The Maslov Index and the Spectral Flow - revisited

We give an elementary proof of a celebrated theorem of Cappell, Lee and Miller which relates the Maslov index of a pair of paths of Lagrangian subspaces to the spectral flow of an associated path of selfadjoint first-order operators. We particularly pay attention to the continuity of the latter path of operators, where we consider the gap-metric on the set of all closed operators on a Hilbert space. Finally, we obtain from Cappell, Lee and Miller's theorem a spectral flow formula for linear Hamiltonian systems which generalises a recent result of Hu and Portaluri.


Introduction
Let ·, · be the Euclidean scalar product on R 2n and ω 0 (·, ·) = J·, · the standard symplectic form, where and I n denotes the identity matrix. Let us recall that an n-dimensional subspace L ⊂ R 2n is called Lagrangian if the restriction of ω 0 to L × L vanishes. The set Λ(n) of all Lagrangian subspaces in R 2n is called the Lagrangian Grassmannian. It can be regarded as a submanifold of the Grassmannian G n (R 2n ) and so it has a canonical topology. In what follows, we denote by I the unit interval [0, 1]. The Maslov index µ Mas (γ 1 , γ 2 ) assigns to any pair of paths γ 1 , γ 2 : I → Λ(n) an integer which, roughly speaking, is the total number of non-trivial intersections of the Lagrangian spaces γ 1 (λ) and γ 2 (λ) whilst the parameter λ travels along the interval I. There are several different approaches to the Maslov index and here we just want to mention [1], [4], [6], [8], [17] and [19], which is far from being exhaustive. Cappell, Lee and Miller introduced in [5] four different ways to define the Maslov index and showed that they are all equivalent. They first construct the Maslov index geometrically by using a stratification of Λ(n) and intersection theory from differential topology following [8]. Their approach also yields a uniqueness theorem for the Maslov index characterising this invariant uniquely by six axioms. The uniqueness theorem is then used to show that the Maslov index can alternatively be defined by determinant line bundles, η-invariants and the spectral flow, respectively. In this paper we focus on the latter invariant and aim to give a more elementary proof of the equality of the Maslov index and the spectral flow of a path of operators as introduced by Cappell, Lee and Miller in [5]. Let us first recall that the spectral flow is a homotopy invariant for paths of selfadjoint Fredholm operators that was invented by Atiyah, Patodi and Singer in [2], and since then has been used in various different settings (see e.g. [21, §5.2]). The spectrum of a selfadjoint Fredholm operator consists only of eigenvalues of finite multiplicity in a neighbourhood of 0 ∈ R and, roughly speaking, the spectral flow of a path of such operators is the net number of eigenvalues crossing 0 whilst the parameter of the path travels along the interval. Let us now consider for a pair of paths (γ 1 , γ 2 ) in Λ(n) the differential operators where D(A λ ) = {u ∈ H 1 (I, R 2n ) : u(0) ∈ γ 1 (λ), u(1) ∈ γ 2 (λ)}.
By an elementary computation, A λ is symmetric, and it is also not difficult to see that it actually is a selfadjoint Fredholm operator. Note that the kernel of A λ is isomorphic to γ 1 (λ) ∩ γ 2 (λ), which suggests that the spectral flow of the path A = {A λ } λ∈I is related to the Maslov index of the pair (γ 1 , γ 2 ). As we already mentioned above, their equality is one of the main achievements of [5]. However, before we formulate this as a theorem, we want to highlight a further issue related to this problem. Above, we have spoken about paths of differential operators and so tacitly assumed continuity. Note that the family (2) has the non-constant domains (3) and so continuity is a non-trivial problem. There are different metrics on spaces of unbounded selfadjoint Fredholm operators on a Hilbert space H and we recommend [11] for an exhaustive discussion (see also [20]). A classical approach is to transform unbounded selfadjoint operators T by functional calculus to the bounded selfadjoint operators and to use the operator norm on L(H) for introducing a distance between unbounded operators. Actually, Atiyah, Patodi and Singer defined the spectral flow in [2] for bounded selfadjoint Fredholm operators and applied it to paths of differential operators by using (4). However, checking continuity along these lines is tedious, if possible at all (see e.g. [13]), and it seems that the continuity of families of unbounded operators has sometimes been ignored in the literature. Every (generally unbounded) selfadjoint operator on a Hilbert space is closed, and there is a canonical metric on the set of all closed operators which is called the gap-metric (see §IV.2 in Kato's monograph [10]). It was shown in [13] (see also [11,Prop. 2.2]) that every path of selfadjoint Fredholm operators that is mapped to a continuous path of bounded operators under (4) is also continuous with respect to the gap-metric. Finally, Booss-Bavnbek, Lesch and Phillips constructed in [3] the spectral flow for paths of selfadjoint Fredholm operators in this more general setting. The main result of this paper now reads as follows (see [5,Thm. 0.4]).
is a pair of paths in Λ(n), then the family of differential operators (2) is continuous with respect to the gap-metric and Let us make a few comments on our proof. Firstly, we want to emphasise that we prove the gap-continuity of the family (2) from first principles just by elementary estimates and standard facts about orthogonal projections that can all be found in the monograph [10]. Secondly, our proof of the spectral flow formula in Theorem 1.1 is surprisingly simple. We assume at first that and show that the Maslov index can be characterised in this case by three axioms. This uniqueness theorem needs nothing else than the elementary properties of the Maslov index and the fact that the fundamental group of Λ(n) is infinitely cyclic, which was known already from Arnold's classical paper [1]. Two of our axioms are trivially satisfied for the spectral flow of (2), and the remaining one only requires the computation of the spectra of two simple examples of differential operators as in (2). The general case when (5) is not assumed, can easily be obtained from the previous case by a simple conjugation by a path of invertible operators. After a brief recapitulation of the Maslov index in Section 2.1, and the gap-metric and spectral flow in Section 2.2, we explain all this in detail in Section 3 where we prove Theorem 1.1. Throughout the paper, we aim our presentation to be rather self-contained, and we will just use some well-known facts from Kato [10]. Finally, we review a recent spectral flow formula for linear Hamiltonian systems by Hu and Portaluri from [9], which they call a new index theory on bounded domains. Firstly, we note that the considered families of Hamiltonian systems are continuous with respect to the gap-metric, which follows easily from our approach to Cappell, Lee and Miller's Theorem. Secondly, we obtain a spectral flow formula in this setting by a conjugation from Cappell, Lee and Miller, and we explain that our result actually is a generalisation of Hu and Portaluri's Theorem.
2 Maslov Index and Spectral Flow -a brief recap

The Maslov Index
The aim of this section is to briefly recall the definition of the Maslov index, where we follow [16]. Let Sp(2n, R) denote the group of symplectic matrices on R 2n , i.e., those A ∈ M (2n, R) satisfying A T JA = J or, alternatively, which preserve ω 0 . If we identify R 2n with C n by (x 1 , . . . , x 2n ) → (x 1 , . . . , x n ) + i(x n+1 , . . . , x 2n ) then the standard hermitian scalar product on C n is x, y C = x, y − iω 0 (x, y).
Hence each unitary matrix U ∈ U (n) preserves ω 0 and so we can regard U (n) as a subset of Sp(2n, R). Also, the orthogonal matrices O(n) can be seen as a subgroup of U (n) by complexification. Then O(n) consists exactly of those A ∈ U (n) which leave R n × {0} invariant. Obviously, AL ∈ Λ(n) if L ∈ Λ(n) and A ∈ Sp(2n, R), and it can be shown that the restriction of this action to U (n) × Λ(n) → Λ(n) is transitive. As the stabiliser subgroup of R n × {0} ∈ Λ(n) is O(n), we see that there is a diffeomorphism Let us now consider the map d : , which descends to the quotient by Note that is a fibre bundle, and it is not difficult to see that ker(d)/O(n) ≃ SU (n)/SO(n), where the latter space is simply connected. It follows from the long exact sequence of a fibre bundle that the induced map is an isomorphism. Consequently, we obtain from (6) an isomorphism which is the Maslov index for closed paths in Λ(n). Roughly speaking, given an arbitrary L 0 ∈ Λ(n), the Maslov index counts the total number of intersections of a loop in Λ(n) with L 0 . This is independent of the particular choice of L 0 , which however is no longer the case if we extend the definition to non closed paths in Λ(n) as follows.
We fix L 0 ∈ Λ(n) and note at first that L 0 yields a stratification From the fact that Λ 0 (L 0 ) is contractible (see e.g. [16,Rem. 2.5.3]) and the long exact sequence of homology, we see that the inclusion induces an isomorphism Also, as π 1 (Λ(n)) is abelian, H 1 (Λ(n)) is isomorphic to π 1 (Λ(n)) and so we obtain a sequence of isomorphisms Finally, every path in Λ(n) having endpoints in Λ 0 (L 0 ) canonically yields an element in H 1 (Λ(n), Λ 0 (L 0 )). The Maslov index of the path is the integer obtained from the sequence of isomorphisms (7). Let us note from the very definition the following three properties of the Maslov index: (i) If γ 1 , γ 2 are homotopic by a homotopy having endpoints in Λ 0 (L 0 ), then Let us point out that (iii) also follows from (i) and (ii) independently of the construction. The Maslov index can easily be generalised to a pair of paths in Λ(n). To this aim let us call a pair of paths (γ 1 , γ 2 ) admissible if In what follows we consider R 2n × R 2n as a symplectic space with respect to the symplectic form Hence it is natural to define the Maslov index for a pair (γ 1 , γ 2 ) of admissible paths in Λ(n) as Note that the basic properties which we previously mentioned carry over immediately, i.e., are homotopic by a homotopy through admissible pairs.
Also, it is not difficult to see from the construction of the Maslov index that Finally, let us define the Maslov index for a non-admissible pair of paths. It is important to note that in this case there are different definitions in the literature. Here we follow [5], and note that given We define the Maslov index as where Θ is such that By the homotopy invariance, it is clear that this definition does not depend on the choice of Θ. Also, it coincides with the previous definition in case that the pair of paths is admissible.

The Paths γ nor and γ ′ nor
The aim of this section is to compute the Maslov index for two elementary paths that will also become important in our proof of Theorem 1.1 below. The examples also show that (7) is very convenient to obtain paths in Λ(n) with a given Maslov index. Let us first consider the path Note that A(0) diag(−1, 1, . . . , 1) = A(1) and so A is a closed curve. Also, as det 2 (A(λ)) = e 2πiλ , we see that the Maslov index of the corresponding path in Λ(n) is 1. Using the identification C n ∼ = R 2n , it is readily seen that Re j ∈ Λ(n).
Let us now consider and note that again the projection B to U (n)/O(n) is a closed path and det 2 (B(λ)) = (−1) n e 2πiλ . Hence

The Gap-Metric and the Spectral Flow
Our first aim of this section is to recall the definition of the gap-metric, where we follow Kato's monograph [10].
Let H be a real Hilbert space and let G(H) denote the set of all closed subspaces of H. For every U ∈ G(H) there is a unique orthogonal projection P U onto U which is a bounded operator on H. We set and note that this is obviously a metric on G(H). The distance between two non-trivial subspaces U, V ∈ G(H) can also be obtained as follows. Let S U denote the unit sphere in U and d(u, which explains why d G (U, V ) is called the gap between U and V . We now consider operators T : D(T ) ⊂ H → H which we assume to be defined on a dense domain D(T ). Let us recall that T is called closed if its graph graph(T ) is closed in H × H. If we denote by C(H) the set of all closed operators, then the gap-metric on H × H induces a metric on C(H) by As the adjoint of a densely defined operator is closed, every selfadjoint operator on H belongs to the metric space C(H). Moreover, let us recall that a closed operator T is called Fredholm if its kernel and cokernel are of finite dimension. In what follows, we denote the subset of C(H) consisting of all T which are selfadjoint and Fredholm by CF sa (H). It is well known that the spectrum σ(T ) of every selfadjoint operator is real. Moreover, if T ∈ CF sa (H) then 0 is either in the resolvent set or an isolated eigenvalue of finite multiplicity (see e.g. [21, Lemma 2.2.5]). It was shown in [3] that for every T ∈ CF sa (H) there is ε > 0 and a neighbourhood N T,ε ⊂ CF sa (H) of T such that ±ε / ∈ σ(S) and the spectral projection χ [−ε,ε] (S) is of finite rank for all S ∈ N T,ε . Let us now consider a path A = {A λ } λ∈I in CF sa (H). There are 0 = λ 0 < λ 1 < . . . < λ N = 1 such that the restriction of the path A to [λ i−1 , λ i ] is entirely contained in a neighbourhood N Ti,εi as above for some T i ∈ CF sa (H) and some ε i > 0. The spectral flow of the path A is defined as It follows by an argument of Phillips [15] that sf(A) only depends on the path A, and that the following fundamental property holds (see also [3]). (iii) if A 1 and A 2 are two paths in CF sa (H) such that A 1 1 = A 2 0 , then Let us finally note two further elementary properties of the spectral flow which play a crucial role in our proof of Theorem 1.1 below. The first of them has been used, e.g., in [14, §7].
Finally, let us note the following stability of the spectral flow under conjugation by invertible operators, where we denote by M T the adjoint of an operator in the real Hilbert space H. Proof. Note that H). By [10, Thm. I. 6.35], we have for the corresponding Consequently, {P graph(M T λ A λ M λ ) } λ∈I is continuous, which shows that M T AM is gap-continuous. For the equality of the spectral flows, we just need to note that M is homotopic inside GL(H) to the constant path given by the identity I H . Let us point out that this does not even require Kuiper's Theorem as we just need to shrink M to a constant path and use that GL(H) is connected. As the conjugation preserves kernel dimensions, we obtain by the homotopy invariance (i) from above sf(M T AM ) = sf(A).

Proof of Theorem 1.1
The proof of Theorem 1.1 falls naturally into two parts. In the first part we deal with the continuity of families of the type (2), where we actually consider a slightly more general setting. In the second part we show the spectral flow formula in Theorem 1.1.

Continuity
To simplify notation, we set E = L 2 (I, R 2n ) and H = H 1 (I, R 2n ). The aim of this step is to prove the following proposition, which we will later apply in the cases X = I and X = I × I. Proposition 3.1. Let X be a metric space and γ 1 , γ 2 : X → Λ(n) two families of Lagrangian subspaces in R 2n . Then is continuous with respect to the gap-metric on CF sa (E).
Note at first that for u ∈ D(A λ ) and v ∈ D(A λ0 ) where we have used that J = 1. Let us recall that the topology of G n (R 2n ) is induced by the metric d(L, M ) = P L − P M , where P L , P M ∈ M (2n, R) are the orthogonal projections onto L and M , respectively. Hence, by the continuity of γ 1 and γ 2 , there are two families of orthogonal projectionsP ,P : X → M (2n, R) such that im(P λ ) = γ 1 (λ), im(P λ ) = γ 2 (λ), λ ∈ X.
Since the point evaluation is continuous in H, there is a constant α > 0 such that for t = 0 and t = 1 where we use that J is an isometry on R 2n . Hence, by (11)- (14), As the unit sphere in graph(A λ ) is given by Note that if we swap λ and λ 0 and repeat the above argument, we also have To finish the proof, we need the following well-known theorem that can be found, e.g., in [10, I.6.34]. Now, as (I 2n −P λ )P λ0 = (I 2n −P λ0 )P λ = 0 for λ = λ 0 , we have for all λ in a neighbourhood of λ 0 (I 2n −P λ )P λ0 = (I 2n −P λ0 )P λ = P λ −P λ0 and likewise (I 2n −P λ )P λ0 = (I 2n −P λ0 )P λ = P λ −P λ0 .

The Spectral Flow Formula
We now prove the spectral flow formula in Theorem 1.1 in two steps.
Step 1: Theorem 1.1 for admissible paths We begin this first step of our proof with the following elementary observation.
is path-connected.
Proof. Let us first recall the well-known fact that Λ 0 (L 0 ) is contractible, and hence pathconnected, for any L 0 ∈ Λ(n) (see [16,Rem. 2.5.3]). Now let (L 1 , L 2 ) and (L 3 , L 4 ) be two transversal pairs. As in the construction of the Maslov index in Section 2.1, L ′ 1 = e ΘJ L 1 is transversal to L 2 and L 4 for any sufficiently small Θ > 0. In particular, we obtain a path connecting (L 1 , L 2 ) and (L ′ 1 , L 2 ) inside (17). Also, as Λ 0 (L ′ 1 ) is path-connected, there is a path connecting (L ′ 1 , L 2 ) and (L ′ 1 , L 4 ) inside (17). Finally, there is a path from This step of the proof is based on the following proposition in which we denote by Ω 2 the set of all admissible pairs of paths in Λ(n) (see (5)). Let us note that by Section 2. Then µ = µ Mas on Ω 2 .
We now define where A is the path of differential operators (2) for the pair (γ 1 , γ 2 ). We aim to use Proposition 3.4 to show Theorem 1.1 and so we need to check the properties (i'), (ii'), (iii') and (N). Let us first note that (i') follows immediately from (ii) in Section 2.2 and the fact that ker(A λ ) = γ 1 (λ)∩γ 2 (λ). Also, (ii') follows from (iii) in Section 2.2. Finally, (ii') is an immediate consequence of the homotopy invariance (i) of the spectral flow and Proposition 3.1. Hence it remains to show that µ(γ nor , L 1 ) = 1 and µ(L 0 , γ ′ nor ) = −1, which will be a direct consequence of the following lemma.
Hence |α| = 1, and the latter equation holds if and only if πλ = π 2 − µ + kπ which finally shows that µ = −πλ + π 2 + kπ. We see from the previous lemma that in both cases there is only one eigenvalue of A λ that crosses the axis whilst the parameter λ travels from 0 to 1. It is now an immediate consequence of the definition of the spectral flow that sf(A) = 1 for (γ 1 , γ 2 ) = (γ nor , L 1 ) and sf(A) = −1 for (γ 1 , γ 2 ) = (L 0 , γ ′ nor ). Hence Theorem 1.1 is shown in the admissible case.
Step 2: The general case Let (γ 1 , γ 2 ) be a pair of paths in Λ(n) which is not necessarily admissible, and let A be the path (2). Let δ > 0 be as in Lemma 2.1 such that We consider the solution Ψ : I → Sp(2n, R) of the differential equation and the operator M ∈ GL(L 2 (I, R 2n )) given by (M u)(t) = Ψ(t)u(t), t ∈ I. Then, as D(A δ0 and given by As Ψ(t) = exp(δ 0 Jt), t ∈ I, we see that Ψ(1)

A Spectral Flow Formula for Hamiltonian Systems
Let γ 1 , γ 2 : I → Λ(n) be two paths of Lagrangian subspaces in R 2n . We note for later reference the following two standard properties of the Maslov index (see e.g. [17]) (vi') If Ψ : I → Sp(2n, R) is a path of symplectic matrices, then Moreover, we need below the following homotopy invariance property which is an immediate consequence of (iii') in Section 2.1 and the definition of the Maslov index for non-admissible pairs of paths: (viii') µ Mas (γ 1 , γ 2 ) = µ Mas (γ 3 , γ 4 ) if γ 1 ≃ γ 3 and γ 2 ≃ γ 4 are homotopic by homotopies with fixed endpoints.
The corollary is now an immediate consequence of the previous theorem.
Consequently, under these assumptions the paths (23) and (2) have the same spectral flow and so the spectral flow of (23) does not depend on the family of matrices S. Note that each S λ is A λ -compact, i.e.
Finally, let us briefly point out that a version of the Morse Index Theorem in semi-Riemannian geometry from [12] can easily be derived from Theorem 4.1 as well. We do not intend to explain the geometric content of the theorem, but just mention that it deals with non-trivial solutions of boundary value problems of the type Ju ′ (t) + S λ (t)u(t) = 0, t ∈ I u(0), u(1) ∈ {0} × R n where J is as in (1) and S λ is again a family of symmetric 2n × 2n matrices. If we consider the operators A λ in (23) for the equations (26), then by Theorem 4.1, where Ψ = {Ψ λ (1)} λ∈I is the path in Sp(2n, R) obtained as in (24). This is Proposition 6.1 in [12]. Note that in this setting the path A = {A λ } λ∈I has the constant domain D(A λ ) = {u ∈ H 1 (I, R 2n ) : u(0), u(1) ∈ {0} × R n }, which allows to compute its spectral flow by crossing forms (see [18] and [22]) and yields the different proof of (27) given in [12].