An index theorem of Callias type for pseudodifferential operators

AN INDEX THEOREM OF CALLIAS TYPE F OR PSEUDODIFFERENTIAL OPERA TORS CHRIS KOTTKE Abstract. W e pro v e an index theorem for families of pseudo differential operators generalizing those studied by C. Callias, N. Anghel and others. Specifically , w e consider operators on a manifold with boundary equipped with an asymptotically conic (scattering) metric, which hav e the form D + i Φ, where D is elliptic pseudodifferential with Hermitian symbols, and Φ is a Hermitian bundle endomorphism which is inv ertible at the boundary and commutes with the symbol of D there. The index of such op erators is completely determined by the symbolic data ov er the b oundary . W e use the scattering calculus of R. Melrose in order to prov e our results usi ng methods of top ological K-theory , and we devote special atten tion to the case in which D is a family of Dirac op erators, in which case our theorem specializes to give families versions of the previously known index formulas. 1. Introduction In [Cal78], C. Callias obtained a formula for the F redholm index of op erators on o dd-dimensional Euclidean space R n ha ving the form P = / D ⊗ 1 + i ⊗ Φ( x ) : C ∞ c ( R n ; V 0 ⊗ V 00 ) − → C ∞ c ( R n ; V 0 ⊗ V 00 ) , where / D is a self-adjoin t spin Dirac op erator (asso ciated to any connection with appropriate flatness at infinit y), Φ is a Hermitian matrix-v alued function which is uniformly inv ertible off a compact set (representing a Higgs p otential in physics), and V 0 and V 00 are trivial vector bundles. According to his formula, the index dep ends only on a top ological in v arian t of Φ restricted to S n − 1 , the sphere at infinit y . In a follo wing paper [BS78], R. Bott and R. Seeley interpreted Callias’ result in terms of a sym b ol map (equiv alent to the total sym b ol defined b elow) σ ( P ) : S 2 n − 1 = ∂  T ∗ R n  − → End( V 0 ⊗ V 00 ) and p oint out that the resulting index form ula has the form of the product of the Chern characters of σ ( / D ) and Φ, each integrated ov er a cop y of S n − 1 . The F redholm index of Dirac op erators coupled to sk ew-Hermitian nonscalar potentials on v arious o dd dimensional manifolds was subsequently studied b y several authors, culminating in a result in [Ang93b] by N. Anghel (also obtained indep endently by J. R ˚ ade in [R ˚ ad94], and U. Bunke in [Bun95] who pro v ed a C ∗ equiv ariant version of the theorem applicable in particular to families of Dirac op erators) for op erators of the ab ov e type on arbitrary o dd-dimensional, complete Riemannian manifolds . Under suitable conditions on / D , a Dirac op erator asso ciated to a vector bundle V − → X , and on the p otential Φ ∈ C ∞ ( X ; End( V )), Anghel prov ed that ind( / D + i Φ) = ind( / ∂ + + ) where / ∂ + + is a related Dirac op erator on a h ypersurface Y ⊂ X , represen ting a suitable “infinity” in X . The pro ofs of these results dep end on the fact that / D is a Dirac op erator – Callias’ original pro of uses lo cal trace form ulas of the integral k ernels, Anghel and Bunke use the relative index theorem of Gromo v and Lawson [GL83], and R ˚ ade uses elliptic b oundary conditions analogous to the Atiy ah-Patodi-Singer conditions to preserv e the index under v arious cutting and gluing pro cedures. In this pap er we shall determine the index of Callias-type op erators via methods in topological K-theory , in the spirit of [AS68] and [AS71], using R. Melrose’s calculus of scattering pseudo differential op erators [Mel94]. In particular, this allows us to consider a class of pseudo differ ential Callias-t yp e op erators, whic h w e dub Callias-Anghel op erators, and to obtain a families v ersion of the index theorem with little additional 2010 Mathematics Subject Classific ation. Primary 58J20; Secondary 19K56, 58J40. Key wor ds and phr ases. index theorem, scattering pseudodifferential op erator, Dirac op erator, scattering manifold, asymp- totically conic manifold, asymptotically locally Euclidean manifold. 1 effort. Our result applies also to even-dimensional manifolds, where Callias-Anghel op erators which are not of Dirac type ma y indeed hav e non trivial index. A Callias-Anghel op erator P on a manifold X with b oundary ∂ X (though t of as “infinit y”) has the form P = D + i Φ, where D ∈ Ψ m sc ( X ; V ) is an elliptic scattering pseudodifferential op erator with Hermitian sym b ols, and Φ ∈ C ∞ ( X ; End( V )) is a compatible p otential, meaning that Φ | ∂ X is Hermitian, in v ertible and commutes with the sym bol of D . Our main result, prov ed in section 5, is the following. Theorem. A Cal lias-Anghel op er ator P = D + i Φ extends to a F r e dholm op er ator on natur al ly define d Sob olev sp ac es, with index ind( P ) = Z S ∗ ∂ X X c h( V + + ) · π ∗ Td( ∂ X ) . wher e V + + ⊂ π ∗ V − → S ∗ ∂ X X is the bund le of jointly p ositive eigenve ctors of σ ( D ) and π ∗ Φ on the c ospher e bund le of X over ∂ X . Note that V + + is well-defined since σ ( D ) and π ∗ Φ are Hermitian and commute, hence are jointly diago- nalizable. The essence of our proof is to note that the index of D + i Φ is determined top ologically b y the symbolic data, which is sho wn to b e trivial ov er the interior of X and completely determined by the coupling b etw een Hermitian (from σ ( D )) and skew-Hermitian (from i Φ) terms on the cosphere bundle ov er infinity ( S ∗ ∂ X X ). Indeed, once the problem is prop erly form ulated, the pro of is a straigh tforw ard computation in K-theory . In using the scattering calculus of pseudo differential operators, we restrict ourselv es to asymptotically conic, or “scattering” manifolds which are compact manifolds with b oundary , equipp ed with metrics of the form dx 2 x 4 + h x 2 , h | ∂ X a metric on ∂ X , where x is a b oundary defining function. Indeed, in order to hav e some reasonable class of pseudo differential op erators at our disp osal, it is necessary to restrict the asymptotic geometry in some wa y , and our choice is motiv ated b y the following considerations. • While this class of manifolds is geometrically more restricted than the general complete Riemannian manifolds considered b y Anghel and others, the conditions for op erators to b e F redholm on these spaces are muc h less restrictive and more easily v erified in practice. Corresp ondingly , w e need to assume less ab out D and Φ to obtain our result. W e discuss a connection betw een our setup and the one considered by Anghel in section 6, and exp ect that using scattering mo dels in the context of certain other noncompact index problems (essen tially situations in which the F redholm data is sufficien tly local near infinit y) may b e possible. • The sym bolic structure of the scattering calculus has a very simple in terpretation in terms of topo- logical K-theory , permitting us to utilize a p ow erful families index theorem (Theorem 1) deriv ed from [AS71]. • The author’s work on this sub ject w as motiv ated by his thesis work on S U (2) monopole mo duli spaces o ver asymptotically conic manifolds, where the dimension of the moduli space is giv en b y the index of a Callias-Anghel t yp e op erator. In that case the potential Φ ma y hav e (constant rank) n ull space ov er ∂ X ; how ev er, the index problem for suc h an op erator, whic h will b e the sub ject of a subsequen t paper, ma y nonetheless b e reduced to one of the type considered here. W e b egin with a brief introduction to the scattering calculus in section 2, culminating with the pro of of the index theorem for families of scattering op erators. W e in troduce Callias-Anghel type op erators in section 3 and pro ve that they extend to F redholm operators. Section 4 is the heart of our result, and consists of the reduction of the symbol by homotopy to the corner S ∗ ∂ X X of the total space ∂ ( T ∗ X ) in K-theory; it is en tirely top ological in nature. W e presen t our results in section 5, with a particular analysis of the imp ortant case of Dirac operators. Finally , we discuss the relation to previous results in section 6. The author would like to thank his thesis advisor Richard Melrose for his supp ort and guidance, and also Pierre Albin for man y helpful conv ersations. 2 2. The Sca ttering Calculus W e briefly recall the imp ortant elemen ts of the scattering calculus of pseudo differential operators. A basic reference for the material in this section is [Mel94], and more generally [Mel]. By a sc attering manifold we shall mean a compact manifold X with b oundary , typically equipp ed with an exact scattering metric as defined b elow. W e refer to ∂ X as “infinit y .” 2.1. Structure algebra and bundles. Given a compact manifold X with b oundary , the algebra of sc at- tering ve ctor fields V sc ( X ) is a Lie subalgebra of the algebra V ( X ) of v ector fields defined by V sc ( X ) = x V b ( X ) . where x is a b oundary defining function ( x ≥ 0 , x − 1 (0) = ∂ X, dx | ∂ X 6 = 0), and V b ( X ) is the subalgebra of v ector fields tangent to the b oundary: V b ( X ) = { V ∈ V ( X ) ; V x ∈ xC ∞ ( X ) } . Both V b ( X ) and V sc ( X ) are indep endent of the c hoice of x . In a coordinate neigh b orho o d near the b oundary , V sc ( X ) is spanned b y x 2 ∂ x , x∂ y 1 , . . . , x∂ y n where y 1 , . . . , y n are lo cal co ordinates on ∂ X . Just as V ( X ) = C ∞ ( X, T X ) is the space of sections of the vector bundle T X , V sc ( X ) and V b ( X ) are the spaces of sections of the sc attering and b tangent bund les sc T X and b T X , resp ectively . In lo cal coordinates x, y 1 , . . . , y n near ∂ X , bases for sc T p X and b T p X at a p oin t p are giv en by sc T p X = span R  x 2 ∂ x , x∂ y 1 , . . . , x∂ y n  , b T p X = span R { x∂ x , ∂ y 1 , . . . , ∂ y n } . The sc attering and b c otangent bund les sc T ∗ X and b T ∗ X are the dual bundles to sc T X and b T X , with bases in lo cal co ordinates giv en by sc T ∗ p X = span R  dx x 2 , dy 1 x , . . . , dy n x  , b T ∗ p X = span R  dx x , dy 1 , . . . , dy n  . While T ∗ X , b T ∗ X and sc T ∗ X (or T X , b T X and sc T X ) are isomorphic in the interior of X , they are not canonically so at the b oundary (any identification ov er ∂ X depends on a c hoice of b oundary defining function x ). How ev er they are homotopy equiv alen t as vector bundles, so for top ological purp oses such as index computations, the difference is often unimp ortant. The reader unfamiliar with the s cattering calculus ma y men tally replace sc T by T without muc h harm. The natural metrics to consider in the context of scattering operators are those which are smooth sections not of Sym 2 ( T ∗ X ) but of Sym 2 ( sc T ∗ X ). W e will further restrict consideration to the case of so-called exact scattering metrics, which hav e the form g = dx 2 x 4 + h x 2 , for some c hoice of a boundary defining function x , and where h restricts to a metric on the compact manifold ∂ X . Scattering operators (defined b elow) naturally extend to op erators on Sob olev spaces asso ciated to suc h metrics. In preparation for section 5.1, w e define sc attering (r esp e ctively b) c onne ctions on a v ector bundle V − → X as cov ariant deriv atives ∇ : C ∞ ( X ; V ) − → C ∞ ( X ; sc T ∗ X ⊗ V )  resp. ∇ : C ∞ ( X ; V ) − → C ∞ ( X ; b T ∗ X ⊗ V )  , whic h satisfy the Leibniz condition ∇ ( f · s ) = d f ⊗ s + f ·∇ s , where f ∈ C ∞ ( X ) and s ∈ C ∞ ( X ; V ). Here we are taking the image of the one-form d f in either C ∞ ( X ; sc T ∗ X ) or C ∞ ( X ; b T ∗ X ), which is p ossible since there are natural bundle maps T ∗ X − → b T ∗ X − → sc T ∗ X induced by the inclusions V sc ( X ) ⊂ V b ( X ) ⊂ V ( X ). F rom the maps T ∗ X − → b T ∗ X − → sc T ∗ X , we also see that true connections (which is what we shall call connections in the ordinary sense) extend naturally to b and scattering c onnections, and that b connections similarly extend to scattering connections. If a scattering connection ∇ is obtained from such a b connection, w e will say that it is the lift of a b c onne ction . A choice of boundary defining function giv es a natural pro duct structure ∂ X × [0 , 1) x on a neighborho o d of the b oundary and hence a wa y to extend v ector fields from ∂ X in to the interior. In this w a y , true and b connections restrict to connections ∇ | ∂ X on the boundary . Note how ever that a scattering connection do es not naturally restrict to a connection on ∂ X unless it is the lift of a true or b connection. F or instance, given 3 an exact scattering metric g , the Levi-Civita connection ∇ LC( g ) is the lift of a b connection 1 whic h restricts to ∇ LC( h ) at ∂ X . 2.2. Op erators and symbols. The algebra of scattering differen tial op erators acting on sections of a v ector bundle V is just the univ ersal en veloping algebra of V sc ( X ) ⊗ C ∞ ( X ; End( V )) o ver C ∞ ( X ); for a giv en k ∈ N 0 , Diff k sc ( X ; V ) =    X 0 ≤ l ≤ k c l ∇ V 1 · · · ∇ V l ; V i ∈ V sc ( X ) , c l ∈ C ∞ ( X ; End( V ))    . and Diff ∗ sc ( X ; V ) forms a filtered algebra of op erators on C ∞ ( X ; V ). This algebra can b e “microlo calized” to pro duce an algebra of sc attering pseudo differ ential op er ators acting on sections of V , denoted Ψ m sc ( X ; V ) , m ∈ R , by constructing their Sch w artz k ernels on an appropriately blo wn up version of the space X 2 (see [Mel94] for details). Giv en D ∈ Ψ m sc ( X ; V ), w e hav e an interior symb ol map analogous to the usual principal symbol, σ int ( D ) : sc T ∗ X − → End( π ∗ V ) , where 2 π : sc T ∗ X − → X . In addition to this sym b ol, w e hav e a b oundary or sc attering symb ol σ sc ( D ) : sc T ∗ ∂ X X − → End( π ∗ V ) whic h, for a differen tial op erator in local co ordinates, tak es the form D = X | α | + j ≤ m a α ( x, y )( x 2 ∂ x ) j ( x∂ y ) α = ⇒ σ sc ( D )(0 , y , ζ , η ) = X | α | + j ≤ m a α (0 , y ) ζ j η α . (Note that the b oundary symbol in v olv es the sum ov er all orders in the op erator, whereas the in terior sym b ol only inv olv es the top order | α | + j = m .) Both symbols hav e asymptotic growth/deca y of order ≤ m along the fibers, and they satisfy the compat- ibilit y condition that, asymptotically , σ int ( D )( p, ξ ) ∼ σ sc ( D )( q , ξ ) as p → q , | ξ | → ∞ , where p ∈ X , q ∈ ∂ X , ξ ∈ sc T ∗ p X . W e will restrict ourselves to so-called “classical” op erators whose sym b ols hav e asymptotic expansions in terms of | ξ | m − k , k ∈ N , as | ξ | → ∞ . Then for 0th order op erators, we can regard the interior symbol as a map σ int ( D ) : sc S ∗ X − → End( π ∗ V ) , where sc S ∗ p X is the boundary of the radially compactified fib er sc T ∗ p X , and the v alue of σ int ( D ) is obtained b y taking the limit of the leading term in the asymptotic expansion. Similarly , we extend σ sc ( D ) to a map σ sc ( D ) : sc T ∗ ∂ X X − → End( π ∗ V ) , again using radial compactification of the fibers. Since D is 0th order, b oth sym b ols are bounded, asymptotic compatibilit y is just equality of the limits, and w e can combine the tw o symbols in to a contin uous total symb ol σ tot ( D ) : ∂ ( sc T ∗ X ) − → End( π ∗ V ) , where sc T ∗ X is the total space of the compactified scattering cotangent bundle. It is a manifold with corners, with boundary ∂ ( sc T ∗ X ) consisting of b oth sc S ∗ X and sc T ∗ ∂ X X , whic h intersect at the corner sc S ∗ ∂ X X (see Figure 1). W e can produce a total symbol in the general case as follo ws. F or every m ∈ R , w e construct a trivial real line bundle N m − → sc T ∗ X whose b ounded sections consist of functions with asymptotic growth/deca y of order m . Giv en a scattering metric, a trivialization o v er the interior is giv en by the section | ξ | m , that is sc T ∗ X × R − → N m : (( p, ξ ) , t ) ∼ = 7− → ( p, ξ , t | ξ | m p ) . 1 Note that ∇ LC( g ) is not , how ever, the lift of the Levi-Civita connection associated to x 2 g . 2 W e will use π to denote the pro jection for v arious bundles related to sc T ∗ X , suc h as the scattering cosphere bundle sc S ∗ X and the radially compactified scattering cotangent bundle sc T ∗ X . The appropriate domain will b e clear from context, and no confusion should arise. 4 Sym b ols of m th order operators define b ounded sections of N m , which take limiting v alues at the boundary as ab ov e, and we define the r enormalize d symb ols as m σ int ( D ) : sc S ∗ X − → N m ⊗ End( π ∗ V ) , m σ sc ( D ) : sc T ∗ ∂ X X − → N m ⊗ End( π ∗ V ) . W e com bine these to obtain the renormalized total symbol m σ tot ( D ) : ∂ ( sc T ∗ X ) − → N m ⊗ End( π ∗ V ) . Giv en D ∈ Ψ m sc ( X ; V ), we sa y D is el liptic when its interior symbol σ int ( D ) is inv ertible, as usual. Elliptic scattering op erators satisfy the usual elliptic regularity conditions, but in general fail to b e F redholm as op erators on any natural Sobolev spaces. D is said to be ful ly el liptic if b oth its interior symbol σ int ( D ) and its b oundary symbol σ sc ( D ) are everywhere inv ertible. This is equiv alent to inv ertibility of the renormalized total symbol m σ tot ( D ), since inv ertibility does not dep end on the chosen trivialization of N m . If D is fully elliptic, it has a unique extension from an operator on C ∞ c ( X ; V ) to a b ounded, F redholm op erator on w eigh ted sc attering Sob olev sp ac es : D : x α H m + k sc ( X ; V ) − → x α H k sc ( X ; V ) is F redholm for all α ∈ R , where for 3 k ∈ N 0 , x α H k sc ( X ; V ) = n v = x α u ; u ∈ L 2 ( X ; V ) , and P u ∈ L 2 ( X ; V ) for all P ∈ Diff k sc ( X ; V ) o . R emark. Note that in discussing the total symbols of pseudo differential op erators, w e use the notation π : ∂ ( sc T ∗ X ) − → X to denote the pro jection. This is not a prop er fib er bundle, as the fib er ov er the interior is a sphere, sc S ∗ p X , while the fib er o v er a boundary p oin t is the (radially compactified) vector space sc T ∗ p X . Nev ertheless, the notation is conv enien t. 2.3. F amilies of op erators. Belo w we shall consider families of scattering pseudo differential op erators, for whic h w e use the follo wing notation. Supp ose X has the structure of a fib er bundle X − → Z , where Z is a compact manifold without boundary , and such that the fiber is a manifold Y with b oundary ∂ Y . W e use the notation X/ Z := Y to denote the fiber, though there is no real such quotient. Thus X has b oundary ∂ X which itself fib ers ov er Z , with fib er ∂ Y . X is asso ciated to a principal Diffeo( Y )-bundle P − → Z , from whic h w e derive additional associated bundles. Supp ose we are given a metric on X whic h restricts to a fixed exact scattering metric on each fib er, for instance by taking a scattering metric on the total space. A family of sc attering op er ators on X − → Z , is an op erator acting on sections of a vector bundle 4 V − → X which is scattering pseudo differential in the fib er directions, and smo othly v arying in the base. It is properly defined as a section of the bundle Ψ m sc ( X/ Z ; V ) = P × Diffeo( Y ) Ψ m sc ( Y ; W ) − → Z , where V and W are related b y V = P × Diffeo( Y ) W . In the simple case that X is a pro duct, X = Y × Z , Z is just a sm o oth parameter space for the op erators, and we recov er the case of a single op erator by taking Z = pt, X/ Z = X = Y . F or a family D ∈ Ψ m sc ( X/ Z ; V ) of op erators, the symbol maps ha ve domain sc T ∗ ( X/ Z ), whic h is the v ertical scattering cotangent bundle sc T ∗ ( X/ Z ) = P × Diffeo( Y ) sc T ∗ Y − → Z, with fib ers isomorphic to the scattering cotangent bundle sc T ∗ Y of the fib er. The renormalized total symbol is a map m σ tot ( D ) : ∂  sc T ∗ ( X/ Z )  − → N m ⊗ End( π ∗ V ) where now everything is fibered o ver Z , and ∂ ( sc T ∗ ( X/ Z )) has fib ers isomorphic to ∂ ( sc T ∗ Y ). Note that π : ∂ ( sc T ∗ ( X/ Z )) − → X is a family of pro jections mo deled on π : ∂ ( sc T ∗ Y ) − → Y , to which the remark at the end of section 2.2 applies. 3 It is straightforw ard to define scattering Sob olev spaces of all real orders in terms of pseudo differential op erators, but we restrict ourselves here to the case of non-negative integer k for simplicity . 4 Any vector bundle V − → X can b e exhibited as a family of v ector bundles V = P × Diffeo( Y ) W , where W − → Y is a fixed vector bundle with the same rank as V . 5 As in the case of ordinary pseudodifferential op erators, a family D of F redholm operators 5 o v er Z has an index ind( D ) ∈ K 0 ( Z ) given b y ind( D ) = [k er( D )] − [coker( D )] ∈ K 0 ( Z ) , whic h is well-defined by a stabilization pro cedure and Kuiper’s theorem [LM89]. sc S ∗ ∂ X ( X/ Z ) X sc S ∗ ( X/ Z ) sc T ∗ ( X/ Z ) ∂ X sc T ∗ ∂ X ( X/ Z ) Figure 1. The total space sc T ∗ ( X/ Z ) and its b oundary 2.4. The index theorem. The follo wing theorem is one of the primary reasons for using the scattering calculus in our treatment. Among calculi of pseudo differential op erators on manifolds with boundary , the scattering calculus is particularly simple since its b oundary symbols are lo c al 6 , and hence give w ell-defined elemen ts in the compactly supp orted top ological K -theory of sc T ∗ X . In particular, this allows for the index to b e computed b y a reduction to the A tiy ah-Singer index theorem for compact manifolds ([AS71], [AS68]). A pro of of Theorem 1 can b e found in [Mel95] and [MR04], and a more explicit v ersion of the cohomological form ula (an extension of F edoso v’s form ula for the classical index theorem) is obtained in [AM09]. As our applications are to self-adjoin t op erators with sk ew-adjoint p otentials, the domain and range bundles of our op erators will alwa ys b e the same, whic h p ermits us to write the index formula below in terms of the o dd Chern character of the total symbol, whic h in this case defines an element of the o dd K-group K 1 ( ∂ ( sc T ∗ X )). First let us introduce the notation we use for K-theory . As usual, we write elements in ev en K-theory as formal differences of v ector bundles up to equiv alence and stabilization, [ V ] − [ W ] ∈ K 0 ( M ) and use the notation [ V , W , σ ] ∈ K 0 ( M , N ) for relative classes, where σ : V | N ∼ = − → W | N is an isomorphism o v er N . This applies in particular to K-theory with c omp act supp ort K 0 c ( M ) = K 0 ( M , ∞ ), which is just K-theory relative to infinity with resp ect to any compactification (the one p oint compactification M ∪ {∞} is typically used, though for vector bundles V , w e use the fib erwise radial compactification V ∪ S V ). Odd K-theory is represented b y homotopy classes of maps M − → lim n →∞ GL( n ), though we use the notation [ V , σ ] ∈ K 1 ( M ) 5 The F redholm prop erty in the families setting is with resp ect to families of scattering Sob olev spaces H ∗ sc ( X/ Z ; V ) = P × Diffeo( Y ) H ∗ sc ( Y ; W ). 6 The trade off is that the condition of full ellipticit y in the scattering calculus is stronger than the corresp onding condition in calculi with less local b oundary data. 6 as shorthand for the element [ V ⊕ V ⊥ , σ ⊕ Id] ∈ K 1 ( M ), where V ⊕ V ⊥ ∼ = M × C N , so σ ⊕ Id : M − → GL( N ). In particular, an element [ V , V , σ ] ∈ K 0 ( M , N ) with iden tical domain and range bundles is the image of an elemen t [ V , σ ] ∈ K 1 ( N ) in the long exact sequence of the pair ( M , N ). Lastly , w e define a top ological index map for scattering pseudo differential op erators analogous to the classical one. F or D ∈ Ψ m sc ( X/ Z ; V , W ) fully elliptic, [ π ∗ V , π ∗ W , m σ tot ( D )] ∈ K 0 c ( sc T ∗ ( ˚ X / Z )) is well-defined 7 , where ˚ X = X \ ∂ X is the in terior of X (so the compact supp ort refers b oth to the fib er and base directions). Giv en an em b edding ˚ X  → R N × Z of fibrations ov er Z , w e hav e an induced, K-oriented em b edding sc T ∗ ( ˚ X / Z ) − → R 2 N × Z in to an even dimensional trivial Euclidean fibration. W e define top-ind( D ) to b e the image of [ π ∗ V , π ∗ W , m σ tot ( D )] under the composition K 0 c ( sc T ∗ ( ˚ X / Z )) − → K 0 c ( N ( sc T ∗ ( ˚ X / Z ))) − → K 0 c ( R 2 N × Z ) ∼ = K 0 ( Z ) , where the first map is the Thom isomorphism onto the normal bundle of sc T ∗ ( ˚ X / Z ) in R 2 N × Z , the second is the pushforward with respect to the open em b edding N ( sc T ∗ ( ˚ X / Z ))  → R 2 N × Z in compactly supported K-theory 8 , and the last is the Bott perio dicity isomorphism (equiv alen tly , the Thom isomorphism for a trivial bundle). That this is a well-defined map indep endent of choices follows exactly as in the classical case in [AS71]. Theorem 1. [Mel95] , [MR04] L et P ∈ Ψ m sc ( X/ Z ; V ) b e a family of ful ly el liptic sc attering pseudo differ ential op er ators. It is ther efor e a F r e dholm family, with wel l-define d index ind( P ) ∈ K 0 ( Z ) , and ind( P ) = top-ind( P ) . F urthermor e, the Chern char acter of this index is given by the c ohomolo gic al formula c h(ind( P )) = p ! (c h( σ tot ( P )) · π ∗ Td( X/ Z )) , wher e p ! : H even c ( sc T ∗ ( ˚ X / Z )) − → H even ( Z ) denotes inte gr ation over the fib ers and ch( σ tot ( P )) is shorthand for ch even ([ π ∗ V , π ∗ V , m σ tot ( P )]) ∈ H even c ( sc T ∗ ( ˚ X / Z )) . Since [ π ∗ V , π ∗ V , m σ tot ( P )] ∈ K 0 c ( sc T ∗ ( ˚ X / Z )) is in the image of K 1 ( ∂ ( sc T ∗ ( X/ Z ))), we can reform ulate the ab ov e in terms of the o dd Chern character as follows. First, w e define a generalized fib er in tegration map q ! : H odd ( ∂ ( sc T ∗ ( X/ Z ))) − → H even ( Z ), where q is the composition q : ∂ ( sc T ∗ ( X/ Z )) π − → X − → Z . Of course, q is not prop erly a fibration as p er the remark at the end of 2.2; it really consists of tw o fibrations q 1 : sc S ∗ ( X/ Z ) − → Z and q 2 : sc T ∗ ∂ X ( X/ Z ) − → Z , with an iden tification of their common b oundary , which is the fibration sc S ∗ ∂ X ( X/ Z ) − → Z . W e define q ! as the sum q ! µ = ( q 1 ) ! µ + ( q 2 ) ! µ pulling back µ ∈ H odd ( ∂ ( sc T ∗ ( X/ Z ))) as appropriate in each of the summands; it is well-defined on coho- mology since if µ = dα is exact (or more generally fib erwise exact), ( q ! dα )( z ) = Z ( sc T ∗ ( X/ Z ) ) z dα + Z ( sc S ∗ ( X/ Z )) z dα = Z ( sc S ∗ ∂ X ( X/ Z ) ) z α − Z ( sc S ∗ ∂ X ( X/ Z ) ) z α = 0 b y Stok es’ Theorem, since sc T ∗ ( X/ Z ) and sc S ∗ ( X/ Z ) share the common b oundary sc S ∗ ∂ X ( X/ Z ) but with opp osite orientation. No w, since q ! factors as the comp osition of the connecting map H odd ( ∂ ( sc T ∗ ( X/ Z ))) − → H even c ( sc T ∗ ( ˚ X / Z )) with the even fib er in tegration map p ! : H even c ( sc T ∗ ( ˚ X / Z )) − → H even ( Z ), and since the connecting maps in ev en/o dd cohomology and K-theory intert wine the even/odd Chern character maps, we obtain the following o dd version of the cohomological formula. 7 This inv olves c hoosing a trivialization of N m , though the element of K-theory obtained is indep endent of this c hoice. 8 Recall that, while cohomology theories (K-theory in particular), are contra v ariant, there is a limited form of cov ariance with resp ect to open embeddings in an y compactly supp orted theory . If i : O  → M is an op en embedding, we obtain a pushforward map i ∗ : K ∗ c ( O ) − → K ∗ c ( M ) via the quotient map M + / ∞ − → M / ( M \ O ) ∼ = O + / ∞ , where M + denotes the one point compactification M ∪ {∞} . 7 Corollary 2. L et P ∈ Ψ m sc ( X/ Z ; V ) b e a family of ful ly el liptic sc attering pseudo differ ential op er ators as ab ove. Then c h(ind( P )) = q ! (c h odd ( σ tot ( P )) · π ∗ Td( X/ Z )) , wher e σ tot ( P ) is short for [ π ∗ V , m σ tot ( P )] ∈ K 1 ( ∂ ( sc T ∗ ( X/ Z ))) , define d using any trivialization of N m . In the sp e cial c ase of a single op er ator P ∈ Ψ m sc ( X ; V ) , the index is an inte ger ind( P ) ∈ Z , and we have ind( P ) = Z ∂ ( sc T ∗ X ) c h odd ( σ tot ( P )) · π ∗ Td( X ) . 3. Callias-Anghel type opera tors W e shall be concerned with pseudodifferential families D whose symbols are Hermitian, coupled to skew- Hermitian p oten tials i Φ. It is actually only necessary that i Φ b e sk ew-Hermitian at infinit y , as well as satisfy some compatibility conditions with D . This is more general than the operators considered in the literature, and we will see in the section follo wing this one why the index is only dependent on these conditions. Let V − → X be a family of Hermitian complex v ector bundles asso ciated to the family of scattering manifolds X − → Z . W e will denote the inner pro duct on V b y h· , ·i . Let D ∈ Ψ m sc ( X/ Z ; V ) , m > 0 be a family of elliptic (but not necessarily ful ly elliptic) scattering op erators with Hermitian symbols, so σ int ( D ) : sc S ∗ ( X/ Z ) − → GL( π ∗ V ) ⊂ End( π ∗ V ) is Hermitian with respect to h· , ·i and σ sc ( D ) : sc T ∗ ∂ X ( X/ Z ) − → End( π ∗ V ) is Hermitian but not necessarily inv ertible. Of particular interest later will b e the case of a family of Dirac op erators, for whic h σ sc ( D )( p, ξ ) = i c  ( ξ ) · v anishes at the 0-section ov er ∂ ( X/ Z ) and is therefore never fully elliptic. Next let Φ be a section of End( V ). Motiv ated b y physics, we refer to Φ as the p otential . W e will assume Φ satisfies the follo wing conditions ov er the b oundary ∂ X , which w e shall dub c omp atibility with D . (1) Φ | ∂ X is Hermitian with respect to h· , ·i (2) Φ | ∂ X is inv ertible (3) Φ | ∂ X comm utes with the b oundary symbol of D , that is [ π ∗ Φ | ∂ X , σ sc ( D )] = 0 ∈ End( π ∗ V ) on sc T ∗ ∂ X X . W e refer to condition 3 as symb olic c ommutativity . Giv en D and a compatible potential Φ, the Cal lias-Anghel typ e op er ator P = D + i Φ ∈ Ψ m sc ( X/ Z ; V ) is fully elliptic (and therefore F redholm on appropriate spaces) b y the following elementary lemma. Lemma 3. L et α and β b e Hermitian se ctions of the bund le End( V ) − → M , and supp ose that, over a subset Ω ⊂ M , we have [ α, β ] = αβ − β α = 0 ∈ Γ(Ω; End( V )) . If either α or β is invertible over Ω , then the c ombination α + iβ ∈ Γ(Ω; GL( V )) is invertible over Ω . In p articular, if b oth α and β ar e invertible over Ω , then the c ombination tα + i sβ ∈ Γ(Ω; End( V )) is invertible for al l ( s, t ) 6 = (0 , 0) ∈ R 2 + . Pr o of. It suffices to consider an arbitrary fib er V p , p ∈ Ω. By the assumption that α and β are Hermitian, α ( p ) has purely real eigenv alues while iβ ( p ) has purely imaginary ones. Since [ α, β ] = 0, there is a basis of V p in which α ( p ) and β ( p ) are simultaneously diagonal; with resp ect to this basis α + iβ acts diagonally with eigen v alues of the form λ j + iµ j with λ j , µ j ∈ R . If either α or β is inv ertible, then either λ j 6 = 0 or µ j 6 = 0 for all j ; therefore λ j + iµ j 6 = 0 ∈ C and α + iβ m ust b e in v ertible.  8 Corollary 4. The family of sc attering op er ators P = D + i Φ extends to a family of F r e dholm op er ators 9 P : x α H k + m sc ( X/ Z ; V ) − → x α H k sc ( X/ Z ; V ) for al l k , α . Pr o of. The interior sym b ol σ int ( P ) = σ int ( D ) is in v ertible on sc S ∗ ( X/ Z ), since D is elliptic. The b oundary sym b ol σ sc ( P ) = σ sc ( D + i Φ) = σ sc ( D ) + iπ ∗ Φ is in v ertible on sc T ∗ ∂ X ( X/ Z ) by sym b olic commutativit y , using Lemma 3 with α = σ sc ( D ), β = π ∗ Φ and Ω = sc T ∗ ∂ X ( X/ Z ). P is therefore fully elliptic, and by the theory of scattering pseudo differential op erators [Mel94], the claim follo ws.  R emark. Note ho w the compatibility of σ int ( P ) and σ sc ( P ) is satisfied. Since D is a family of op erators of order m > 0, the leading term in the asymptotic expansion of σ sc ( P ) = σ sc ( D ) + iπ ∗ Φ as | ξ | − → ∞ is that of σ sc ( D ), whic h grows lik e | ξ | m , whereas π ∗ Φ is constant. In terms of the renormalized sym bols and a c hoice of radial co ordinate | ξ | , m σ tot ( P ) = m σ tot ( D ) + i | ξ | − m π ∗ Φ and the latter term v anishes on sc S ∗ ( X/ Z ). 4. Reduction to the corner By Corollary 2, the index of P is determined b y the element in the o dd K-theory of ∂ ( sc T ∗ ( X/ Z )) defined b y the (renormalized) total symbol m σ tot ( P ). The remainder of our work consists of reducing this top ological datum to one supp orted at the corner, sc S ∗ ∂ X ( X/ Z ). T o this end, we will abstract the situation somewhat, in order to simplify the notation and clarify the concepts in volv ed. Th us we shall forget, for the time b eing, that our K-class is coming from the symbol of a family of pseudo differen tial op erators, as w ell as most of the structure of ∂ ( sc T ∗ ( X/ Z )). Let M = ∂ ( sc T ∗ ( X/ Z )), and let N = sc S ∗ ∂ X ( X/ Z ) b e the corner. The imp ortant feature of N is that it is a hypersurface, separating M \ N in to disjoint comp onents M 1 = sc S ∗ ( ˚ X / Z ) and M 2 = sc T ∗ ∂ X ( X/ Z ). Actually , the fact that it is a corner is indistinguishable top ologically , and we consider it just as a top ological h yp ersurface in M . W e assume a trivialization of the line bundle N m has b een chosen, so we iden tify m σ tot ( P ) and σ tot ( P ) and consider the index to be determined by the element [ π ∗ V , σ tot ( P )] ∈ K 1 ( M ). Also, for notational con v enience, w e will write V instead of π ∗ V for the remainder of this section. Prop osition 5 clarifies the fundamental symbolic structure of P . W e see that its sym b ol essen tially consists of an inv ertible Hermitian term from D ov er M 1 and an inv ertible skew-Hermitian term from i Φ ov er M 2 , whose supp orts ov erlap in a neighborho o d of the corner N . The tw o terms are fundamentally coupled there, in that w e cannot separate their supp orts via an y homotopy in GL( V ). Also note that, w ere the total sym b ol either entir ely Hermitian or entir ely skew-Hermitian, it would b e homotopic to the identit y and P would therefore hav e index 0. Hence the nontrivialit y of ind( P ) must b e enco ded by the coupling of the terms near the corner. Prop osition 6 confirms this, and identifies an element in K 0 ( N ) which captures this coupling. Prop osition 5. M is c over e d by two op en sets f M 1 and f M 2 such that f M 1 ∩ f M 2 ∼ = N × I wher e I is a c onne cte d, op en interval. F urthermor e, [ V , σ tot ( P )] = [ V , χA + i (1 − χ ) B ] ∈ K 1 ( M ) , wher e A and B ar e unitary 10 , Hermitian se ctions of GL( V ) such that [ A, B ] = 0 on f M 1 ∩ f M 2 , and wher e χ : M − → [0 , 1] is a cutoff function such that supp χ ⊂ f M 1 and supp(1 − χ ) ⊂ f M 2 . The p ositive and ne gative eigenbund les 11 of A and B c oincide, r esp e ctively, with those of σ tot ( D ) and π ∗ Φ . Pr o of. As remarked at the end of Section 3, σ tot ( P ) is equal to σ sc ( D ) on M 1 and to σ int ( D ) + φπ ∗ Φ on M 2 , where φ ∼ | ξ | − m is a nonnegativ e real-v alued function v anishing on the closure of M 1 . In particular, φπ ∗ Φ has the same ± eigenbundles as Φ wherev er φ 6 = 0. 9 See the footnote on page 5 for the definition of this family of Sob olev spaces. 10 at least on supp χ and supp(1 − χ ), resp ectively . 11 Meaning the bundles of positive and negative eigenvectors. 9 Since σ tot ( D ) is in vertible on M 1 = M 1 ∪ N , b y ellipticity , it must b e inv ertible on a sligh tly larger neigh b orho o d f M 1 . W e set f M 2 = M 2 , on which Φ is self-adjoint, in v ertible, and commutes with σ sc ( D ) by the compatibility assumption. Shrinking either if necessary , we can assume that f M 1 ∩ f M 2 ∼ = N × I . Let χ b e a cutoff function with prop erties as ab ov e. Recall that C ∈ GL( n, C ) is homotopic in GL( n, C ) to its unitarization U ( C ) via C t = C  t ( √ C ∗ C ) − 1 + (1 − t )Id  U ( C ) = ( C t ) t =1 . Let A and B b e the (generalized) unitarizations of σ tot ( D ) and φπ ∗ Φ, resp ectively; thus A is giv en by A ( p ) = ( U ( σ tot ( D )( p )) if σ tot ( D )( p ) is inv ertible 0 otherwise, and similarly for B . Note that while A and B are not necessarily contin uous sections of End( V ) (as σ tot ( D ) ma y fail to b e inv ertible off of f M 1 and φπ ∗ Φ v anishes a w a y from f M 2 ), χA and (1 − χ ) B ar e contin uous and ha v e support, resp ectively , where σ tot ( D ) (resp. φπ ∗ Φ) is inv ertible. W e claim that the homotopy σ t = (1 − t ) σ tot ( P ) + t ( χA + i (1 − χ ) B ) is through inv ertible endomorphisms. Indeed, at a general p oint p ∈ M , σ t ( p ) = [(1 − t ) σ tot ( D )( p ) + tχ ( p ) A ( p )] + i [(1 − t ) φ ( p ) π ∗ Φ( p ) + t (1 − χ )( p ) B ( p )] . where the tw o brack eted terms commute with one another due to symbolic commutativit y (where the latter is nonzero), and at least one term is inv ertible for an y p and all t . In vertibilit y of σ t ( p ) is then immediate from Lemma 3.  In what follows we will identify N × I with the set f M 1 ∩ f M 2 , and denote its inclusion b y j : N × I  → M . Note that ov er N × I , V splits as V = V + ⊕ V − in to ± 1 eigenbundles for A (since A is in vertible here), and similarly V = V + ⊕ V − in to ± 1 eigenbundles for B . Since A and B commute ov er N × I , these splittings are compatible, giving V | N × I ∼ = V + + ⊕ V − − ⊕ V + − ⊕ V − + . The follo wing makes use of the pushforward with resp ect to op en embeddings in compactly supp orted K-theory , and also the Bott p erio dicity isomorphism K 0 ( N ) ∼ = − → K 1 ( N ∧ S 1 ) = K 1 c ( N × I ) . Prop osition 6. L et V | N × I ∼ = V + + ⊕ V − − ⊕ V + − ⊕ V − + b e the splitting into joint eigenbund les of A and B wher e V ± denotes the ± eigenbund le of A and V ± the ± eigenbund le of B . Identify [ V + + ] = [ V + + ] − [0] ∈ K 0 ( N ) with its image in K 1 c ( N × I ) under the Bott isomorphism, and denote by j ∗ ([ V + + ]) the image in K 1 ( M ) of [ V + + ] under the pushforwar d j ∗ : K 1 c ( N × I ) − → K 1 ( M , M \ ( N × I )) and the long exact se quenc e map K 1 ( M , M \ ( N × I )) − → K 1 ( M ) . Then [ V , χA + i (1 − χ ) B ] = j ∗  [ V + + ]  ∈ K 1 ( M ) , Al ternatively, we c ould have use d any of the bund les V ± ± / ∓ , which ar e r elate d by j ∗  [ V + + ]  = j ∗  [ V − − ]  = − j ∗  [ V + − ]  = − j ∗  [ V − + ]  . Pr o of. Let σ = χA + i (1 − χ ) B . W e first trivialize σ aw a y from N × I , so that the K-class it defines is compactly supp orted 12 in N × I . Let { ρ 0 , . . . , ρ 4 } b e a partition of unity satisfying supp( ρ i ) ∩ supp( ρ j ) = ∅ unless i = j or i = j + 1, supp( ρ 0 ) ∩ supp( χ ) = ∅ , and, for i ∈ { 1 , 2 , 3 } , s upp( ρ i ) b N × I = f M 1 ∩ f M 2 . In particular, ρ 0 ≡ 1 aw ay from f M 1 with supp( ρ 0 ) ⊂ f M 2 , and ρ 4 ≡ 1 a wa y from f M 2 with supp( ρ 4 ) ⊂ f M 1 . W e claim there is a homotop y through inv ertible endomorphisms σ ∼ σ 0 = − ρ 0 Id + ρ 1 iB + ρ 2 A − ρ 3 i Id − ρ 4 Id , 12 Recall that an element α ∈ K 1 ( M ) in odd K-theory has supp ort in a set A if it is in the image of K 1 ( M , M \ A ) with respect to the long exact sequence of the pair ( M , A ), and can therefore b e represented by an element [ V , σ ] with σ | M \ A ≡ Id. 10 so that σ 0 ≡ − Id on the complement of N × I . Indeed, such a homotopy is given by σ t =                − tρ 0 Id + (1 − χ ) iB + χA 0 ≤ t ≤ 1 − ρ 0 Id + (2 − t )(1 − χ ) iB + ( t − 1) ρ 1 iB + χA 1 ≤ t ≤ 2 − ρ 0 Id + ρ 1 iB + χA − ( t − 2)( ρ 3 + ρ 4 ) i Id 2 ≤ t ≤ 3 − ρ 0 Id + ρ 1 iB + (4 − t ) χA + ( t − 3) ρ 2 A − ( ρ 3 + ρ 4 ) i Id 3 ≤ t ≤ 4 − ρ 0 Id + ρ 1 iB + ρ 2 A − ρ 3 i Id − ρ 4 ((5 − t ) i Id + ( t − 4)Id) 4 ≤ t ≤ 5 Note that for eac h t , σ t is inv ertible by Lemma 3 and the supp ort conditions on { χ, ρ 0 , . . . , ρ 4 } . Since σ 0 ≡ − Id on M \ ( N × I ) (while we ha v e trivialized σ 0 b y − Id aw a y from N × I instead of +Id, the t w o are equiv alen t up to homotop y; indeed σ 0 ∼ − σ 0 for an y clutc hing function), it is now eviden t that [ V , σ ] = [ V , σ 0 ] is in the image of K 1 ( M , M \ ( N × I )). By identifying ends of the interv al I , w e see that σ 0 defines a map σ 0 | N × S 1 =     σ 1 ( θ ) σ 2 ( θ ) σ 3 ( θ ) σ 4 ( θ )     σ i : S 1 − → C \ { 0 } . whic h is diagonal with resp ect to the splitting V | N × I ∼ = V + + ⊕ V − − ⊕ V + − ⊕ V − + , with scalar entries (since A and B are unitary) independent of N , whose winding n um b ers are easily determined. Indeed, by considering the effect of m ultiplication by σ i ( t ) as σ 0 ( t ) passes from − Id, to iB , to A , to − i Id and then back to − Id, we see that wn( σ 2 ) = wn( σ 3 ) = wn( σ 4 ) = 0 and wn( σ 1 ) = − 1. Th us there are homotopies σ i ∼ ˜ σ i ≡ 1 , i = 2 , 3 , 4 , and σ 1 ( θ ) ∼ ˜ σ 1 ( θ ) = e − iθ . whic h, taken to b e the diagonal elements of a matrix, define a homotopy σ 0 ∼ ˜ σ . Restricting to N × I , we see K 1 c ( N × I ) = K 1 ( N ∧ S 1 ) 3 j ∗ [ V , ˜ σ ] = [ V + + ⊕ V − − ⊕ V + − ⊕ V − + , e − iθ ⊕ Id ⊕ Id ⊕ Id] = [ V + + , e − iθ ] , whic h is just the image [ V + + ] · β ∈ K 1 c ( N × I ) of [ V + + ] ∈ K 0 ( N ) under Bott p erio dicity , where β = [ C , e − iθ ] ∈ K 1 ( S 1 ) ∼ = K 1 c ( I ) is the Bott element. Finally , since [ V , σ ] = [ V , ˜ σ ] ∈ K 1 ( M )) is in the image of j ∗ : K 1 c ( N × I ) − → K 1 ( M , M \ ( N × I )) − → K 1 ( M ), we obtain [ V , σ ] = j ∗ ([ V + + ]) as claimed. Similar pro ofs, using initial trivializations to σ ∼ σ 00 = ρ 0 Id + ρ 1 iB + ρ 2 A + ρ 3 i Id + ρ 4 Id , σ ∼ σ 000 = ρ 0 Id + ρ 1 iB + ρ 2 A − ρ 3 i Id + ρ 4 Id , and σ ∼ σ 0000 = − ρ 0 Id + ρ 1 iB + ρ 2 A + ρ 3 i Id − ρ 4 Id , giv e [ V , σ ] = j ∗ ([ V − − ]), [ V , σ ] = − j ∗ ([ V − + ]), and [ V , σ ] = − j ∗ ([ V + − ]), resp ectively .  5. Resul ts W e now presen t our main results. T o simplify notation, we drop the “sc” lab els in the remainder of the pap er, identifying sc T ∗ ( X/ Z ) with T ∗ ( X/ Z ) via a (non-canonical) isomorphism, which is unique up to homotop y . Theorem 7. Given an el liptic family of sc attering pseudo differ ential op er ators D ∈ Ψ m sc ( X/ Z ; V ) with Hermitian symb ols, and a c omp atible family of p otentials Φ ∈ C ∞ ( X ; End( V )) as define d in se ction 3, the family P = D + i Φ is ful ly el liptic, and extends to a F r e dholm family with index satisfying c h(ind( P )) = p ! (c h( V + + ) · π ∗ Td( ∂ X/ Z )) , 11 wher e p ! : H even ( S ∗ ∂ X ( X/ Z )) − → H even ( Z ) denotes inte gr ation over the fib ers, V + + − → S ∗ ∂ X ( X/ Z ) is the family of ve ctor bund les c orr esp onding to the jointly p ositive eigenve ctors of σ tot ( D ) | S ∗ ∂ X ( X/ Z ) and π ∗ Φ | ∂ X , and σ tot ( D ) is obtaine d fr om m σ tot ( D ) using any trivialization of N m . R emark. In the case of a single op erator P = D + i Φ ∈ Ψ m sc ( X ; V ), the index formula can be written ind( P ) = Z S ∗ ∂ X X c h( V + + ) · π ∗ Td( ∂ X ) . Pr o of. By Corollary 2, c h(ind( P )) = q ! (c h( σ tot ( P )) · π ∗ Td( X/ Z )) . F rom Prop ositions 5 and 6, K 1 ( ∂ ( T ∗ ( X/ Z ))) 3 [ π ∗ V , σ tot ( P )] = j ∗ [ V + + ] with [ V + + ] ∈ K 0 ( S ∗ ∂ X ( X/ Z )) ∼ = K 1 c ( S ∗ ∂ X ( X/ Z ) × I ), where V + + is the jointly p ositive eigenbundle of σ tot ( D ) and π ∗ Φ. No w, since the Chern character is a natural mapping c h : K ∗ − → H ∗ , we obtain c h( σ tot ( P )) = c h( j ∗ [ V + + ]) = j ∗ c h( V + + ) , where j ∗ is the comp osition H even ( S ∗ ∂ X ( X/ Z )) ∼ = − → H odd c ( S ∗ ∂ X ( X/ Z ) × I ) − → H odd ( ∂ ( T ∗ ( X/ Z ))). Since j ∗ c h( V + + ) is supp orted on S ∗ ∂ X ( X/ Z ), the in tegration ov er the fib ers reduces to c h(ind( P )) = p !  c h( V + + ) · π ∗ Td( X/ Z )  , where no w p : S ∗ ∂ X ( X/ Z ) − → Z . F urthermore, since the T o dd class is natural, π ∗ Td( X/ Z ) factors through S ∗ ∂ X ( X/ Z ) − → ∂ X  → X (all ov er Z ) and we obtain Td( X/ Z ) | ∂ X = Td( ∂ X/ Z ); this can alternatively b e seen by taking a product metric at the b oundary .  An interesting case of the ab ov e is when the family V = E ⊗ F is a tensor pro duct of vector bundles, with D ∈ Ψ m sc ( X/ Z ; E ) and Φ ∈ C ∞ ( X ; End( F )). In this case, σ sc ( D ) ⊗ 1 and i ⊗ Φ commute automatically , it is sufficien t that Φ | ∂ X b e inv ertible and self-adjoint in order to b e compatible with D . Theorem 8. Given D ∈ Ψ m sc ( X/ Z ; E ) el liptic with self-adjoint symb ols, and a c omp atible p otential Φ ∈ C ∞ ( X ; End( F )) , the family P = D ⊗ 1 + i ⊗ Φ ∈ Ψ m sc ( X/ Z ; E ⊗ F ) is ful ly el liptic, with F r e dholm index satisfying c h(ind( P )) = p ! (c h( E + ) · ch( F + ) · π ∗ Td( ∂ X/ Z )) , wher e p ! : H even ( S ∗ ∂ X ( X/ Z )) − → H even ( Z ) denotes inte gr ation over the fib ers, and π ∗ E = E + ⊕ E − and π ∗ F = F + ⊕ F − ar e the splittings over S ∗ ∂ X ( X/ Z ) into p ositive and ne gative eigenbund les of σ ( D ) and π ∗ Φ , r esp e ctively. R emark. Note that the splitting of F is actually coming from the base: F | ∂ X = F + ⊕ F − , and π ∗ F = π ∗ F + ⊕ π ∗ F − . Pr o of. The pro of is as ab ov e, noting that the splitting of π ∗ V = π ∗ ( E ⊗ F ) o v er S ∗ ∂ X ( X/ Z ) into π ∗ V | S ∗ ∂ X ( X/ Z ) ∼ = V + + ⊕ V − − ⊕ V + − ⊕ V − + corresp onds to π ∗ ( E ⊗ F ) | S ∗ ∂ X ( X/ Z ) ∼ = ( E + ⊗ F + ) ⊕ ( E − ⊗ F − ) ⊕ ( E + ⊗ F − ) ⊕ ( E − ⊗ F + ) , with π ∗ E = E + ⊕ E − and π ∗ F = F + ⊕ F − split into ± eigenbundles of σ tot ( D ) and π ∗ Φ, resp ectively . Then w e note that in K -theory , [ E + ⊗ F + ] − [0] = ([ E + ] − [0]) · ([ F + ] − [0]) ∈ K 0 ( S ∗ ∂ X ( X/ Z )) . Since the pushforward j ∗ : K ∗ c ( S ∗ ∂ X ( X/ Z ) × I ) − → K ∗ ( ∂ ( T ∗ ( X/ Z ))) behav es naturally with respect to pro ducts in K -theory , and since ch([ E + ] · [ F + ]) = c h( E + ) · ch( F + ), we hav e c h(ind( P )) = p ! (c h( E + ) · ch( F + ) · π ∗ Td( ∂ X/ Z )) , as claimed.  12 5.1. Dirac case. W e further sp ecialize to the case where D = / D is a family of (self-adjoint) Dirac op erators, acting on sections of a family of Clifford mo dules V . In this case, our index formula further reduces to one o v er T ∗ ( ∂ X/ Z ) rather than S ∗ ∂ X ( X/ Z ), and is given in terms of a related family of Dirac op erators on ∂ X . Let C  ( X/ Z ) denote the Clifford bundle C  ( sc T ( X/ Z ) , g ). Supp ose that V − → X is a family of Clifford mo dules with unitary , skew-Hermitian action c  : C  ( X/ Z ) − → End( V ), and a compatible Cliffor d c onne ction ∇ : C ∞ ( X ; V ) − → C ∞ ( X ; sc T ∗ ( X/ Z ) ⊗ V ), i.e. ∇ ( φ · u ) = ∇ LC( g ) φ · u + φ · ( ∇ u ) , φ ∈ C ∞ ( X ; C  ( X/ Z )) , u ∈ C ∞ ( X ; V ) , where ∇ LC( g ) : C ∞ ( X ; C  ( X/ Z )) − → C ∞ ( X ; sc T ∗ ( X/ Z ) ⊗ C  ( X/ Z )) is the natural extension of the Levi- Civita connection to C  ( X/ Z ). In analogy to the case of compact manifolds [LM89], these data lead to the construction of a canonical scattering Dirac operator / D ∈ Diff 1 sc ( X/ Z ; V ), defined at p ∈ X by / D p = X j c  ( e j ) · ∇ e j : C ∞ c ( ˚ X ; V ) − → C ∞ c ( ˚ X ; V ) , { e j } n j =1 an orthonormal basis for sc T p ( X/ Z ) , whic h is essentially self-adjoint with respect to the L 2 ( X ; V ) pairing ( / D u, v ) = ( u, / D v ) . Note that σ sc ( / D )( p, ξ ) = i c  ( ξ ) · . There is a splitting of V ov er ∂ X coming from the Clifford mo dule structure. T o see this, recall the isomorphism C  ( R n − 1 ) ∼ = C  0 ( R n ) , C  0 denoting the even graded part of the algebra, which is generated by R n − 1 3 e i 7− → e i · e n . Similarly , giv en a choice of normal section ν = x 2 ∂ x : ∂ X − → sc N ( ∂ X/ Z ), we hav e a bundle isomorphism C  ( sc T ( ∂ X/ Z ) , g ) ∼ = C  0 ( sc T ( X/ Z ) , g ) ∂ X = C  0 ( X/ Z ) ∂ X , and, by a c hoice of b oundary defining function x , we can further iden tify C  ( sc T ( ∂ X/ Z ) , g ) and C  ( ∂ X/ Z ) ≡ C  ( T ( ∂ X/ Z ) , h ). The family of Clifford mo dules V | ∂ X has the structure of a Z / 2 Z -graded module ov er this subalgebra C  ( ∂ X/ Z ). Explicitly , if V | ∂ X = V 0 ⊕ V 1 is the splitting according to ± 1 eigenspaces of the Hermitian endomorphism i c  ( ν ), then by the anticomm u- tativit y of sc T ( ∂ X/ Z ) and sc N ( ∂ X/ Z ) within C  ( X/ Z ), it follows that C  ( ∂ X/ Z ) acts by graded endomor- phisms, so C  j ( ∂ X/ Z ) : V i − → V i + j , i, j ∈ Z / 2 Z . W e denote this induced C  ( ∂ X/ Z ) action by c  0 : C  ( ∂ X/ Z ) − → End gr ( V 0 ⊕ V 1 ) . If the Clifford connection ∇ is the lift of a b connection (so ∇ restricts to a connection on ∂ X ), we define the induc e d b oundary Dir ac op er ator / ∂ ∈ Diff 1 ( ∂ X/ Z ; V ) b y / ∂ p = X j c  0 ( e j ) ∇ e j , { e j } n − 1 j =1 an orthonormal basis for T p ( ∂ X/ Z ). No w assume Φ ∈ C ∞ ( X ; End( V )) is a compatible family of p otentials. In particular, symbolic commu- tativit y at ∂ X implies that the positive/negativ e eigenbundles of V | ∂ X with resp ect to Φ are themselves Clifford mo dules: [ π ∗ Φ , σ sc ( / D )] = [ π ∗ Φ , c  ( · )] = 0 ∈ C ∞ ( sc T ∗ ( ∂ X/ Z ); End( V )) = ⇒ V | ∂ X = V + ⊕ V − , C  ( X/ Z ) | ∂ X : V ± − → V ± . F urthermore, it is possible to choose the Clifford connection c ompatible with this splitting, so that the restriction of the connection to ∂ X preserv es V ± . By symbolic commutativit y , the splittings V | ∂ X = V + ⊕ V − and V | ∂ X = V 0 ⊕ V 1 are compatible, so we ha v e V | ∂ X = V 0 + ⊕ V 0 − ⊕ V 1 + ⊕ V 1 − , 13 with resp ect to whic h the induced boundary Dirac operator / ∂ takes the form / ∂ =     0 / ∂ − + 0 0 / ∂ + + 0 0 0 0 0 0 / ∂ − − 0 0 / ∂ + − 0     . In particular, the op erators / ∂ ± ± are families of Dirac op erators on ∂ X , a fibration of closed manifolds ov er Z , whose principal symbols are given by the induced Clifford action c  0 , for instance, σ ( / ∂ + + ) = i c  0 ( · ) : T ∗ ( ∂ X/ Z ) − → Hom( π ∗ V 0 + , π ∗ V 1 + ) . It remains to show how the splitting of π ∗ V | S ∗ ∂ X ( X/ Z ) = V + ⊕ V − in to eigenbundles of σ tot ( / D ) is related to the splitting V | ∂ X = V 0 ⊕ V 1 . Lemma 9. We have K 0 ( S ∗ ∂ X ( X/ Z )) ∼ = K 0 c ( T ∗ ( ∂ X/ Z )) ⊕ K 0 ( ∂ X ) , with r esp e ct to which [ V + ± ] = [ π ∗ V 0 ± , π ∗ V 1 ± , i c  0 ] + [ V 1 ± ] . Pr o of. W e can identify S ∗ ∂ X ( X/ Z ) with tw o copies of T ∗ ( ∂ X/ Z ), glued along their common b oundary S ∗ ( ∂ X/ Z ). With this iden tification, the exact sequence of the pair ( S ∗ ∂ X ( X/ Z ) , T ∗ ( ∂ X/ Z )) splits since there is an ob vious retraction S ∗ ∂ X ( X/ Z ) − → T ∗ ( ∂ X/ Z ) (pro jecting one hemisphere of each fib er onto the other). Hence we hav e a split short exact sequence 0 − → K 0 ( S ∗ ∂ X ( X/ Z ) , T ∗ ( ∂ X/ Z )) − → K 0 ( S ∗ ∂ X ( X/ Z )) − → K 0 ( T ∗ ( ∂ X/ Z )) − → 0 Along with the isomorphisms K 0 ( S ∗ ∂ X ( X/ Z ) , T ∗ ( ∂ X/ Z )) ∼ = K 0 ( T ∗ ( ∂ X/ Z ) , S ∗ ( ∂ X/ Z )) = K 0 c ( T ∗ ( ∂ X/ Z )) , and K 0 ( T ∗ ( ∂ X/ Z )) ∼ = K 0 ( ∂ X ) (by contractibilit y of the fibers), w e obtain K 0 ( S ∗ ∂ X ( X/ Z )) ∼ = K 0 c ( T ∗ ( ∂ X/ Z )) ⊕ K 0 ( ∂ X ) , as claimed. W e will exhibit the decomp osition of [ V + + ] under this splitting; the case of [ V + − ] is similar. W e claim that, as a vector bundle, V + + ∼ = π ∗ V 0 + ∪ i c ` 0 ( · ) π ∗ V 1 + , that is, V + + is isomorphic to the gluing of the vector bundles π ∗ V 0 + − → T ∗ ( ∂ X/ Z ) and π ∗ V 1 + − → T ∗ ( ∂ X/ Z ) via the clutching function i c  0 : S ∗ ( ∂ X/ Z ) − → Hom( π ∗ V 0 + , π ∗ V 1 + ). T o see this, observe that the t w o copies of T ∗ ( ∂ X/ Z ) in S ∗ ∂ X ( X/ Z ) retract, resp ectively , onto the images of ∂ X under the in w ard and outw ard p oin ting conormal sections ν : ∂ X − → N ∗ ( ∂ X/ Z ) ⊂ S ∗ ∂ X ( X/ Z ) and − ν : ∂ X − → N ∗ ( ∂ X/ Z ) ⊂ S ∗ ∂ X ( X/ Z ) . Recall that ( V + + ) ξ is the p ositive eigenspace of Clifford multiplication i c  ( ξ ) at the p oint ξ , whereas ( π ∗ V 0 + ) ξ (resp. ( π ∗ V 1 + ) ξ ) is the positive (resp. negative) eigenspace of Clifford multiplication b y the corresp onding in w ard pointing normal, i c  ( ν ( π ( ξ ))). Thus, o v er a p oint ν ( p ) ∈ S ∗ ∂ X ( X/ Z ), we hav e ( V + + ) ν ( p ) = π ∗ ( V 0 + ) ν ( p ) , while ov er the antipo dal point − ν ( p ), we hav e ( V + + ) ( − ν ( p )) = π ∗ ( V 1 + ) ( − ν ( p )) , since, for v ∈ ( V + + ) ( − ν ( p )) , v = i c  ( − ν ( p )) v = − i c  ( ν ( p )) v . The bundle V + + can therefore be identified with π ∗ V 0 + and π ∗ V 1 + o v er the in ward and outw ard directed copies of T ∗ ( ∂ X/ Z ), respectively , and (since c  0 ( ξ ) = − i c  ( ξ ) ∼ c  ( ξ ) on π ∗ V 0 when ξ ∈ S ∗ ( ∂ X/ Z )), it is clear that i c  0 ∼ i c  : S ∗ ( ∂ X/ Z ) − → Hom( π ∗ V 0 + , π ∗ V 1 + ) is the transition function gluing them together to pro duce V + + , which finishes the claim. 14 Consider then the elemen t [ V + + ] = [ V + + ] − [ π ∗ V 1 + ] + [ π ∗ V 1 + ] ∈ K 0 ( S ∗ ∂ X ( X/ Z )) . F rom the ab ov e, w e see that [ V + + ] − [ π ∗ V 1 + ] v anishes o v er the outw ard facing copy of T ∗ ( ∂ X/ Z ), and so maps to the element [ π ∗ V 0 + , π ∗ V 1 + , i c  0 ] ∈ K 0 c ( T ∗ ( ∂ X/ Z )) in the decomp osition ab ov e. Clearly K 0 ( T ∗ ( ∂ X/ Z )) 3 [ π ∗ V 1 + ] ∼ = [ V 1 + ] ∈ K 0 ( ∂ X ) under contraction along the fib ers, and we therefore hav e [ V + + ] ∼ = [ π ∗ V 0 + , π ∗ V 1 + , i c  0 ] + [ V 1 + ] , as claimed.  The element [ π ∗ V 0 + , π ∗ V 1 + , i c  0 ] ∈ K 0 c ( T ∗ ∂ X ) corresp onds precisely to the sym bol of / ∂ + + , and we obtain the following: Theorem 10. L et / D ∈ Diff 1 sc ( X/ Z ; V ) b e a family of sc attering Dir ac op er ators acting on a family of Cliffor d mo dules V − → X , c onstructe d fr om a lifte d b Cliffor d c onne ction, and supp ose Φ ∈ C ∞ ( X ; End( V )) is a c omp atible p otential. L et / ∂ + + ∈ Diff 1 ( ∂ X/ Z ; V 0 + , V 1 + ) b e the gr ade d p art of the induc e d b oundary Dir ac op er ator / ∂ acting on the p ositive eigenbund le ( V + ) | ∂ X = V 0 + ⊕ V 1 + of Φ | ∂ X . Then P = / D + i Φ ∈ Diff 1 sc ( X/ Z ; V ) extends to a family of F r e dholm op er ators and ind( P ) = ind( / ∂ + + ) ∈ K 0 ( Z ) . In p articular, we have the index formula c h(ind( P )) = p ! (c h( σ ( / ∂ + + )) · π ∗ Td( ∂ X/ Z )) wher e p ! : H even c ( T ∗ ( ∂ X/ Z )) − → H even ( Z ) denotes inte gr ation over the fib ers and, in the c ase of a single op er ator ind( P ) = ind( / ∂ + + ) = Z T ∗ ( ∂ X ) c h( σ ( / ∂ + + )) · π ∗ Td( ∂ X ) . Pr o of. W e’v e seen that [ π ∗ V , π ∗ V , σ tot ( P )] ∈ K 0 c ( T ∗ ( ˚ X / Z )) is the image of [ V + + ] under the comp osition K 0 ( S ∗ ∂ X ( X/ Z )) j ∗ − → K 1 ( ∂ ( T ∗ ( X/ Z ))) − → K 0 c ( T ∗ ( ˚ X / Z )). In fact, it follo ws that this image is supported in a set homeomorphic to a small product neigh borho o d of the corner, S ∗ ∂ X ( X/ Z ) × R 2  → T ∗ ( ˚ X / Z ), and that [ π ∗ V , π ∗ V , σ tot ( P )] ∼ = β · [ V + + ] ∈ K 0 c ( S ∗ ∂ X ( X/ Z ) × R 2 ) where β ∈ K 0 c ( R 2 ) is the Bott element. F rom Lemma 9, [ V + + ] is a sum of tw o terms, [ V + + ] = [ π ∗ V 0 + , π ∗ V 1 + , σ ( / ∂ + + )] + [ V 1 + ] , and we define σ 1 , σ 2 ∈ K 0 c ( T ∗ ( ˚ X / Z )) to b e the images of these terms. As ab o v e, σ 1 and σ 2 are supp orted in our set S ∗ ∂ X ( X/ Z ) × R 2 , where they ha v e the form σ 1 ∼ = β · [ π ∗ V 0 + , π ∗ V 1 + , σ ( / ∂ + + )] , and σ 2 ∼ = β · [ π ∗ V 1 + ] . T o prov e the theorem, it suffices to show that top-ind( σ 1 ) = top-ind( / ∂ + + ) and that top-ind( σ 2 ) = 0. In fact, w e will show that σ 2 v anishes iden tically . T o see the latter, note that σ 2 ∼ = β · [ π ∗ V 1 + ] = β · π ∗ [ V 1 + ] where π ∗ : K 0 ( ∂ X ) − → K 0 ( S ∗ ∂ X ( X/ Z )). It follows that σ 2 ∈ K 0 c ( T ∗ ( ˚ X / Z )) is inv ariant with resp ect to the action of the rotation group O ( n ) ( n = dim( X/ Z )) on the fib ers of T ∗ ( ˚ X / Z ), since O ( n ) acts fib erwise on the first factor of S ∗ ∂ X ( X/ Z ) × R 2 , and π ∗ [ V 1 + ] is constant on these fib ers. Th us σ 2 is obtained by pullback of an element σ 0 2 ∈ K 0 c ( L ) , where L is the radial R + bundle L = T ∗ ( X/ Z ) /O ( n ) − → X . Ho w ever, K 0 c ( L ) ≡ 0 for any suc h bundle, since it is equiv alent to the (reduced) K-theory of C X , the cone on X , which is a contractible space. Therefore σ 0 2 = 0 which implies σ 2 = 0. 15 It remains to show top-ind( σ 1 ) = top-ind( / ∂ + + ). Supp ose we are given a K-orien ted em b edding of fibrations T ∗ ( X/ Z )  → R 2 N × Z coming from an em b edding X − → R N × Z . Let g : S ∗ ∂ X ( X/ Z ) × R 2  → R 2 N × Z b e the induced em b edding of our product neighborho o d and let h = g | S ∗ ∂ X ( X/ Z ) ×{ (0 , 0) } : S ∗ ∂ X ( X/ Z )  → R 2 N × Z b e the induced embedding of S ∗ ∂ X ( X/ Z ). W e denote the normal bundles of S ∗ ∂ X ( X/ Z ) × R 2 and S ∗ ∂ X ( X/ Z ) in R 2 N × Z b y N g ( S ∗ ∂ X ( X/ Z ) × R 2 ) and N h ( S ∗ ∂ X ( X/ Z )), resp ectively , emphasizing the corresp onding em- b eddings. Of course N h ( S ∗ ∂ X ( X/ Z )) = R 2 × N g ( S ∗ ∂ X ( X/ Z ) × R 2 ) since our neighborho o d is a pro duct. Let f : T ∗ ( ∂ X/ Z )  → S ∗ ∂ X ( X/ Z ) b e the op en em b edding onto the op en in w ard facing op en disk bundle as in Lemma 9, so σ 1 ∼ = β · f ∗ [ σ ( / ∂ + + )] , where [ σ ( / ∂ + + )] will b e shorthand for [ π ∗ V 0 + , π ∗ V 1 + , σ ( / ∂ + + )]. Finally , note that h ◦ f is a K-oriented embedding of T ∗ ( ∂ X/ Z ) in to a trivial Euclidean fibration whic h is homotopic (by stereographic pro jection) to the em b edding induced by the map ∂ X − → X − → R N × Z , and hence is suitable for computing top-ind( / ∂ + + ). W e will mak e use of the follo wing general fact. If V − → B is an orien ted complex v ector bundle and i : A  → B is an op en embedding of spaces (which induces an op en embedding ˜ i : V | A − → V ), then for an y α ∈ K ∗ c ( A ), we hav e i ∗ ( α ) · T ( V ) = ˜ i ∗  α · T ( V | A )  ∈ K ∗ c ( V ) , where T ( V ) ∈ K 0 c ( V ) is the K-orientation class (Thom class) generating K 0 c ( V ) as a mo dule o ver K 0 ( B ). As a consequence, w e need only to show that top-ind( σ 1 ) = σ 1 · T ( N g ( S ∗ ∂ X ( X/ Z ) × R 2 )) ∈ K 0 c ( R 2 N × Z ) is equiv alen t to top-ind( / ∂ + + ) = ˜ f ∗  [ σ ( / ∂ + + )] · T ( N h ◦ f ( T ∗ ( ∂ X/ Z )))  = f ∗ [ σ ( / ∂ + + )] · T ( N h ( S ∗ ∂ X ( X/ Z ))) ∈ K 0 c ( R 2 N × Z ) . Ho w ever this follo ws immediately since, on the one hand, σ 1 ∼ = β · f ∗ [ σ ( / ∂ + + )]; and on the other hand, b ecause N h ( S ∗ ∂ X ( X/ Z )) = R 2 × N g ( S ∗ ∂ X ( X/ Z ) × R 2 ), we hav e T ( N h ( S ∗ ∂ X ( X/ Z ))) = β · T ( N g ( S ∗ ∂ X ( X/ Z ) × R 2 ) , b y multiplicativit y of the Thom class and the fact that the Thom class of a trivial R 2 bundle is exactly β . Th us, w e obtain that ind( P ) = ind( / ∂ + + ) since top-ind( P ) = top-ind( / ∂ + + ), and the rest of the pro of follo ws by taking the Chern c haracter of b oth sides.  Finally , we consider the pro duct Dirac case; that is, assume V = E ⊗ F − → X where / D ∈ Diff 1 sc ( X/ Z ; E ) acts on E and the compatible p otential Φ ∈ C ∞ ( X ; End( F )) acts on F , and / D ⊗ 1 = / D F is obtained by equipping E ⊗ F with a tensor pro duct connection. W e form the Callias-Anghel t ype family P = / D ⊗ 1 + i ⊗ Φ ∈ Diff 1 sc ( X/ Z ; E ⊗ F ) . As ab ov e, the Clifford module E splits ov er the boundary in to E 0 ⊕ E 1 , with / ∂ =  0 / ∂ − / ∂ + 0  , and F | ∂ X = F + ⊕ F − splits into p ositive and negative eigen bundles of Φ | ∂ X . Theorem 11. L et P = / D ⊗ 1 + i ⊗ Φ ∈ Diff 1 sc ( X/ Z ; E ⊗ F ) as ab ove. Then P extends to a F r e dholm family with index ind( P ) = ind  / ∂ + F +  wher e / ∂ + F + is the twiste d Dir ac op er ator obtaine d by twisting / ∂ + ∈ Diff 1 sc ( ∂ X/ Z ; E 0 , E 1 ) by F + , the p ositive eigenbund le of Φ | ∂ X . 16 R emark. In particular, when X is an odd-dimensional spin manifold and / D is the (self-adjoin t) spin Dirac op- erator (i.e. constructed using the fundamental representation of C  ( X ) on spinors), then / ∂ + ∈ Diff 1 ( ∂ X ; S 0 , S 1 ) is the graded spin Dirac op erator o v er the b oundary , and w e obtain ind( / D ⊗ 1 + i ⊗ Φ) = Z T ∗ ∂ X c h( F + ) · ˆ A( ∂ X ) , since ch( σ ( / ∂ + )) · Td( ∂ X ) = ˆ A( ∂ X ) (compare to the formula obtained b y R ˚ ade in [R ˚ ad94]). Pr o of. The result follows from the previous one, after noting that V 0 + and V 1 + are giv en by E 0 ⊗ F + and E 1 ⊗ F + , resp ectively , and that the clutc hing function i c  0 : S ∗ ( ∂ X/ Z ) − → Hom( π ∗ E 0 ⊗ F + , π ∗ E 1 ⊗ F + ) is given by σ ( / ∂ + ) ⊗ 1 = σ ( / ∂ + F + ).  There are a few final remarks to b e made: • First, regarding ev en/o dd dimensionalit y: in the case of (families of ) Dirac op erators, P will only ha v e a nonzero index when the dimension dim( X/ Z ) of the fib er is o dd . Since the index of P reduces to the index of a family of differ ential op erators on ∂ X − → Z , it must v anish when dim( ∂ X/ Z ) = dim( X/ Z ) − 1 is odd for the usual reason. Because of this, previous literature on the sub ject was limited to the index problem on odd-dimensional manifolds, though w e emphasize that, if D is allo w ed to b e pseudo differen tial, P may hav e non trivial index even when dim( X/ Z ) is even. • Our analysis of clutching data in the Dirac case, which related [ V + + ] ∈ K 0 ( S ∗ ∂ X ( X/ Z )) to the symbol [ π ∗ V 0 + , π ∗ V 1 + , σ ( / ∂ + + )] ∈ K 0 c ( T ∗ ( ∂ X/ Z ) of an operator / ∂ + + on ∂ X is equally v alid when D is pseudo- differen tial. Indeed, V + + can alw a ys be written as the clutc hing of bundles π ∗ V 0 + and π ∗ V 1 + com- ing from the base, with resp ect to some clutching function f , and then ind( P ) = ind( δ ), where δ ∈ Ψ m ( ∂ X/ Z ; V 0 , V 1 ) is any elliptic pseudo differen tial operator whose sym b ol σ ( δ ) = f . How ever, suc h a choice of δ is far from canonical without the additional structure of the Clifford bundles. 6. Rela tion to previous resul ts In [Ang93b], N. Anghel generalized Callias’ original index theorem to the follo wing situation 13 (adapted to our notation): Let X be a general o dd-dimensional, non-compact, complete Riemannian manifold (with no particular structure assumed at infinity), with a Clifford mo dule V − → X . Let / D : C ∞ c ( X ; V ) − → C ∞ c ( X ; V ) b e a self-adjoin t Dirac operator, and Φ ∈ C ∞ ( X ; End( V )) a p otential which is assumed to be uniformly in v ertible aw a y from a compact set K b X and such that [ / D , Φ] is a uniformly b ounded, 0th order operator (in particular, Φ comm utes with Clifford m ultiplication). First he pro ves that, for sufficien tly large λ > 0, P λ = / D + iλ Φ is F redholm, essen tially by showing that P λ and P ∗ λ satisfy what the author likes to call “ inje ctivity ne ar infinity ” condi- tions: k P λ u k L 2 ≥ c k u k L 2 for all u ∈ C ∞ c ( X \ K ; V ) and similarly for P ∗ λ . In [Ang93a] Anghel shows ho w such conditions are equiv alen t to F redholmness for self-adjoin t Dirac op erators, but it is easy to see that his pro of generalizes to show that an y differential op erator P , which is injectiv e near infinity along with its adjoin t, extends to be F redholm. In any case, as in Section 5.1, V splits o v er X \ K into p ositive and negative eigenbundles of Φ: V | X \ K = V + ⊕ V − , and choosing a compact set L b X suc h that K ⊂ ˚ L with ∂ L = Y a separating hypersurface (compare our earlier situation in which Y = ∂ X ), w e hav e further compatible splitting ( V ± ) | Y = V 0 ± ⊕ V 1 ± according to the decomp osition C  ( Y ) ∼ = C  ( X ) 0 | Y , with − i c  ( ν ) ≡ ( − 1) i : V i ± − → V i ± where ν is a unit normal section. Cho osing appropriate connections, we can again construct an induced Dirac op erator on Y , / ∂ Y =     0 / ∂ − + 0 0 / ∂ + + 0 0 0 0 0 0 / ∂ − − 0 0 / ∂ + − 0     , 13 See also [R ˚ ad94] for an indep endently obtained pro of which addresses the Dirac pro duct case as in section 5.1. 17 and Anghel prov es that ind( / D + iλ Φ) = ind( / ∂ + + ) . His pro of consists of index preserving deformations, along with the relative index theorem of Gromov and Lawson in [GL83] (discussed further in [Ang93a]) to reduce to a pro duct type Dirac op erator e / D on a Riemannian pro duct Y × R , e / D = i c  ( ν ) ∂ ∂ t + / ∂ + + iλχ ( t ) : C ∞ c ( Y × R ; V + ) − → C ∞ c ( Y × R ; V + ) with equiv alen t index. Here χ : R − → [ − 1 , 1] is a smo oth function such that χ ≡ − 1 near −∞ and χ ≡ 1 near + ∞ . Direct computation then shows that ind( e / D ) = ind( / ∂ + + ). W e p oin t out that the steps in his pro of could just as easily reduce to a sc attering pro duct (i.e. Y × R , but with lo cally Euclidean ends instead of cylindrical ones), with a scattering type Dirac operator e / D 0 = i c  ( ν ) ∂ ∂ t + 1 t / ∂ + + iλχ ( t ) : C ∞ c ( Y × R ; V + ) − → C ∞ c ( Y × R ; V + ) whose index is equiv alen t to ind( / ∂ + + ) b y our own Theorem 10. This is really ov erkill in this case, since the index of e / D is determined simply enough; how ev er, it raises the p oint that scattering-t ype infinite ends ( ∂ L × [0 , ∞ ), where L b X as ab ov e) may b e utilized for the purp ose of computing the index of Dirac op erators satisfying Anghel’s F redholm conditions (injectivity near infinit y for P and its adjoin t). The author an ticipates that cutting and gluing constructions, similar to those used in [GL83] to prov e the relative index form ula, may b e able exhibit such equiv alences for arbitrary F redholm differential operators satisfying injectivity near infinit y conditions. References [AM09] P . Albin and R. Melrose, R elative chern character, b oundaries and index formulæ , Journal of T opology and Analysis 1 (2009), no. 3, 207–250. [Ang93a] N. Anghel, An abstr act index the orem on non-c omp act Riemannian manifolds , Houston Journal of Mathematics 19 (1993), no. 2, 223–237. 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Michelsohn, Spin geometry , Princeton Universit y Press, 1989. [Mel] R. Melrose, Differ ential analysis on manifolds with c orners. Book in pr epar ation . [Mel94] , Sp e ctr al and scattering the ory for the L aplacian on asymptotic al ly Euclidian sp ac es , Spectral and scattering theory: pro ceedings of the T aniguchi international workshop, CRC, 1994, pp. 85–130. [Mel95] R.B. Melrose, Ge ometric sc attering the ory , Cambridge University Press, 1995. [MR04] R. Melrose and F. Ro chon, F amilies index for pseudo differential oper ators on manifolds with b oundary , International Mathematics Research Notices 2004 (2004), no. 22, 1115. [R ˚ ad94] J. R ˚ ade, Cal lias’index the or em, el liptic boundary c onditions, and cutting and gluing , Communications in Mathemat- ical Physics 161 (1994), no. 1, 51–61. Massachusetts Institute of Technology, Dep ar tment of Ma thema tics, Cambridge, MA 02139 Curr ent addr ess : Brown Universit y, Department of Mathematics, Providence, RI 02912 E-mail address : ckottke@math.brown.edu 18