The Maslov index and the spectral flow—revisited
 101 Downloads
Abstract
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 firstorder operators.
We particularly pay attention to the continuity of the latter path of operators, where we consider the gapmetric 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.
Keywords
Maslov index Spectral flow Hamiltonian systemsMSC
53D12 58J30 37J05 58E101 Introduction
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. [22, §5.2]). The spectrum of a selfadjoint Fredholm operator consists only of eigenvalues of finite multiplicity in a neighbourhood of \(0\in \mathbb{R}\) 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.
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 gapmetric (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 gapmetric. Finally, BoossBavnbek, 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]).
Theorem 1.1
If\((\gamma _{1},\gamma _{2})\)is a pair of paths in\(\varLambda (n)\), then the family of differential operators (2) is continuous with respect to the gapmetric and\(\operatorname {sf}(\mathcal{A})=\mu _{\mathrm{Mas}}(\gamma _{1},\gamma _{2})\).
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 gapmetric, 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
2.1 The Maslov index
The aim of this section is to briefly recall the definition of the Maslov index, where we follow [16].
 (i)If \(\gamma _{1}\), \(\gamma _{2}\) are homotopic by a homotopy having endpoints in \(\varLambda _{0}(L_{0})\), then$$ \mu _{\mathrm{Mas}}(\gamma _{1},L_{0})=\mu _{\mathrm{Mas}}( \gamma _{2},L_{0}). $$
 (ii)If \(\gamma _{1}\), \(\gamma _{2}\) are such that \(\gamma _{1}(1)= \gamma _{2}(0)\), then$$ \mu _{\mathrm{Mas}}(\gamma _{1}\ast \gamma _{2},L_{0})= \mu _{\mathrm{Mas}}(\gamma _{1},L_{0})+ \mu _{\mathrm{Mas}}( \gamma _{2},L_{0}). $$
 (iii)
If \(\gamma (\lambda )\in \varLambda _{0}(L_{0})\) for all \(\lambda \in I\), then \(\mu _{\mathrm{Mas}}(\gamma ,L_{0})=0\).
 (i′)

\(\mu _{\mathrm{Mas}}(\gamma _{1},\gamma _{2})=0\) if \(\gamma _{1}( \lambda )\cap \gamma _{2}(\lambda )=\{0\}\) for all \(\lambda \in I\).
 (ii′)

\(\mu _{\mathrm{Mas}}(\gamma _{1}\ast \gamma _{3},\gamma _{2}\ast \gamma _{4})=\mu _{\mathrm{Mas}}(\gamma _{1},\gamma _{2})+\mu _{\mathrm{Mas}}(\gamma _{3}, \gamma _{4})\) if \(\gamma _{1}(1)=\gamma _{3}(0)\) and \(\gamma _{2}(1)= \gamma _{4}(0)\).
 (iii′)

\(\mu _{\mathrm{Mas}}(\gamma _{1},\gamma _{2})=\mu _{\mathrm{Mas}}(\gamma _{3}, \gamma _{4})\) if \(\gamma _{1}\simeq \gamma _{3}\) and \(\gamma _{2}\simeq \gamma _{4}\) are homotopic by a homotopy through admissible pairs.
 (iv′)

\(\mu _{\mathrm{Mas}}(\gamma _{1},\gamma _{2})=\mu _{\mathrm{Mas}}(\gamma _{1},L _{0})\) in the case that \(\gamma _{2}(\lambda )=L_{0}\) for some \(L_{0}\in \varLambda (n)\) and all \(\lambda \in I\),
 (v′)

\(\mu _{\mathrm{Mas}}(\gamma _{1},\gamma _{2})=\mu _{\mathrm{Mas}}(\gamma _{2}, \gamma _{1})\) for any admissible pair \((\gamma _{1},\gamma _{2})\).
The final 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 \(\varLambda (n)\) with a given Maslov index.
2.2 The gapmetric and the spectral flow
Our first aim of this section is to recall the definition of the gapmetric, where we follow Kato’s monograph [10].
 (i)Let \(h:I\times I\rightarrow \mathcal{CF}^{\mathrm{sa}}(H)\) be a homotopy such that the dimensions of the kernels of \(h(s,0)\) and \(h(s,1)\) are constant for all \(s\in I\). Then$$ \operatorname {sf}\bigl(h(0,\cdot ) \bigr)=\operatorname {sf}\bigl(h(1,\cdot ) \bigr). $$
 (ii)
if the dimension of the kernel of \(\mathcal{A}_{\lambda }\) is constant for all \(\lambda \in I\), then \(\operatorname {sf}(\mathcal{A})=0\);
 (iii)if \(\mathcal{A}^{1}\) and \(\mathcal{A}^{2}\) are two paths in \(\mathcal{CF}^{\mathrm{sa}}(H)\) such that \(\mathcal{A}^{1}_{1}= \mathcal{A}^{2}_{0}\), then$$ \operatorname {sf}\bigl(\mathcal{A}^{1}\ast \mathcal{A}^{2} \bigr)=\operatorname {sf}\bigl(\mathcal{A}^{1} \bigr)+\operatorname {sf}\bigl( \mathcal{A}^{2} \bigr). $$
Lemma 2.1
Proof
We note at first that the operators \(\mathcal{A}^{\delta }_{\lambda }\) are selfadjoint and Fredholm for δ sufficiently small, which follows from standard stability theory (see e.g. [10]). Moreover, the path \(\mathcal{A}^{\delta }\) is gapcontinuous by [10, Thm. IV.2.17], and so \(\operatorname {sf}(\mathcal{A}^{\delta })\) is well defined.
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.
Lemma 2.2
Let\(\mathcal{A}:I\rightarrow \mathcal{CF}^{\mathrm{sa}}(H)\)be a gapcontinuous path and\(M:I\rightarrow \operatorname{GL}(H)\)a continuous family of bounded invertible operators. Then\(\{M^{T}_{\lambda }\mathcal{A}_{ \lambda }M_{\lambda }\}_{\lambda \in I}\)is gapcontinuous and\(\operatorname {sf}(M^{T}\mathcal{A}M)=\operatorname {sf}(\mathcal{A})\).
Proof
2.3 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.
2.3.1 Continuity
To simplify notation, we set \(E=L^{2}(I,\mathbb{R}^{2n})\) and \(H=H^{1}(I,\mathbb{R}^{2n})\). The aim of this step is to prove the following proposition, which we will later apply in the cases \(X=I\) and \(X=I\times I\).
Proposition 2.3
Theorem 2.4
2.3.2 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.
Lemma 2.5
Proof
Let us first recall the wellknown fact that \(\varLambda _{0}(L_{0})\) is contractible, and hence pathconnected, for any \(L_{0}\in \varLambda (n)\) (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 Sect. 2.1, \(L'_{1}=e^{\varTheta J}L_{1}\) is transversal to \(L_{2}\) and \(L_{4}\) for any sufficiently small \(\varTheta >0\). In particular, we obtain a path connecting \((L_{1},L_{2})\) and \((L'_{1},L_{2})\) inside (17). Also, as \(\varLambda _{0}(L'_{1})\) is pathconnected, there is a path connecting \((L'_{1},L_{2})\) and \((L'_{1},L_{4})\) inside (17). Finally, there is a path from \((L'_{1},L_{4})\) to \((L_{3},L_{4})\) inside (17) as \(\varLambda _{0}(L_{4})\) is pathconnected. □
This step of the proof is based on the following proposition, in which we denote by \(\varOmega ^{2}\) the set of all admissible pairs of paths in \(\varLambda (n)\) (see (5)). Let us note that by (v′) in Sect. 2.1, \(\mu _{\mathrm{Mas}}(\gamma _{\mathrm{nor}},L_{1})=1\) and \(\mu _{\mathrm{Mas}}(L_{0},\gamma '_{\mathrm{nor}})=1\), where \(L_{0}=\mathbb{R}^{n} \times \{0\}\), \(L_{1}=\{0\}\times \mathbb{R}^{n}\) and \(\gamma _{\mathrm{nor}}\), \(\gamma '_{\mathrm{nor}}\) are the paths that we introduced at the end of Sect. 2.1.
Proposition 2.6
 (N)
\(\mu (\gamma _{\mathrm{nor}},L_{1})=1\)and\(\mu (L_{0},\gamma '_{\mathrm{nor}})=1\), where\(L_{0}=\mathbb{R}^{n}\times \{0 \}\)and\(L_{1}=\{0\}\times \mathbb{R}^{n}\).
Proof
Remark 2.7
We now define \(\mu :\varOmega ^{2}\rightarrow \mathbb{Z}\), \(\mu (\gamma _{1},\gamma _{2})=\operatorname {sf}(\mathcal{A})\), where \(\mathcal{A}\) is the path of differential operators (2) for the pair \((\gamma _{1}, \gamma _{2})\). We aim to use Proposition 2.6 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 Sect. 2.2 and the fact that \(\ker ( \mathcal{A}_{\lambda })=\gamma _{1}(\lambda )\cap \gamma _{2}(\lambda )\). Also, (ii′) follows from (iii) in Sect. 2.2. Finally, (ii′) is an immediate consequence of the homotopy invariance (i) of the spectral flow and Proposition 2.3.
Hence it remains to show that \(\mu (\gamma _{\mathrm{nor}},L_{1})=1\) and \(\mu (L_{0},\gamma '_{\mathrm{nor}})=1\), which will be a direct consequence of the following lemma.
Lemma 2.8
 (i)for \((\gamma _{1},\gamma _{2})=(\gamma _{\mathrm{nor}},L_{1})\)$$ \sigma (\mathcal{A}_{\lambda }) = \biggl\{ \pi \lambda \frac{\pi }{2}+ \pi k: k\in \mathbb{Z} \biggr\} \cup \biggl\{ \frac{\pi }{2}+k\pi : k \in \mathbb{Z} \biggr\} , $$
 (ii)for \((\gamma _{1},\gamma _{2})=(L_{0},\gamma '_{\mathrm{nor}})\)$$ \sigma (\mathcal{A}_{\lambda }) = \biggl\{ \pi \lambda +\frac{\pi }{2}+ \pi k: k\in \mathbb{Z} \biggr\} \cup \biggl\{ \frac{\pi }{2}+k\pi : k \in \mathbb{Z} \biggr\} . $$
Proof
We see from the previous lemma that in both cases there is only one eigenvalue of \(\mathcal{A}_{\lambda }\) 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 \(\operatorname {sf}( \mathcal{A})=1\) for \((\gamma _{1},\gamma _{2})=(\gamma _{\mathrm{nor}},L_{1})\) and \(\operatorname {sf}(\mathcal{A})=1\) for \((\gamma _{1},\gamma _{2})=(L_{0},\gamma '_{\mathrm{nor}})\). Hence Theorem 1.1 is shown in the admissible case.
Step 2: the general case
Let \((\gamma _{1},\gamma _{2})\) be a pair of paths in \(\varLambda (n)\) which is not necessarily admissible, and let \(\mathcal{A}\) be the path (2). Let \(\delta >0\) be as in Lemma 2.1 such that \(\operatorname {sf}(\mathcal{A})=\operatorname {sf}( \mathcal{A}^{\delta _{0}})\) for all \(0\leq \delta _{0}\leq \delta \).
3 A spectral flow formula for Hamiltonian systems
 (vi′)
 If \(\varPsi :I\rightarrow \operatorname {Sp}(2n,\mathbb{R})\) is a path of symplectic matrices, then$$\begin{aligned} \mu _{\mathrm{Mas}}(\varPsi \gamma _{1},\varPsi \gamma _{2})=\mu _{\mathrm{Mas}}(\gamma _{1},\gamma _{2}). \end{aligned}$$(20)
 (vii′)
 If \(\gamma '_{1},\gamma '_{2}:I\rightarrow \varLambda (n)\) denote the reverse paths defined by \(\gamma '_{1}(\lambda )=\gamma _{1}(1\lambda )\) and \(\gamma '_{2}(\lambda )=\gamma _{2}(1\lambda )\), then$$\begin{aligned} \mu _{\mathrm{Mas}} \bigl(\gamma '_{1}, \gamma '_{2} \bigr)=\mu _{\mathrm{Mas}}(\gamma _{1},\gamma _{2}). \end{aligned}$$(21)
 (viii′)

\(\mu _{\mathrm{Mas}}(\gamma _{1},\gamma _{2})=\mu _{\mathrm{Mas}}(\gamma _{3}, \gamma _{4})\) if \(\gamma _{1}\simeq \gamma _{3}\) and \(\gamma _{2}\simeq \gamma _{4}\) are homotopic by homotopies with fixed endpoints.
Theorem 3.1
Under the assumptions above, \(\mathcal{A}\)is a gapcontinuous path of selfadjoint Fredholm operators on\(L^{2}(I,\mathbb{R}^{2n})\)and\(\operatorname {sf}(\mathcal{A})=\mu _{\mathrm{Mas}}(\varPsi \gamma _{1},\gamma _{2})\).
Proof
Corollary 3.2
Proof
Corollary 3.3
Proof
4 Conclusion
We have given an elementary proof of a classical 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 firstorder operators. In our work, we have particularly paid attention to the continuity of the latter path of differential operators, as we felt that this point was somewhat neglected in the past. Finally, we have considered families of linear Hamiltonian systems and generalised a recent result of Hu and Portaluri that is based on Cappell, Lee and Miller’s theorem.
Notes
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Funding
M. Izydorek and J. Janczewska are supported by the grant BEETHOVEN2 of the National Science Centre, Poland, no. 2016/23/G/ST1/04081.
Competing interests
The authors declare that they have no competing interests.
References
 1.Arnold, V.I.: On a characteristic class entering into conditions of quantization. Funkc. Anal. Prilozh. 1, 1–14 (1967) MathSciNetCrossRefGoogle Scholar
 2.Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry III. Math. Proc. Camb. Philos. Soc. 79, 71–99 (1976) MathSciNetCrossRefGoogle Scholar
 3.BoossBavnbek, B., Lesch, M., Phillips, J.: Unbounded Fredholm operators and spectral flow. Can. J. Math. 57, 225–250 (2005) MathSciNetCrossRefGoogle Scholar
 4.Bott, R.: On the iteration of closed geodesics and the Sturm intersection theory. Commun. Pure Appl. Math. 9, 171–206 (1956) MathSciNetCrossRefGoogle Scholar
 5.Cappell, S.E., Lee, R., Miller, E.Y.: On the Maslov index. Commun. Pure Appl. Math. 47, 121–186 (1994) MathSciNetCrossRefGoogle Scholar
 6.Duistermaat, J.J.: On the Morse index in variational calculus. Adv. Math. 21, 173–195 (1976) MathSciNetCrossRefGoogle Scholar
 7.Fitzpatrick, P.M., Pejsachowicz, J., Recht, L.: Spectral flow and bifurcation of critical points of strongly indefinite functionals—part I: general theory. J. Funct. Anal. 162, 52–95 (1999) MathSciNetCrossRefGoogle Scholar
 8.Guillemin, V., Sternberg, S.: Geometric Asymptotics, Mathematical Surveys, vol. 14. Am. Math. Soc., Providence (1977) CrossRefGoogle Scholar
 9.Hu, X., Portaluri, A.: Index theory for heteroclinic orbits of Hamiltonian systems. Calc. Var. Partial Differ. Equ. 56, 167 (2017) MathSciNetCrossRefGoogle Scholar
 10.Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1995). Reprint of the 1980 edition, Classics in Mathematics CrossRefGoogle Scholar
 11.Lesch, M.: The uniqueness of the spectral flow on spaces of unbounded selfadjoint Fredholm operators. In: Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds. Contemp. Math., vol. 366, pp. 193–224. Am. Math. Soc., Providence (2005) CrossRefGoogle Scholar
 12.Musso, M., Pejsachowicz, J., Portaluri, A.: A Morse index theorem for perturbed geodesics on semiRiemannian manifolds. Topol. Methods Nonlinear Anal. 25, 69–99 (2005) MathSciNetCrossRefGoogle Scholar
 13.Nicolaescu, L.: On the space of Fredholm operators. An. Ştiinţ. Univ. ‘Al.I. Cuza’ Iaşi, Mat. 53, 209–227 (2007) MathSciNetzbMATHGoogle Scholar
 14.Pejsachowicz, J., Waterstraat, N.: Bifurcation of critical points for continuous families of \(C^{2}\)functionals of Fredholm type. J. Fixed Point Theory Appl. 13, 537–560 (2013) MathSciNetCrossRefGoogle Scholar
 15.Phillips, J.: Selfadjoint Fredholm operators and spectral flow. Can. Math. Bull. 39, 460–467 (1996) MathSciNetCrossRefGoogle Scholar
 16.Piccione, P., Tausk, D.V.: A Student’s Guide to Symplectic Spaces, Grassmannians and Maslov Index. IMPA Mathematical Publications, Rio de Janeiro (2008) zbMATHGoogle Scholar
 17.Robbin, J., Salamon, D.: The Maslov index for paths. Topology 32, 827–844 (1993) MathSciNetCrossRefGoogle Scholar
 18.Robbin, J., Salamon, D.: The spectral flow and the Maslov index. Bull. Lond. Math. Soc. 27, 1–33 (1995) MathSciNetCrossRefGoogle Scholar
 19.Salamon, D., Zehnder, E.: Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. Commun. Pure Appl. Math. 45, 1303–1360 (1992) MathSciNetCrossRefGoogle Scholar
 20.Wahl, C.: A new topology on the space of unbounded selfadjoint operators, Ktheory and spectral flow. In: \(C^{*}\)Algebras and Elliptic Theory II. Trends. Math., pp. 297–309. Birkhäuser, Basel (2008) CrossRefGoogle Scholar
 21.Waterstraat, N.: Spectral flow, crossing forms and homoclinics of Hamiltonian systems. Proc. Lond. Math. Soc. (3) 111, 275–304 (2015) MathSciNetCrossRefGoogle Scholar
 22.Waterstraat, N.: Fredholm operators and spectral flow. Rend. Semin. Mat. (Torino) 75, 7–51 (2017) MathSciNetGoogle Scholar
Copyright information
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.