | ω ∈ H pext,rel ( X ) , (cid:107) ω (cid:107) ≤ } = sup (cid:8) ( F, ψ ) ext | ψ ∈ H n − pext,abs ( X ) , (cid:107) ψ (cid:107) ≤ (cid:9) . (113)Choose a representative φ of the cohomology class R c ( F ) ∈ H pc ( X ) with supp( φ ) ⊂ M .Then η rel ( F ) = [ φ ]. Since the diagramm (72) commutes, we get (cid:104) φ, ψ (cid:105) = ( R c ( φ ) , R ( ∗ ψ )) = ([ φ ] , [ ∗ ψ | M ]) , φ, ψ ∈ H pext,rel ( X ) . Using Lemma 5.1, it follows from (113) that (cid:107) F (cid:107) = sup { ([ φ ] , [ ψ | M ]) | ψ ∈ H n − pext,abs ( X ) , (cid:107) ψ (cid:107) ≤ }≥ C ( n, p ) − / Vol ∗ ( M ) − / sup {(cid:104) [ φ ] ∪ α, [ M ] (cid:105) | α ∈ H n − p ( M, R ) , (cid:107) α (cid:107) ∞ ≤ } . Since the stable norm is dual to the comass norm, we finally get (cid:107) F (cid:107) ≥ C ( n, p ) − / Vol ∗ ( M ) − / (cid:107) [ M ] ∩ [ φ ] (cid:107) st , (114)Note that the Poincare-Lefschetz dual [ M ] ∩ [ φ ] of [ φ ] is in H p ( M, R ) and its stable normequals its stable norm as an element of H p ( M, R ). Lemma 5.2 gives an upper bound for (cid:107) F (cid:107) and we get Proposition 6.2.
Let F ∈ H pext,rel ( X ) and φ = η rel ( F ) ∈ H p ( M, Y, R ) . Then we have C ( n, p ) − / Vol ∗ ( M ) − / (cid:107) [ M ] ∩ φ (cid:107) st ≤ (cid:107) F (cid:107) ≤ C ( n, p ) / Vol( M ) / (cid:107) φ (cid:107) ∞ . (115) CATTERING AT LOW ENERGIES 31
Using Theorem 4.7, we obtain Theorem 1.1.If we denote by ι the inclusion map Y → M and by ι ∗ : H p ( Y, R ) → H p ( M, R ) theinduced map in homology, then [ M ] ∩ ∂φ = ι ∗ ([ Y ] ∩ φ ) . (116)That is [ M ] ∩ ∂φ coincides with the image of the Poincare dual of the class φ in H n − p − ( M, R ).The stable norm of ∂φ can be calculated in many cases explicitly in terms of geometricdata using the fact that the stable mass is dual to the comass norm. In order to makestatements about the spectrum of the map T (0) one can combine these estimates with theestimates of the L -norms of the cohomology classes on Y (see Proposition 6.1).7. Examples
The scattering length for functions.
Note that the extended harmonic functionsare exactly the constant functions. Therefore, H ( Y ) ∩ ker ∂ is spanned by the functionequal to 1 on Y . Thus, the +1 eigenspace to S (0) is spanned by 1. Moreover, since S anticommutes with the Hodge ∗ operator we have S n − (0) ∗ − ∗
1. By Stokes formula( ∂ [ ∗ M ] = [ ∗ Y ](117)and therefore ∂ [ ∗
1] = Vol( Y )Vol( M ) [ ∗ M T n − (0) | (ker ∂ ) ⊥ = 2 Vol( M )Vol( Y ) . (119)This in turn implies that T (0) | ker ∂ = 2 Vol( M )Vol( Y ) . (120)7.2. Y has only one connected component. In this case H ( Y ) consists of the constantfunctions only. By the above we have T n − (0) = 2 Vol( M )Vol( Y ) . (121)This in turn implies that T (0) = 2 Vol( M )Vol( Y ) . (122)Note that if Y = S n − the only non-vanishing cohomology groups are H ( Y ) and H n − ( Y ).Thus, in this case the above formulas determine T (0) completely. Y = Y ∪ Y has two boundary components. Now we have H ( Y ) ∼ = R andtherefore on H ( Y ) is a direct sum of the one dimensional spaces ker ∂ and ker ∂ ⊥ . Underthe splitting H ( Y ) = ker ∂ ⊕ ker ∂ ⊥ the operator T (0) is of the form T (0) = (cid:18) t t (cid:19) , (123)where we have already seen, that t = 2 Vol( M )Vol( Y ) . (124)Our formula now allows us to give an estimate of t in purely geometric terms. Theharmonic functions in ker ∂ are multiples of the function 1 on Y . The complement ker ∂ ⊥ has dimension 1 and is spanned by the function χ ( x ) = (cid:26) Vol( Y ) for x ∈ Y − Vol( Y ) for x ∈ Y (125)Clearly, (cid:107) χ (cid:107) = Vol( Y ) Vol( Y ) + Vol( Y ) Vol( Y ) . (126)It remains to estimate the L -norm of the class ∂χ in H ( M, Y, R ). The comass normof ∂χ may be calculated using the duality between the stable norm and the comass norm.Let c be a Lipschitz continuous chain whose boundary is homologuous to y − y , where y ∈ Y and y ∈ Y . Then c defines a relative cycle [ c ] in H ( M, Y, R ) and we have ∂χ ([ c ]) = (cid:90) c ∂χ = Vol( Y ) + Vol( Y ) = Vol( Y ) . (127)We observe that the cycle c can be written as a linear combination of curves which areeither closed or whose end points are in the boundary. This implies (cid:107) [ c ] (cid:107) st ≥ dist( Y , Y ) , (128)with equality if for example [ c ] is in the same homology class as a shortest curve connecting Y with Y .Duality implies (cid:107) ∂χ (cid:107) ∞ = dist( Y , Y ) − Vol( Y ) , (129)and therefore (cid:107) ∂χ (cid:107) ≤ Vol( M ) / dist( Y , Y ) − Vol( Y ) . (130)To get the estimate from above we need to look at (cid:107) [ M ] ∩ ∂χ (cid:107) st , that is the stable normof the Poincare dual of the class ∂χ . By Equation (116) we have [ M ] ∩ ∂χ = ι ∗ ([ Y ] ∩ χ ).Of course, ι ∗ ([ Y ] ∩ χ ) = ι ∗ (Vol( Y )[ Y ] − Vol( Y )[ Y ]) = Vol( Y ) ι ∗ ([ Y ]) , (131) CATTERING AT LOW ENERGIES 33 where we have used that ι ∗ ([ Y ]) + ι ∗ ([ Y ]) = 0. Therefore, (cid:107) M ∩ ∂χ (cid:107) st = Vol( Y ) (cid:107) ι ∗ ([ Y ]) (cid:107) st . (132)Combining these two estimates we obtain C ≤ t ≤ C (133)with C = 2Vol ∗ ( M ) Vol( Y )Vol( Y ) (cid:107) ι ∗ ([ Y ]) (cid:107) st (Vol( Y ) + Vol( Y )) , (134) C = 2Vol( M ) − dist( Y , Y ) Vol( Y )Vol( Y )Vol( Y ) + Vol( Y ) . (135)Note that with respect to this basis, the scattering matrix at zero has the form S (0) = (cid:18) − (cid:19) . (136)In order to interpret this in terms of reflection and transmission coefficients we chooseanother basis ( χ , χ ) of H ( Y ), where χ i is constant equal to 1 on Y i and equal to zero onthe other boundary component. In this basis we get S (0) = 1Vol( Y ) + Vol( Y ) (cid:18) Vol( Y ) − Vol( Y ) 2Vol( Y )2Vol( Y ) Vol( Y ) − Vol( Y ) (cid:19) . (137)The reflection coefficient r and the transmission coefficient r of a wave of low energythat comes in at the end Y and is scattered in M therefore are r = Vol( Y ) − Vol( Y )Vol( Y )+Vol( Y ) and r = Y )Vol( Y )+Vol( Y ) . Transforming T (0) into this basis gives T (0) = 1Vol( Y ) + Vol( Y ) (cid:18) t Vol( Y ) + t Vol( Y ) ( t − t )Vol( Y )( t − t )Vol( Y ) t Vol( Y ) + t Vol( Y ) (cid:19) . (138)As a remark we would like to add here that in physics the scattering matrix for onedimensional scattering problems has the transmission coefficients on the diagonal and thereflection coefficients off the diagonal. Our notation differs here as we consider the operatorwith Neumann boundary conditions at each end as the unperturbed operator.7.4. The full-torus.
Let M be the full torus D × S with boundary Y = T = S × S .We view both D and S as subsets of the complex planes and use coordinates z = re i y on D and z = e i x on S . We assume that we are given a metric on M which has productstructure in a small neighborhood of the boundary and that the metric on the boundaryis equal to the product metric (cid:96) dx + (cid:96) dy , (139)with positive real numbers (cid:96) , (cid:96) . Then, the volume is Vol( Y ) = 4 π (cid:96) (cid:96) . Moreover, H ( M, R ) ∼ = R and it is gerated by the class of the one form dx . The group H ( Y, R ) is α β Figure 2.
The cycles α ∈ H ( M, Y ) and β ∈ H ( M ) on M .isomorphic to R and is generated by the classes of the two harmonic 1-forms dx and dy .The L -norms of these forms is easily calculated. || dx || L = (cid:90) Y dx ∧ ∗ dx = 4 π (cid:96) (cid:96) , (140) || dy || L = (cid:90) Y dy ∧ ∗ dy = 4 π (cid:96) (cid:96) . (141)and, moreover, dx and dy are orthogonal to each other. The restriction of the form dx on M to Y is the form dx regarded as a form on Y . Therefore, the kernel of the connectinghomomorphism ∂ is spanned by [ dx ]. Hence, (ker ∂ ) ⊥ is spanned by [ dy ]. Since the Hodgestar operator commutes with T (0) the map T (0) has the form T (0) = (cid:18) t t (cid:19) , (142)with respect to the decomposition H ( T , R ) = ker ∂ ⊕ (ker ∂ ) ⊥ . In order to get an estimatefor t we need to calculate (cid:107) ∂ [ dy ] (cid:107) ∞ and (cid:107) [ M ] ∩ ∂ [ dy ] | st . The stable norm of (cid:107) ∂ [ dy ] (cid:107) ∞ canagain be calculated by duality. For this note that H ( M, Y, R ) ∼ = R by Alexander dualityand it is generated by the relative cycle α = D × { } ⊂ M . Now, of course (cid:104) ∂ [ dy ] , [ α ] (cid:105) = (cid:90) ∂D ×{ } dy = 2 π. (143)Duality of comass norm and stable norm now implies that (cid:107) ∂ [ dy ] (cid:107) ∞ = 2 π (cid:107) [ D × { } ] (cid:107) − st . (144)The class [ M ] ∩ ∂ [ dy ] is given by − πι ∗ ( { } × S ). So π (cid:107) [ M ] ∩ ∂ [ dy ] (cid:107) st is equal to theinfimum of the lengths of all representatives of the cycle β = { } × S in H ( M, R ). Thegeometric picture is described in Fig. 2. Theorem 1.1 now gives an estimate of t :2 (cid:96) (cid:96) (cid:107) α (cid:107) st Vol( M ) ≤ t ≤ (cid:96) (cid:96) Vol ∗ ( M ) (cid:107) β (cid:107) st . (145)Since µ = min { (cid:96) − , (cid:96) − } we have Vol ∗ ( M ) = Vol( M ) + 4 π (cid:96) (cid:96) max { (cid:96) , (cid:96) } . CATTERING AT LOW ENERGIES 35
Appendix A. Dynamical approach and spectral decomposition
In this appendix we discuss the relation between the stationary and the dynamicalapproach to scattering theory and we establish the Eisenbud-Wigner formula for manifoldswith cylindrical ends. For details concerning scattering theory we refer to [Ya92]. Fororiginal papers on the Eisenbud-Wigner formula see [Eis48] and [Wi55]. In order to simplifythe relation of the scattering length to the time-delay operator we consider scattering theoryfor the square root of the Laplace Beltrami operator, i.e. we consider scattering theoryand the time-delay in relativistic quantum mechanics.Let ∆ p be the closure in L of the operator ∆ p with domain Λ pc ( X ). Denote by ∆ p, theself-adjoint extension of ∆ p with respect to Neumann boundary conditions along Y . Notethat ∆ p, is the self-adjoint operator associated to the quadratic form φ (cid:55)→ (cid:90) M q x ( φ ) dx + (cid:90) Z q x ( φ ) ,q x ( φ ) = dφ ( x ) ∧ ∗ dφ ( x ) + δφ ( x ) ∧ ∗ δφ ( x )on H ( M, Λ p T ∗ M ) ⊕ H ( Z, Λ p T ∗ Z ) ⊂ L Λ p ( X ). With respect to the decomposition L Λ p ( X ) = L Λ p ( M ) ⊕ L Λ p ( Z )we have ∆ p, = ∆ p,M ⊕ ∆ p,Z , where ∆ p,M and ∆ p,Z are the corresponding self-adjoint extensions of ∆ p | M and ∆ p | Z ,respectively, with Neumann boundary conditions imposed. Let H be the square root of ∆ p defined by spectral calculus and similarly define H = (∆ p, ) / , H M = ∆ / p,M , and H Z =∆ / p,Z . Since M is compact, H M has purely discrete spectrum. The spectral resolution of H Z can be determined, using separation of variables. The spectrum is absolutely continuous.A complete set of generalized eigensections is given by F ( φ, λ ) = (cid:16) e +i √ λ − µ φ u + e − i √ λ − µ φ u (cid:17) j p ( φ ) , (146)where φ runs through an orthonormal basis of eigensection of the operator ∆ (cid:48) p ⊕ ∆ (cid:48) p − witheigenvalue µ φ .Denote by P ac (resp. P ac ) the orthogonal projection onto the absolutely continuous sub-space of H (resp. H ). Denote by H ac and H ,ac the restrictions of H and H , respectively,to the absolutely continuous subspaces. Furthermore, for an open interval ( a, b ) we denoteby P ac ( a, b ) (resp. P ac ( a, b )) the projection onto the continuous part of the spectral sub-space of the interval. As explained above, the absolutely continuous subspace for H is L Λ p ( Z ). Thus P ac is the orthogonal projection of L Λ p ( X ) onto the subspace L Λ p ( Z ).By [Gui89, Th´eor`em 3.6] and the Birman-Kato invariance principle the wave operators W ± = s − lim t →±∞ e i tH e − i tH P ac (147)exist and are complete. This means that the strong limit exists and and the operators W ± define isometries of P ac L Λ p ( X ) onto P ac L Λ p ( X ), intertwining H ,ac and H ac . In this context the scattering operator S is defined as S = W ∗ + W − . (148)This is a unitary operator in P ac L Λ p ( X ) which commutes with H ,ac . Let σ = σ ac ( H ) bethe absolutely continuous spectrum of H . It equals [ µ, ∞ ) with µ ≥
0. Let { E ( λ ) } λ ∈ σ be the spectral family of H ,ac . Since S commutes with H ,ac , we have S = (cid:90) σ S ( λ ) dE ( λ ) , where S ( λ ) = S ( λ ; H, H ) acts in the finite-dimensional Hilbert space H ( λ ) = (cid:77) µ ≤ λ Eig µ (∆ (cid:48) p ) ⊕ Eig µ (∆ (cid:48) p − ) . (149)For the purposes of this paper we need only to investigate the structure of the continuousspectrum of H in the interval [0 , µ ) in the situations when µ = 0. The generalizedeigensections F ( φ, λ ) constructed in Theorem 2.1 are well defined as distributions on (0 , µ )with values in L Λ p ( X ). This follows easily from the properties of the Fourier transformand the expansion of F ( φ, λ ) on Z . Of course, in the weak topology on the space ofdistributions we have (cid:104) F ( φ, λ ) , F ( ψ, λ (cid:48) ) (cid:105) = lim a →∞ (cid:90) M a F ( φ, λ )( x ) ∧ ∗ F ( ψ, λ (cid:48) )( x ) . (150)The smooth function (cid:90) M a F ( φ, λ )( x ) ∧ ∗ F ( ψ, λ (cid:48) )( x )(151)can be determined as the continuous extension of1 λ − ( λ (cid:48) ) (cid:90) M a ∆ p F ( φ, λ ) ∧ ∗ F ( ψ, λ (cid:48) )( x ) − F ( φ, λ ) ∧ ∗ ∆ p F ( ψ, λ (cid:48) )( x ) . (152)This integral can be simplified using Green’s formula and the limit a → ∞ can be explicitlyevaluated (see Equ. (172)). A simple exercise in distribution theory shows that indeed wehave the orthogonality relations (cid:104) F ( φ, λ ) , F ( ψ, λ (cid:48) ) (cid:105) = 2 π (cid:104) φ, ψ (cid:105) δ ( λ − λ (cid:48) )(153)as distributions on (0 , µ ) × (0 , µ ). Moreover, in the same way as in [Gui89], Theorem 6.2,one shows that W + F ( φ, λ ) = F ( C ∗ ( λ ) φ, λ ) , (154) W − F ( φ, λ ) = F ( φ, λ ) . (155)Therefore, in the distributional sense for λ ∈ (0 , µ ) S F ( φ, λ ) = F ( C p ( λ ) , λ )(156)which shows that the dynamical and the stationary scattering matrices coincide S ( λ ) = C p ( λ ) . (157) CATTERING AT LOW ENERGIES 37
The time delay operator T is defined in the following way. If φ ∈ P ac L Λ p ( X ) then,according to the laws of quantum mechanics, the probability of finding the particle withwave function φ in M a at time t is given by (cid:90) M a (cid:107) e − i Ht φ (cid:107) x dx = (cid:107) χ M a e − i Ht φ (cid:107) . (158)The total time spent in M a is then given by (cid:90) ∞−∞ (cid:107) χ M a e − i Ht φ (cid:107) dt. (159)This expression is not necessarily finite for all φ . Now, according to scattering theory, for φ ∈ P ac L Λ p ( X ) the states e − i tH W − φ and e − i H t φ are asymptotically the same for t → −∞ .Thus, the time excess due to the interaction (the presence of M) is (cid:90) ∞−∞ (cid:0) (cid:107) χ M a e − i Ht W − φ (cid:107) − (cid:107) χ M a e − i H t φ (cid:107) (cid:1) dt. (160)The Eisenbud-Wigner time-delay operator T is formally defined by (cid:104) φ, T φ (cid:105) = lim a →∞ (cid:90) ∞−∞ (cid:0) (cid:107) χ M a e − i Ht W − P ac φ (cid:107) − (cid:107) χ M a e − i H t P ac φ (cid:107) (cid:1) dt. (161)In many situations in potential scattering it can be shown that the above defines a closablequadratic form and T is a self-adjoint operator that commutes with H and the Eisenbud-Wigner formula T = (cid:90) σ ac ( H ) T ( λ ) dE ( λ ) , (162) T ( λ ) = − i S ( λ ) − d S dλ ( λ )(163)holds. Since we are only interested in the spectrum near 0, we prove this formula only forelements in P ac (0 , µ ). Proposition A.1.
Suppose that g ∈ C ∞ (0 , µ ) , φ ∈ ker∆ (cid:48) p ⊕ ∆ (cid:48) p − . Let F ( φ, g ) := (cid:82) R F ( φ, λ ) g ( λ ) dλ . Then, (cid:90) ∞−∞ (cid:107) χ M a e − i Ht W − F ( φ, g ) (cid:107) dt < ∞ , (cid:90) ∞−∞ (cid:107) χ M a e − i H t F ( φ, g ) (cid:107) dt < ∞ . Moreover, (cid:104) F ( φ, g ) , T F ( φ, g ) (cid:105) = 2 π (cid:90) σ T ( λ ) g ( λ ) dE ( λ ) , (164) where T ( λ ) = − i S ( λ ) − S (cid:48) ( λ ) . (165) Hence, T is a self-adjoint operator on P ac (0 , µ ) L Λ p ( X ) which has the form T = (cid:90) σ T ( λ ) dE ( λ )(166) with respect to the spectral family { E ( λ ) } λ ∈ σ of H ,ac .Proof. We can do this calculation in the explicit spectral decomposition. (cid:90) ∞−∞ (cid:107) χ M a e − i Ht F ( φ, g ) (cid:107) dλ == (cid:90) ∞−∞ (cid:90) (cid:90) (cid:90) M a e − i( λ − λ (cid:48) ) t (cid:104) F ( φ, λ ) , F ( φ, λ (cid:48) ) (cid:105) x g ( λ ) g ( λ (cid:48) ) dxdλdλ (cid:48) dt =(167) = 2 π (cid:90) (cid:104) F ( φ, λ ) , χ M a F ( φ, λ ) (cid:105) g ( λ ) dλ. and similarly, (cid:90) ∞−∞ (cid:107) χ M a e − i Ht F ( φ, g ) (cid:107) dλ == 2 π (cid:90) (cid:104)(cid:104) F ( φ, λ ) , χ M a F ( φ, λ ) (cid:105) g ( λ ) dλ. (168)In the limit a → ∞ both integrals can be computed up to exponentially small errors sincewe have the expansions on Z : F ( φ, λ ) | Z = e − i λu j p ( φ ) + e +i λu j p ( S ( λ ) φ ) + R, (169) F ( φ, λ ) | Z = e − i λu j p ( φ ) + e +i λu j p ( φ ) . (170)Therefore, integration by parts yields (cid:104) F ( φ, λ ) , χ M a F ( φ (cid:48) , λ (cid:48) ) (cid:105) == 1 λ − ( λ (cid:48) ) (cid:90) M a (cid:104) ∆ p F ( φ, λ ) , F ( φ (cid:48) , λ (cid:48) ) (cid:105) x − (cid:104) F ( φ, λ ) , ∆ p F ( φ (cid:48) , λ (cid:48) ) (cid:105) x =(171) = 2 λ − λ (cid:48) sin (( λ − λ (cid:48) ) a ) (cid:104) φ, φ (cid:48) (cid:105) + 2 λ + λ (cid:48) sin (( λ + λ (cid:48) ) a ) (cid:104) φ, φ (cid:48) (cid:105) . Similarly, we have (cid:104) F ( φ, λ ) , χ M a F ( φ (cid:48) , λ (cid:48) ) (cid:105) == 1 λ − ( λ (cid:48) ) (cid:90) M a (cid:104) ∆ p F ( φ, λ ) , F ( φ (cid:48) , λ (cid:48) ) (cid:105) x − (cid:104) F ( φ, λ ) , ∆ p F ( φ (cid:48) , λ (cid:48) ) (cid:105) x == 2 λ − λ (cid:48) sin (( λ − λ (cid:48) ) a ) (cid:104) φ, φ (cid:48) (cid:105) + (cid:18) i λ − λ (cid:48) e i( λ − λ (cid:48) ) a (cid:19) (cid:104) φ, (1 I − S ∗ ( λ ) S ( λ (cid:48) )) φ (cid:48) (cid:105)− (172) − i λ + λ (cid:48) (cid:16) e i( λ + λ (cid:48) ) a (cid:104) φ, S ∗ ( λ ) φ (cid:48) (cid:105) − e − i( λ + λ (cid:48) ) a (cid:104) φ, S ( λ (cid:48) ) φ (cid:48) (cid:105) (cid:17) + O ( e − µ a ) . CATTERING AT LOW ENERGIES 39
Therefore, (cid:104) F ( φ, λ ) , χ M a F ( φ (cid:48) , λ (cid:48) ) (cid:105) − (cid:104) F ( φ, λ ) , χ M a F ( φ (cid:48) , λ (cid:48) ) (cid:105) == (cid:18) i λ − λ (cid:48) e i( λ − λ (cid:48) ) a (cid:19) (cid:104) φ, (1 I − S ∗ ( λ ) S ( λ (cid:48) )) φ (cid:48) (cid:105) − λ + λ (cid:48) sin (( λ + λ (cid:48) ) a ) (cid:104) φ, φ (cid:48) (cid:105)−− i λ + λ (cid:48) (cid:16) e i( λ + λ (cid:48) ) a (cid:104) φ, S ∗ ( λ ) φ (cid:48) (cid:105) − e − i( λ + λ (cid:48) ) a (cid:104) φ, S ( λ (cid:48) ) φ (cid:48) (cid:105) (cid:17) + O ( e − µ a ) , and one obtainslim a →∞ lim λ (cid:48) → λ (cid:104) F ( φ, λ ) , χ M a F ( φ (cid:48) , λ ) (cid:105) − (cid:104) F ( φ, λ ) , χ M a F ( φ (cid:48) , λ ) (cid:105) = (cid:104) φ, T ( λ ) φ (cid:48) (cid:105) , (173)where the second limit is in the distributional sense and T ( λ ) = − i S ( λ ) − S (cid:48) ( λ ) . (174) (cid:3) By the properties of the scattering matrix T ( λ ) commutes with the Hodge star operatorand leaves the summands in ker∆ (cid:48) p ⊕ ∆ (cid:48) p − invariant (see (32)). We therefore have T p ( λ ) = (cid:18) T p ( λ ) 00 T p − ( λ ) (cid:19) , (175)where T p ( λ ) is the time delay operator for coclosed forms defined by T p ( λ ) = − i S p ( λ ) − S (cid:48) p − ( λ ) . (176)This operator describes the time-delay of coclosed forms of energy λ < µ scattered in M .In the physics literature the time delay of the (cid:96) = 0 partial wave for potential scatteringin R is called the scattering length (e.g. [RS79]). As the (cid:96) = 0 partial wave correspondsto the constant function on the sphere at infinity we call T p (0) in analogy with this thescattering length. Note however, that in the physics literature the time-delay is usuallyconsidered for non-relativistic Schr¨odinger mechanics so that it differs from the relativistictime-delay by an energy dependent factor. A simple relation that equates time-delay andscattering length can for dimensional reasons only hold in relativistic theories. References [AuB06] F. Auer and V. Bangert,
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