Decomposition of spectral flow and Bott-type iteration formula

1.   Introduction

In this paper, we consider the decomposition of the spectral flow for a path of self-adjoint Fredholm operators. Let ${\cal H}$ be a separable Hilbert space, and we denote by $\cal{FS} ({\cal H})$ the set of all densely defined self-adjoint Fredholm operators on ${\cal H}$. We always equip $\cal{FS} ({\cal H})$ with the gap topology. For a continuous path $A(t)\in \mathcal{FS}({\cal H})$, $t\in[a, b]$, the spectral flow $sf(A(t); t\in[a, b])$ is an integer that counts the net number of eigenvalues that change sign. This notation is first introduced by Atiyah-Patodi-Singer [2] in their study of index theory on manifolds with boundaries. Since then it had found many significant applications, see [27,4] and references therein.

Some basic properties of spectral flow such as homotopy invariance, path additivity, direct sum e.t. are well known, please refer to the Appendix. We give the proof for another basic property which is called cogredient invariance property of spectral flow. For convenience, we first introduce some notations. Let ${\cal H}_1, {\cal H}_2$ be separable Hilbert space, we denote by $\cal{L}({\cal H}_1, {\cal H}_2)$ and $\cal{C}({\cal H}_1, {\cal H}_2)$ the set of bounded and closed operators from ${\cal H}_1\to {\cal H}_2$. Let $\cal{S}({\cal H})$ be the set of self-adjoint operators on ${\cal H}$. For convenience, we denote by $\cal{L}^*({\cal H}_1, {\cal H}_2), \cal{C}^*({\cal H}_1, {\cal H}_2), \cal{S}^*({\cal H}), \cal{FS}^* ({\cal H})$ the invertible subsets.

Lemma 1.1. Let $M_s\in C([a, b], \cal{L}^*({\cal H}_1, {\cal H}_2))$, $A_s\in C([a, b], \mathcal{FS}({\cal H}_2))$, then

and we have

Remark 1.2. The cogredient invariance property is a nature property. In the case ${\cal H}_1 = {\cal H}_2$, we can get it by the homotopy invariant property, but the general case is not so easy. In a preprint paper [13], Fitzpatrick-Stuar-Pejsachowicz proved (1.2) in the case that $M_s$ is constant, the domain of $A_s$ is fixed and both$A_a, A_b$ are invertible. Lemma 1.1 can be considered as a generalization of their result. For reader's convenience, we give the detail proof in Section $2$.

Our main result is the decomposition formula based on the cogredient invariance property. Let ${\cal H}_i$ be closed subspaces of ${\cal H}$ for $i = 1, \cdots, m$, then we define

which is the subspace spanned by ${\cal H}_i$, $i = 1, \cdots, m$. Suppose $g\in\cal{L}({\cal H})$, we call $g$ is a matrix-like operator if $\sigma(g) = \{\lambda_1, \cdots, \lambda_n\}$ is finite and there exist $m>0$, such that

We denote by $\mathcal{M}({\cal H})$ the set of matrix-like operators. For $g\in\cal{M}({\cal H})$, $\lambda\in \sigma(g)$, we set

and denote

Then we have

Moreover, let $\hat{F} = span\{F_\lambda, \lambda\in\sigma(g)\cap\mathbb{U}^c \}$, we have the next theorem.

Theorem 1.3. Let $A_s\in C([0, 1], \mathcal{FS}({\cal H}))$. Suppose $g\in \cal{M}({\cal H})$ is invertible and preserve the domain of $A_s$, $\sigma(g)\cap \mathbb{U} = \{\lambda_1, \cdots, \lambda_j \}$. Assume

then we have

In [18], by assuming $g$ is unitary and $\sigma(g)$ is finite, Hu-Sun proved the decomposition formula

under the condition

Obviously, we give a generalization of (1.7). In fact, a significant difference is that we do not assume $g$ is unitary in Theorem 1.3, hence the subspaces ${\cal H}_\lambda$ are not orthogonal. To overcome this difficulty, we develop a new technique (Lemma 2.2) to prove the equality of spectral flow.

The second main result is a generalization for the Bott-type iteration formula which is a powerful tool to study the multiplicity and stability of periodic orbits in Hamiltonian systems. In 1956, Bott got his celebrated iteration formula for the Morse index of closed geodesics [5], and it was generalized by [3,10,9,11]. The precise iteration formula of the general Hamiltonian system was established by Long [22,23]. In fact, the iteration could be regarded as a unitary group action. Motivated by the symmetry orbits in $n$-body problem [12], Hu-Sun [18] use this opinion to give generalization of Bott-type iteration formula to the system under a circle-type symmetry or reversible symmetry group action, and prove the stability of Figure Eight orbit [8]. The case of the reversible symmetry was deeply studied in [24,20,21,16].

Based on Theorem 1.3, we prove the Bott-type iteration formula which cover all the previous cases and moreover give some new generalizations. Our generalized formula could be applied to the closed geodesics on Semi-Riemanian manifold and heteroclinic orbits with reversible symmetry.

Now we consider the linear Hamiltonian system

where $J = \begin{pmatrix} 0 & -I_n\\ I_n & 0 \end{pmatrix}$, $\mathcal{I}\subset \mathbb{R}$ is a connected subinterval, $B(t)\in C(\mathcal{I}, \mathcal{S}(\mathbb{R}^{2n}))$. In the case $\mathcal{I}$ is finite, the boundary conditions are given by the Lagrangian subspaces. Let $(\mathbb{C}^{2n}, \omega)$ be the standard symplectic space with $\omega(x, y) = (Jx, y)$. A Lagrangian subspace $V$ is a $n$-dimensional subspace with $\omega|_V = 0$. We denote the set of Lagrangian subspace by $Lag(2n)$. It is obvious $(\mathbb{C}^{2n}\oplus\mathbb{C}^{2n}, -\omega\oplus\omega)$ is a $4n$-dimensional symplectic space, then for $\mathcal{I} = [a, b]$, the boundary condition is given by

In the case $\mathcal{I} = \mathbb{R}$, we always assume $B(\pm\infty) = \lim_{t\to\pm\infty}B(t)$ exist and $JB(\pm\infty)$ are both hyperbolic, that is

Let ${\cal H} = L^2(\mathcal{I}, \mathbb{C}^{2n})$ and $E$ is $W^{1, 2}(\mathcal{I}, \mathbb{C}^{2n})$ which satisfied some boundary conditions. Let

and $B:{\cal H}\to {\cal H}$ be the multiplicity operator of $B(t)$. Let $A_s = A-sB$ for $s\in{\mathbb R}$, then $A_s\in\mathcal{FS}({\cal H})$. For $g\in {\cal M}({\cal H})$ which satisfies

we have $g^*A_sg = A_s$. By constructing $g$, we get the spectral flow decomposition of $sf(A_s)$. We list $6$-cases which are common in applications of Hamiltonian systems. Our results generalize all the previous results, especially for the reversible symmetry of heteroclinic orbits (Case $5$ and $6$). Our result is new. Please see Section 4 for the detail.

It is well known that spectral flow is equal to Maslov index, and this is also true for the unbounded domain, see [6,27,26,7,15] and reference therein. The Maslov index is associated integer to a pair of continuous path $f(t) = (L_1(t), L_2(t))$, $t\in\mathcal{I}$, in $Lag(2n)\times Lag(2n)$ [6]. From the decomposition of spectral flow, we get the decomposition of Maslov index. Please refer to Section 5 for the detail. For reader's convenience, we give a brief review for the Maslov index and spectral flow in the Appendix.

This paper is organized as follows. We prove Lemma 1.1 in Section 2 and Theorem 1.3 in Section 3. In Section 4, we list $6$-cases of decompositions in Hamiltonian systems. In Section 5, we give some cases of the Bott-type iteration formulas. At last, we briefly review the basic properties of spectral flow and Maslov index in the Section 6.

2.   Spectral flow is preserved under cogredient

Let $V$ be a closed subspace of ${\cal H}$, and $P_V$ be the orthogonal projection from ${\cal H}$ to $V$. For $A\in\cal{C}({\cal H})$, we denote the operator $P_VAP_V:V\to V$ by $A_V$. Obviously, if $A\in \cal{S}({\cal H})$ then $A_V\in \cal{S}(V)$.

Definition 2.1. Let $A:[a, b]\rightarrow \cal{FS} ({\cal H})$ be a continuous curve. We call $A(t)$ is a positive curve if $\{t, \ker A(t)\neq0\}$ is a distinct set and

Let $A\in \cal{FS}({\cal H})$ and $B\in \cal{L}({\cal H})\cap \cal{S}({\cal H})$, then $A+tB\in \cal{FS}({\cal H})$ with $t\in {\mathbb R}$. Note that it is positive if $B|_{\ker(A+tB)>0}$ for any $t\in \{t|\ker(A+tB)\neq 0\}$. For example, $A+t I$ is a positive curve with $A\in \cal{FS}(H)$ for $t\in{\mathbb R}$.

Let $S\subset\cal{FS}({\cal H})$ be a path connected subset. We assume there exists $K\in\cal{L}({\cal H})$ such that $(Kx, x)>0, \, \forall x\in {\cal H}$ and for any $A\in S$, there is a neighborhood $U$ of $A$, and $\epsilon>0$ such that $B+tK\in S, \, t\in [0, \epsilon]$ for each $B\in U$. Then $A+tK, \, t\in [0, \epsilon]$ is a positive curve in $\cal{FS}({\cal H})$. Let $\{{\cal H}_k\}, 1\leq k\leq n$ be a family of Hilbert spaces and $f_k: S\rightarrow \cal{FS}({\cal H}_k)$ be a family of continuous maps.

We assume that

(a) For any $A\in S$, $f_k(A+tK), t\in [0, \epsilon]$ is a positive path in $\cal{FS}({\cal H}_k)$.

(b) For any $A\in S$, $\sum_{1\leq k\leq n}{{\rm{dim ~ }}}\ker f_k(A) = {{\rm{dim ~ }}}\ker A$.

Then we have the following lemma.

Lemma 2.2. Let $A \in C([0, 1], \mathcal{FS}({\cal H}))$ and satisfies condition (a) and (b), we have

Proof. Since the spectral flow satisfies the path additivity property, we only need to prove (2.2) locally. Let $h_k(s, t) = f_k(A(t)+s K)$, $t\in [0, 1], s\in [0, \epsilon]$, then for any $t\in [0, 1]$, $h_k(s, t)$ is a positive curve with $1\leq k\leq n$. Let $t_0\in [0, 1]$, since $(Kx, x)>0$ for $x\in \ker A(t_0)$, there is $\delta>0$ such that

It follows that ${{\rm{dim ~ }}}\ker (h_k(\delta, t_0)) = 0$ for $1\leq k\leq n$. Note that $A(t_0)+\delta K$ is a Fredholm operator, so there is $\delta_1>0$ such that

It follows that ${{\rm{dim ~ }}}\ker (h_k(\delta, t)) = 0$ for $t\in[t_0-\delta_1, t_0+\delta_1]$, $1\leq k\leq n$. Then we have

By homotopy invariance of spectral flow, we have

and

Note that $A(t_0\pm\delta_1)+sK$, $h_k(s, t_0\pm\delta_1), 1\leq k\leq n$ are positive paths. It follows that

This completes the proof.

Please note that Lemma 2.2 can be considered as a generalization of direct sum property of spectral flow.

In the next, we will prove that the spectral flow is invariant under the cogredient. The next Lemma is contained in [13], but for reader's convenience, we give details here.

Lemma 2.3. Let $E$ be the domain of $A\in\cal{FS}({\cal H}_2)$. If $M\in \cal{L}({\cal H}_1, {\cal H}_2)$ is invertible, then $M^*AM\in \cal{FS}({\cal H}_1)$ with domain $M^{-1}(E)$.

Proof. Since $A\in\cal{FS}({\cal H}_2)$ with domain $E$, we have ${{\rm{dim ~ }}} \ker A, \, {{\rm{dim ~ }}} ({\cal H}_2/{\rm im}~A) <+\infty$. Since $M$ is invertible, we have $\ker(M^*AM) = \ker(AM) = M^{-1}\ker A$. Then $M^{-1}$ induce an isomorphism from $\ker A$ to $\ker (M^*AM)$. Note that ${\rm im}(M^*A M) = M^*{\rm im}(A)$. Then $M^*$ induce an isomorphism from ${\cal H}_2/{\rm im}(A)$ to ${\cal H}_1/{\rm im}(M^*AM)$. So $M^*AM$ is a Fredholm operator.

Since $A \in \cal{FS}({\cal H}_2)$ with domain $E$, we see that for each $x\in M^{-1}$, $(AMx, My) = (Mx, AMy)$ if and only if $y\in M^{-1}E$. It follows that $(M^*AM)^* = M^*AM$ with domain $M^{-1}E$. Then we can conclude that $M^*AM\in \cal{FS}({\cal H}_1)$.

Recall that the gap topology can be induced by the gap distance $\hat{\delta}$. Let $X$ be a Banach space. Let $M, N$ be two closed linear subspaces of $X$. Denote by $S_M$ the unit sphere of $M$. Then gap distance is defined as

where

The gap distance has the following properties:

Lemma 2.4. Let $X, Y$ be two Hilbert spaces. Let $M, N$ be two closed linear subspaces of $X$. Let $P, Q\in \cal{L}^*(X, Y)$. Then $\hat\delta(PM, QN)\le \hat\delta(M, N)\max\{\|P\|, \|Q\|\}+\|P-Q\|\max\{\|P^{-1}\|, \|Q^{-1}\|\}$.

Proof. Without loss of generality, we assume that $M, N\neq \{0\}$, and let $d_1 = \hat\delta(M, N)$. Let $x\in PM$ with $\|x\| = 1$, we choose $y\in N$ such that $\|P^{-1}x-y\| = {\rm dist}(P^{-1}x, N)$, then we have $\|y\|\leq \|P^{-1}x\|\leq \|P^{-1}\|$. Note that

It follows that $\delta(PM, QN)\leq \|P\|\delta(M, N)+\|Q-P\|\|P^{-1}\|$. Similarly, we have $\delta(QN, PM)\leq \|Q\|\delta(N, M)+\|Q-P\|\|Q^{-1}\|$. This conclude the proof.

Lemma 2.5. Suppose $M_s\in C([0, 1], \cal{L}^*({\cal H}_1, {\cal H}_2))$, $A_s\in C([0, 1], \cal{FS}({\cal H}_2))$, then $M_s^*A_sM_s\in C([0, 1], \cal{FS}({\cal H}_1))$.

Proof. We only need to show that $M_s^*A_sM_s$ is a continuous curve with the gap topology. Let $E_s$ be the domain of $A_s$. Note that

Let $Q_s:{\cal H}_2\oplus {\cal H}_2 \to {\cal H}_1\oplus {\cal H}_1$ be $\begin{pmatrix} M_s^* & 0\\ 0& M_s^{-1}\\ \end{pmatrix}$, then $Q_s \in C([0, 1], \cal{L}^*({\cal H}_2\oplus {\cal H}_2, {\cal H}_1\oplus{\cal H}_1))$, and we also have ${\rm Gr}(M_s^*A_sM_s) = Q_s{\rm Gr}(A_s)$. Since $\|Q_s\|$ and $\|Q_s^{-1}\|$ are continuous functions on $[0, 1]$, we have $\|Q_s\|>0$, $\|Q_s^{-1}\|>0$ for $s\in [0, 1]$. Let $C_1 = \sup(\|Q_s\|)$, $C_2 = \sup(\|Q_s^{-1}\|)$. For $s_0, s\in [0, 1]$, by Lemma 2.4, we have

By the continuity of $A_s$ and $Q_s$, we see that for any $\epsilon >0$ there is $\delta_1>0$ such that for any $s\in (s_0-\epsilon, s_0-\epsilon)$, we have $\hat\delta({\rm Gr}(A_{s_0}), {\rm Gr}(A_s))<\epsilon/(2C_1)$ and $\|Q_s-Q_{s_0}\|<\epsilon/(2C_2)$. Then we have $\hat\delta({\rm Gr}(M_{s_0}^*A_{s_0}M_{s_0}, M_{s}^*A_{s}M_{s})<\epsilon.$ This completes the proof.

Now we give the proof of Lemma 1.1.

Proof of Lemma 1.1. Please note that (1.1) is from Lemma 2.5. We first prove the case $M_s\equiv M$. Let $S = \cal{FS}({\cal H}_2)$, $K = I$, $f(A) = M^*AM$. Please note that

for each $A\in \cal{FS}({\cal H}_2)$. Furthermore, we have $\frac{d}{dt} M^*(A+tI)M| = M^*M>0$, so $M^*(A+tI)M$ is a positive curve. Then by Lemma 2.2, we have

Now we consider the two family $M_{a+t(s-a)}^*A_sM_{a+t(s-a)}$, $(t, s)\in [0, 1]\times[a, b]$. By the homotopy invariance property of spectral flow, we have

Note that ${{\rm{dim ~ }}} \ker M_{a+t(b-a)}^*A_bM_{a+t(b-a)}$ is a constant which implies

It follows that

This completes the proof.

As an example, we consider the one parameter family of linear Hamiltonian systems

where $B(t)\in C([0, 1]\times[0, T], \cal{S}({\mathbb R}^{2n}))$. The boundary condition is given by

where we assume $\Lambda_s$ is continuous depend on $s$.

Let $A_s = -J{\dfrac{d}{dt}}|_{E(\Lambda_s)}$. It is a path of self adjoint operators on ${\cal H}: = L^2([0, T], \mathbb C^{2n})$ with domain

We define $B_s$ by $(B_sx)(t) = B_s(t)x(t)$. It is well known that $A_s, A_s-B_s\in \cal{FS}({\cal H})$ with domain $E_s$. Let ${\gamma}_s(t)$ be the fundamental solution of (2.6), i.e.

then

which implies $Gr({\gamma}_s(T))\in Lag(4n )$. The following formula which gives the relation of spectral flow and Maslov index (please refer to Theorem 6.1)

Let $P_s(t)\in C^1([0, 1]\times[0, T], {\mathrm {Sp}}(2n))$, then $P_s\in C^1([0, 1], \cal{L}^*({\cal H}))$, hence

with domain

where $\hat{P}_s(t) = diag(I_n, P_s(t))$. Direct calculation shows that

where $\hat{B}_s(t) = -J\dot{P}_s(t)P^{-1}_s(t)+(P^{*}_s(t))^{-1}B(t)P^{-1}_s(t)$. From Lemma 1.1, we have

From (6.6), we can express the left of (2.8) as Maslov index. In fact, the fundamental solution is $P_s(t){\gamma}_s(t)$, and the boundary conditions is given by $(\hat{P}_s(T)\Lambda_s$. Hence we have

Formula (2.8) implies that

which is just the symplectic invariance property (6.4) of Maslov index.

3.   Decomposition of spectral flow under cogredient invariant

In this section, we will prove the decomposition formula for spectral flow. Suppose $g\in\cal{M}({\cal H})$ with $\sigma(g) = \{\lambda_1, \cdots, \lambda_n\}$, then

where ${\cal H}_{\lambda_i}: = \ker(g-\lambda_i)^m$ for large enough $m$. Note that $(\lambda-\lambda_1)^m$ and $(\lambda-\lambda_2)^m$ are coprime, then there are polynomials $p_1, p_2$ such that $p_1(\lambda)(\lambda-\lambda_1)^m+p_2(\lambda)(\lambda-\lambda_2)^m = 1$. For each $x\in {\cal H}_{\lambda_1}\cap {\cal H}_{\lambda_2}$, we have

Similarly we have ${\cal H}_{\lambda_i}\cap {\cal H}_{\lambda_j} = 0$ with $i\neq j$. So the decomposition (3.1) is a inner direct sum.

Lemma 3.1. $g\in\mathcal{M}({\cal H})$ if and only if there exist $\lambda_1, \cdots, \lambda_n\in\mathbb{C}$ such that $\Pi_{i = 1}^n (g-\lambda_i)^m = 0$.

Proof. We only need to show that $g\in\mathcal{M}({\cal H})$ if $\Pi_{i = 1}^n (g-\lambda_i)^m = 0$. Let $G_l(\lambda)$ be the polynomial $\Pi_{i = 1}^{l-1} (\lambda-\lambda_i)^m\Pi_{i = l+1}^n(\lambda-\lambda_i)^m$. Then $G_1, G_2, \cdots, G_n$ are coprime polynomials. It follows that there are polynomials $a_i(\lambda), (1\le i\le n)$, such that

It follows that $\sum_{i = 1}^n a_i(g)G_i(g) = {\rm Id}$. Then we can conclude that

We also have $(g-\lambda_i)^mG_i(g){\cal H} = \Pi_{i = 1}^n (g-\lambda_i)^m{\cal H} = 0$, which implies (3.1).

We have the following lemmas.

Lemma 3.2. Let $A\in\cal{FS}({\cal H})$ with domain $E$. Suppose $g\in\cal{M}({\cal H})$, which satisfied

then ${\cal H}_\lambda, {\cal H}_\mu$ are $A$-orthogonal if $\lambda\bar{\mu}\neq1$, i.e.

Proof. Let $x\in \ker(g-\lambda)^m \cap E$, $y\in \ker(g-\mu)^n\cap E$ with $m, n\ge 1$. We see that $(Ax, y) = 0$ if $m+n = 2$. In fact,

implies $(Ax, y) = 0$ since $\lambda\bar\mu\neq1$. Assume that $(Ax, y) = 0$ if $m+n\le k$. Note that $(g-\lambda)x\in \ker(g-\lambda)^{m-1}\cap E$, $(g-\mu)x\in \ker(g-\mu)^{n-1}\cap E$, $gx\in \ker(g-\lambda)^{m}\cap E$ and $gy\in \ker (g-\mu)^m\cap E$. If $m+n = k+1$, We have

Since $\lambda\bar\mu\neq 1$, we have $(Ax, y) = 0$. By induction, we have $(Ax, y) = 0$ with $x\in {\cal H}_{\lambda}\cap E$ and $y\in {\cal H}_{\mu}\cap E$. This complete the proof.

Lemma 3.3. Under the condition of Lemma 3.2, we have $\ker A = \sum_{1\le i\le n}\ker A\cap {\cal H}_i$ and $E = \sum_{1\le i\le n}E\cap {\cal H}_{i}$.

Proof. Note that $E$ is a invariant subspace of $g$. Then $\Pi_{1\le i\le n}(g-\lambda_i)^m = 0$ on $E$. It follows that $E = \sum_{1\le i\le n}\ker (g|_E-\lambda_i)^m = \sum_{1\le i\le n}E\cap {\cal H}_{i}$. We have

It follows that $g(\ker A)\subset \ker A$. So $\ker A$ is a invariant subspace of $g$. Similarly, we have $\ker A = \sum_{1\le i\le n}\ker A\cap {\cal H}_i$. This complete the proof.

For $A\in\cal{C}({\cal H})$, assume that ${\cal H} = \sum_{1\le i\le k}{\cal H}_i$, where all of ${\cal H}_i$ are closed subspaces of ${\cal H}$. Let $E$ be the domain of $A$ and assume that $E = \sum_{1\le i\le k} E\cap {\cal H}_i$. ${\cal H}_i$, ${\cal H}_j$ are $A$-orthogonal if $i\neq j$. Recall that we set

then we have ${\cal H} = \sum_{1\leq i\leq k}F_{\lambda_i}$ and $F_{\lambda_i}, F_{\lambda_j}$ are $A$-orthogonal if $i\neq j$.

Let $X = \bigoplus_{1\le i \le k} F_{\lambda_i}$, we define an inner product on $X$:

where $(x_i, y_i)$ is the inner product in ${\cal H}$. Then $X$ is a Hilbert space and the map

is a homeomorphism from $X$ to ${\cal H}$.

Please note that $A|_{F_{\lambda_i}}$ is the map $M^*AM:M^{-1}F_{\lambda_i}\to M^{-1}( F_{\lambda_i})$. It is a self-adjoint Fredholm operator on $M^{-1}F_{\lambda_i}$ with domain $M^{-1}(E\cap F_{\lambda_i})$. It follows that

Proposition 3.4. Suppose $g\in \cal{M}^*({\cal H})$, $A_s\in C([0, 1], \cal{FS}({\cal H}))$ with fixed domain $E$ and $gE = E$. We assume $g^* A_sg = A_s$ of $s\in[0, 1]$, then we have

Proof. By Lemma 1.1, we have $M^*A_sM\in C([0, 1], \cal{FS}(X))$, and

Note that $X = \bigoplus_{1\le i\le k} F_{\lambda_i}$ is an orthogonal decomposition. By the Direct sum property of spectral flow, we have

This complete the proof.

Lemma 3.5. If $\lambda\notin\mathbb{U}$ then we have

Proof. Recall that $A_s|_{F_\lambda}$ is the operator $M^*A_sM: M^{-1}(F_\lambda)\to M^{-1}(F_{\lambda})$ and $M^{-1}(F_{\lambda}) = M^{-1}{\cal H}_\lambda+ M^{-1}{\cal H}_{\bar\lambda^{-1}}$. We also have $M^{-1}{\cal H}_\lambda\perp M^{-1}{\cal H}_{\bar\lambda^{-1}}$. Let $Q$ be the map $x+y\to -x+y$ with $x\in M^{-1}{\cal H}_\lambda, y\in M^{-1}{\cal H}_{\bar\lambda^{-1}}$. Then $Q$ is invertible and $Q^* = Q$. Let $x_1, x_2\in M^{-1}(F_{\lambda}\cap E)$, $y_1, y_2\in M^{-1}(F_{\bar\lambda^{-1}}\cap E)$. We have

It follows that $-A_s|_{F_\lambda} = Q(A_s|_{F_\lambda})Q$. Then by Lemma 1.1, we have

The lemma then follows.

Proof of Theorem 1.3. By Proposition 3.4 and Lemma 3.5, we only need to show that

In fact $\ker A_1|_{\hat{F}} = \ker A_1\cap \hat{F}$. By Lemma 3.3, we see that

It follows that $dim\ker (A_1|_{\hat{F}}) = \sum_{\lambda\notin \mathbb{U}}dim \ker(A_1|_{F_{\lambda}})$. It is also true for $A_0$. The theorem then follows.

Corollary 3.6. Under the condition of Theorem 1.3, if $\sigma(M)\cap \mathbb{U} = \emptyset$, then

If the path is closed, then

Remark 3.7. In the case $B$ is compact with respect to $A$, the spectral flow $A-sB$ is only depend on the end points, thus we define the relative Morse index by (follows [27])

Especially, when $A$ is positive, then $I(A, A-B) = m^-(A-B)$ is just the Morse index of $A-B$, i.e. the total number of negative eigenvalues. It is obvious that Theorem 1.3 and Corollary 3.6 give the decomposition formula of relative Morse index and Morse index.

4.   Applications to Hamiltonian systems

In this section, we will give the applications for Hamiltonian systems. We list $6$ cases which are common in applications.

For $\Lambda\in Lag(4n)$, we consider the solution of the flowing linear Hamiltonian systems

where $B(t)\in C([0, T], \cal{S}({\mathbb R}^{2n}))$. Recall that $A = -J{\dfrac{d}{dt}}$ is self adjoint operator on ${\cal H}: = L^2([0, T], \mathbb C^{2n})$ with domain

then $A, A-B\in \cal{FS}({\cal H})$. We will construct $g\in\mathcal{M}({\cal H})$ such that

In order to make $g E_\Lambda = E_\Lambda$, $g$ is always assumed to preserve the boundary condition, that is $g\Lambda = \Lambda$ which means

Hence we have

and get the decomposition formula (1.7).

It is well known that for $P\in{\mathrm {Sp}}(2n)$, if $\lambda\in\sigma(P)$, then $\bar{\lambda}, \lambda^{-1}, \bar{\lambda}^{-1}\in\sigma(P)$ and possess the same geometric and algebraic multiplicities [23]. Case 1 is given by symplectic matrix.

Case 1. For $P\in{\mathrm {Sp}}(2n)$, and satisfied $P\Lambda = \Lambda$ which means if $(x(0), x(T))\in\Lambda$, then $(Px(0), Px(T))\in\Lambda$. Let

then it is obvious that $(g^*x)(t) = P^*x(t)$, $g^*Ag = A$, $g\Lambda = \Lambda$. Moreover, we assume $P^*B(t)P = B(t)$, then $g^*Bg = g$, hence we have (4.2). It is obvious that $g\in\mathcal{M}({\cal H})$ and

Let $V_\lambda = \ker(P-\lambda)^{2n}$, then ${\cal H}_\lambda = L^2([0, T], V_\lambda)$.

Case 2. For $S\in{\mathrm {Sp}}(2n)$, we consider the $S$-periodic solution of (4.1), that is

and moreover we assume

We assume (4.1) with $S$-periodic boundary conditions admits a $\mathbb Z_k$ symmetry. More precisely, let $P\in{\mathrm {Sp}}(2n)$ and $PS = SP$, the group generator $g$ is defined by

Easy computation show that $g\in \cal{L}({{\cal H}})$ and $g E = E$. By direct computation, we get the adjoint operator $g^*$.

Lemma 4.1. The adjoint operator $g^*$ is given by

Proof. Let $y\in L^2([0, T], \mathbb{C}^{2n})$. We see that

and

Then we have checked $\langle gx, y\rangle_{L^2} = \langle x, g^*y\rangle_{L^2}$ for each $x, y\in L^2([0, T], \mathbb{C}^{2n})$. We assume $B(t)$ satisfied

Please note that (4.8) implies (4.5), and (4.2) is satisfied. Since

which is a multiplicity operator on ${{\cal H}}$. Then

To simplify the notation, for $\Omega\in\mathbb C$, we define

By this notation, we have $\sigma(g)\in(\sigma(S^{-1}P^k))^{\frac{1}{k}}$. For $\lambda\in\sigma(g)$, ${{\cal H}}_\lambda = \ker(g-\lambda)^{2n}$.

Case 3. We consider the generalized reversible symmetry. We call a matrix $M$ anti-symplectic if it satisfied

We denote by ${{\mathrm {Sp}}}_a(2n)$ the set of anti-symplectic matrices. For $M_1, M_2\in {{\mathrm {Sp}}}_a(2n)$ and $M_3\in{{\mathrm {Sp}}}(2n)$, then it is obvious that

We list some basic property of ${{\mathrm {Sp}}}_a(2n)$ follows.

Lemma 4.2. If $M\in{{\mathrm {Sp}}}_a(2n)$, $\lambda\in\sigma(M)$, then $\bar{\lambda}, -\lambda^{-1}, -\bar{\lambda}^{-1}\in\sigma(M)$ and possess the same geometric and algebraic multiplicities.

Proof. Note that $M^* = -JM^{-1}J^{-1}$. Let $\lambda\in \mathbb{C}\backslash\{0\}$. It follows that $(M^*-\lambda) = -J(M^{-1}+\lambda)J^{-1}$. Then we have

And we also have

It follows that ${{{\rm{dim }}}}\ker(M-\bar\lambda)^{2n} = {{{\rm{dim ~ }}}}\ker(M+\lambda^{-1})^{2n}$. So $\lambda, -\lambda^{-1}\in \sigma(M)$ and posses the same geometric and algebraic multiplicities. Specially, if $M$ is a real matrix, $\lambda, \bar\lambda, -\lambda^{-1}, -\bar\lambda^{-1}\in \sigma(M)$ and posses the same geometric and algebraic multiplicities.

Similar with the symplectic matrix, we have the following results.

Lemma 4.3. Let $M\in {{\mathrm {Sp}}}_a(2n)$. Let $\lambda, \mu \in \sigma(M)$. Let $V_\lambda = \ker(M-\lambda)^{2n}$, $V_\mu = \ker(M-\mu)^{2n}$. Then we have $(Jx, y) = 0$ if $\lambda\bar{\mu}\neq-1$.

Proof. Let $x\in \ker(M-\lambda)^p$, $y\in \ker(g-\mu)^q$ with $p, q\ge 0$. We see that $(Jx, y) = 0$ if $p+q = 0$. Assume that $(Jx, y) = 0$ if $p+q\le k$. Note that $(M-\lambda)x\in \ker(M-\lambda)^{p-1}$, $(M-\mu)x\in \ker(M-\mu)^{q-1}$. If $p+q = k+1$, We have

Since $\lambda\bar\mu\neq - 1$, we have $(Ax, y) = 0$. By induction, we have $(Jx, y) = 0$ with $x\in \ker(M-\lambda)^{2n}$ and $y\in \ker(M-\mu)^{2n}$. This completes the result.

We assume (4.1) admits a $N$-reversible symmetry. More exactly, for $N\in{{\mathrm {Sp}}}_a(2n)$, let

We assume $g\Lambda = \Lambda$, that is

then $gE = E$. Obviously, $(g^*x)(t) = N^*x(T-t)$. We assume

then (4.2) is satisfied.

Please note that for the $S$-periodic boundary conditions, $NS^{-1} = SN$ implies $g\Lambda = \Lambda$. Separated boundary conditions is another kind of important boundary conditions. More precisely, we consider solution of (4.1) under the boundary conditions

where $V_0, V_1\in Lag(2n)$. In this case $g$ is defined by (4.10), for $N\in{{\mathrm {Sp}}}_a(2n)$ which satisfied

then $g\Lambda = \Lambda$.

Obviously, we have

hence $g\in\mathcal{M}({{\cal H}})$ and

For $\lambda\in\sigma(g)$, ${{\cal H}}_\lambda = \ker(g-\lambda)^{2n}$.

From theorem 1.3, we get the decomposition of spectral flow. Since on the finite interval $B$ is relative compact with respect to $A$, then from Remark 3.7, we have

All the above discussions can be applied to Sturm-Liouville systems, so we don't give the details in all cases, instead we only consider the following two cases which have clear background.

Case 4. We consider the one parameter family Sturm-Liouville system

where $S\in\cal{L}^*({{\mathbb R}}^n)$. We suppose $G_s(t), R_s(t)\in \cal{S}(n)$, instead of the Legendre convex condition we only assume $G_s(t)$ is invertible. Let $P\in\cal{L}^*({{\mathbb R}}^n)$ and $PS = SP$, the group generator $g$ is defined as same form of (4.6). We assume

Then

and we could give the decomposition of spectral flow from Theorem 1.3.

This case includes the Bott-type formula of Semi-Riemann manifold [17]. Let $c$ be a space-like or time-like closed geodesic on $n+1$ dimension Semi-Riemann manifold $(M, \mathfrak{g})$ with period $T$. We choose a parallel $\mathfrak{g}$-orthonormal frame $e_i(t)$ alone $c$, and satisfied $\mathfrak{g}(e_i(t), \dot{c}(t)) = 0$. Assume

and

then $P^TGP = G$ with $G = diag(I_{n-\nu}, -I_\nu)$.

Writing the $\dot{c}$ $\mathfrak{g}$-orthogonal Jacobi vector field alone $c$ as $J(t) = \sum_{i = 1}^nu_i(t)e_i(t)$, then we get the linear second order system of ordinary differential equations

where $R$ is symmetry matrices which is get by the curvature. A period solution is satisfied

For $\omega\in\mathbb{U}$, let

then

are self-adjoint Fredholm operators on $L^2([0, T], \mathbb{C}^n)$ with domain $E^2_{\omega, T}$.

It has proved in [17] that there exist $s_0$ sufficiently large such that for $s\geq s_0$, $A^\omega_s$ is non-degenerate. The $\omega$ spectral index of $c$ is defined by

Let $c^{(m)}$ be the $m$-th iteration of $c$, then

Let $S = P^m$, $G_s = G$, $R_s = R(t)+sG$, $(gu)(t) = Pu(t+T)$, then from Case 4. we get the decomposition of spectral flow. Since $g^m = \omega$, then

Let $\omega_j$ be the $m$-th root of $\omega$, then

We have

Hence we get the Bott-type iteration formula [17]

Obviously, we can consider the case of reversible symmetry, since it is similar, we omit the detail.

Case 5. Now we consider the case of heteroclinic orbits, for the one parameter family linear Hamiltonian system

Let $B_s(\pm\infty) = \lim_{t\to\pm\infty}B_s(t)$ exist and satisfied the hyperbolic condition, i.e.

Let ${{\cal H}} = L^2({{\mathbb R}}, \mathbb C^{2n})$, it is well known that $A-B_s \in\cal{FS}({{\cal H}})$ with domain $E = W^{1, 2}({{\mathbb R}}, \mathbb{C}^{2n})$.

We assume

then $g^*B_s g = g$. Easy computation show that $g^*Ag = A$, then we have

Obviously, we have

hence $g\in\mathcal{M}({{\cal H}})$ and then

For $\lambda\in\sigma(g)$, ${{\cal H}}_\lambda = \ker(g-\lambda)^{2n}$, then we get the decomposition formula from Theorem 1.3.

In the case $N^2 = I$, let

we have

Now we consider the case of homoclinic orbits. For the linear Hamiltonian system

assume $\lim_{t\to\pm\infty}B(t) = B_*$ and $JB_*$ is hyperbolic. In this case, $B-B_*$ is relative compact with respect to $A-B_*$, where $A = -J\frac{d}{dt}$. The relative index is defined by

In the case that (4.22) is a linear system of homoclinic orbits $z$, the index of $z$ is defined by [7]

Assume $N^*B(-t)N = B(t)$ and $N^2 = I$, from (4.21), we have

Case 6. We consider the one parameter Sturm-Liouville system on ${{\mathbb R}}$

where $G(t), R(t)\in \cal{S}(n)$. We assume there exist $\delta>0$, such that $G(t)>\delta$ for $t\in{{\mathbb R}}$, and there exist $T, \delta_1, \delta_2>0$ such that

The Morse index of $\mathcal{A}: = -(G(t)\frac{d}{dt})'+R(t)$ is defined by the maximum dimension of the subspace such that $\mathcal{A}$ restricted on it is negative definite. It is well known that $m^-(\mathcal{A})$ is finite under the condition (4.25). Obviously

We assume there exist $N\in{{\mathrm {Sp}}}_a(n)$, such that

Let

then $g^*(\mathcal{A}+sG(0))g = \mathcal{A}+sG(0)$. Since $g^2 = N$, then $\sigma(g) = (\sigma(N))^{\frac{1}{2}}$, and we get the decomposition formula of Morse index.

In the case $N^2 = I$, we have

5.   Relation with the Maslov index

In this section, we will give some Bott-type iteration formulas of Maslov-index. In what follows $g$ is pointed as the Matrix-like operator which appears in cases 2, 3, 5. To avoid discussing too many technique details, we only consider the case

for some $\omega\in\mathbb{U}$. Let $\omega_1, \cdots, \omega_m$ be the $m$-th roots of $\omega$, and let ${{\cal H}}_i = \ker(g-\omega_i)$.

In cases 2, 3, 5, ${{\cal H}}(\mathcal{I}) = L^2(\mathcal{I}, \mathbb{C}^{2n})$ where $\mathcal{I}$ is some finite interval or $\mathbb{R}$, and $E$ is $W^{1, 2}(\mathcal{I}, \mathbb{C}^{2n})$ which satisfies some boundary conditions. We choose a subinterval $\hat{\mathcal{I}}\subset \mathcal{I}$, and let $\mathcal{T}$ be the restriction map from ${{\cal H}}$ to ${{\cal H}}(\hat{\mathcal{I}}): = L^2(\hat{\mathcal{I}}, \mathbb{C}^{2n})$, that is

$\hat{\mathcal{I}}$ is called a fundamental domain if for any $i = 1, \cdots, m$, $\mathcal{T}$ is a bijection from ${{\cal H}}_i$ to $L^2(\hat{\mathcal{I}}, \mathbb{C}^{2n})$. Recall that $E_i = E\cap{{\cal H}}_i$ is domain of $A_s|_{{{\cal H}}_i}$, then $\mathcal{T}E_i$ is closed in the $W^{1, 2}$ norm. Let $\hat{A}^{i}_s = -J\frac{d}{dt}-B_s$ be the operator on ${{\cal H}}(\hat{\mathcal{I}})$ with domain $\mathcal{T}E_i$.

Lemma 5.1. Suppose for $s\in[a, b]$, $\hat{A}^{i}_s$ is self-adjoint and ${{{\rm{dim ~ }}}}\ker(A_s|_{{{\cal H}}_i}) = {{{\rm{dim ~ }}}}\ker(\hat{A}^{i}_s)$, then

Proof. Note that $(A_s+t{\rm Id})|_{{{\cal H}}_i} = P_{{{\cal H}}_i}(A_s+t{\rm Id}) P_{{{\cal H}}_i} = P_{{{\cal H}}_i}A_s P_{{{\cal H}}_i}+tP_{{{\cal H}}_i}$. Since $A_s\in \mathcal{FS}({{\cal H}})$, there is $\epsilon>0$, such that for each $t\in [0, \epsilon]$, $(A_s+t{\rm Id})\in \mathcal{FS}({{\cal H}})$. Then $(A_s+t{\rm Id})|_{{{\cal H}}_i}$ is a positive curve on $\mathcal{FS}(H_i)$ with $t\in [0, \epsilon]$. Note that $-J\frac{d}{dt}-B_s+t{\rm Id}$ is also a positive curve on $\mathcal {FS}({{\cal H}}(\hat{I}))$, then (5.2) is from Lemma 2.2. This completes the proof.

Now we consider Case 2. We assume $\omega S = P^m$, then

We choose $\hat{\mathcal{I}} = [0, \frac{T}{m}]$ be the fundamental domain, then

From Corollary 6.2, we have

Then we have

Remark 5.2. In the case $P = I_{2n}$, (5.3) is the standard Bott-type iteration formula for Hamiltonian systems, please refer [22], [23] for the detail. In the case $P\in{{\mathrm {Sp}}}(2n)\cap \mathbb{O}(2n)$, (5.3) is established by Hu and Sun [18], the general case is proved by Liu and Tang [19].

For Case 3. We assume $N\in{{\mathrm {Sp}}}_a(2n)$ and $N^2 = I$. Since $N$ is anti-symplectic, we have $(Jx, x) = -(JNx, Nx) = -(Jx, x) = 0$ with $x\in \ker(N-I)$. So $\ker(N-I)$ is a Lagrange subspace of the symplectic space $(\mathbb{R}^{2n}, J)$. Recall that $(gx)(t) = Nx(T-t)$ and $g\Lambda = \Lambda$. Let

then $\hat{\mathcal{I}} = [0, T/2]$ is a fundamental domain.

For the $S$-periodic boundary conditions, that is $x(0) = Sx(T)$, then

We have

Similarly, if the boundary condition is given by $x(0)\in V_0, \, x(T)\in V_1$ for $V_0, V_1\in Lag(2n)$ and $NV_0 = V_1$, $NV_1 = V_0$. Similar discussion with above, we have

Remark 5.3. To our knowledge, in the case $S = I_{2n}$, $N^2 = I$, (5.4) is first established by Long Zhang and Zhu [24]. A deep study is given by Liu and Zhang [20] [21]. Hu and Sun had established the case of $S, N\in\mathbb{O}(2n)$, for the case of dihedral group please refer [16].

Now we consider Case 5. For $\lambda\in[0, 1]$, let $\gamma_\lambda(\tau, t)$ be the fundamental solution of (4.19), that is

Let

be the stable and unstable paths, then $V^s_\lambda(\tau), V^u_\lambda(\tau)\in Lag(2n)$.

Recall that in this case, ${{\cal H}} = L^2({{\mathbb R}}, {{\mathbb R}}^{2n})$ and

Let $\mathbb{R}^-$ be the fundamental domain, then

Let $A^\pm_\lambda$ be the restricted operators on ${{\cal H}}(\mathbb{R}^-)$ with domain $\mathcal{T}E_\pm$.

From Prop 3.7 of [15], we have

Similarly

Then we have

In the case of homoclinic orbits, let $A_\lambda = A-B_*-\lambda(B-B_*)$, then from [7] or [15] the index satisfied

We have

Compare (5.7) and (5.8), we have

or equivalently

Obviously, we can use $\mathbb{R}^+$ as the fundamental domain, for reader's convenience, we list the formulas below. Here we let $A_\lambda^{\pm}$ be the restricted operators on ${{\cal H}}(\mathbb{R}^+)$ with domain ${{\cal T}}(E_{\pm})$.

In the case of homoclinic orbits,

In study the stability problem of homographic solution in planar $n$-body problem, Hu and Ou [14] use the McGehee blow up method to get linear heteroclinic system, this system with reversible symmetry if the corresponding central configurations with a symmetry property. Please refer [14] for the detail.

6.   Appendix: Spectral flow and Maslov index

Spectral flow was introduced by Atiyah, Patodi and Singer in their study of index theory on manifold with boundary [2]. Let $\{A_t , t\in [0, 1]\}$ be a continuous path of self-adjoint Fredholm operators on a Hilbert space ${{\cal H}}$. The spectral flow $sf\{A_t\}$ of $A_t$ counts the algebraic multiplicities of the spectral points of $A_t$ cross the line $\lambda = -\epsilon$ with some small positive number $\epsilon$. For reader's convenience, we list some basic properties of spectral flow.

(Stratum homotopy relative to the ends) If $A_{s, \lambda}\in C([a, b]\times[0, 1], \mathcal{FS}({{\cal H}}))$, such that ${{{\rm{dim ~ }}}}\ker A_{a, \lambda}$ and ${{{\rm{dim ~ }}}}\ker A_{b, \lambda}$ is constant, then

(Path additivity) If $A^1, A^2\in C\big([a, b];\mathcal{FS}({{\cal H}}))\big)$ such that $A^1(b) = A^2(a)$, then

where $*$ denotes the usual catenation between the two paths.

(Direct sum) If ${{\cal H}}_i$ are Hilbert space for $i = 1, 2$, and $A^i\in C\big([a, b];\mathcal{FS}({{\cal H}}_i))\big)$, then

(Nullity) If $A\in C\big([a, b];\mathcal{FS}({{\cal H}}))\big)$, then $sf( A_t; t \in [a, b]) = 0$;

(Reversal) Denote the same path travelled in the reverse direction in $\mathcal{FS}({{\cal H}})$ by $\widehat A(t) = A(-t)$. Then

The spectral flow is related to Maslov index in Hamiltonian systems. We now briefly reviewing the Maslov index theory [1,6,25]. Let $(\mathbb{R}^{2n}, \omega)$ be the standard symplectic space and $Lag(2n)$ the Lagrangian Grassmanian. For two continuous paths $L_1(t), L_2(t)$, $t\in[a, b]$ in $Lag(2n)$, the Maslov index $\mu(L_1, L_2)$ is an integer invariant. Here we use the definition from [6]. We list several properties of the Maslov index. The details could be found in [6].

(Reparametrization invariance) Let $\phi:[c, d]\rightarrow [a, b]$ be a continuous and piecewise smooth function with $\phi(c) = a$, $\phi(d) = b$, then

(Homotopy invariant with end points) For two continuous family of Lagrangian path $L_1(s, t)$, $L_2(s, t)$, $0\leq s\leq 1$, $a\leq t\leq b$, and satisfies $dim L_1(s, a)\cap L_2(s, a)$ and $dim L_1(s, b)\cap L_2(s, b)$ is constant, then

(Path additivity) If $a<c<b$, then then

(Symplectic invariance) Let ${{\gamma}}(t)$, $t\in[a, b]$ is a continuous path in ${{\mathrm {Sp}}}(2n)$, then

(Symplectic additivity) Let $W_i$, $i = 1, 2$ be symplectic space with

then

The next Theorem give the relation of spectral flow and Maslov index.

Theorem 6.1.

The above Theorem is well known [15], we like to give a direct proof here.

Proof. We use the idea of Lemma 2.2. We only need to prove the theorem locally. Let $s_0\in [0, 1]$. $A_{s_0}-B_{s_0}+rI, r\in [0, 1]$ is a positive path in $\cal{FS}({{\cal H}})$. There is $\epsilon>0$ such that $\ker(A_{s_0}-B_{s_0}+\epsilon I) = {0}$. Then there is $\delta>0$ such that $\ker(A_{s}-B_{s}+\epsilon I) = {0}$, $\forall s\in [s_0-\delta, s_0+\delta]$. Without loss of generality, we can assume that

Let $\gamma_{s, r}(t)$ be the fundamental solution of the equation

Recall that ${{{\rm{dim ~ }}}}({\rm Gr}(\gamma_{s, r}(T))\cap \Lambda_s) = {{{\rm{dim ~ }}}}\ker(A_s-B_s+rI)$, then we have

Then use the homotopy invariant property of spectral flow and Maslov index, we have

Note that $A_0-B_0+rI$ is a positive path in $\cal{FS}({{\cal H}})$, then we have

Let $Q_t((x, \gamma(t) x), (y, \gamma(t) y)) = <-J\gamma(t)^{-1}\dot{\gamma}(t)x, y>$ which is a quadratic form on ${\rm Gr}(\gamma(t))$. Recall that the crossing form of the Lagrangian pair $(\Lambda, {\rm Gr}(\gamma(t)))$ is given by $Q_t|_{Gr(\gamma(t))\cap \Lambda}$.

Note that

Then we have $\frac{\partial}{\partial t}(-J\gamma_{0, r}(t)^{-1}\frac{\partial}{\partial r}\gamma_{0, r}(t)) = -I$, and it follows that

Thus we have

Similarly

Then by (6.8), we get (6.6).

From the homotopy invariance of Maslov index, we have

Corollary 6.2.

Acknowledgments

We would like to thank Professor Alessandro Portaluri some helpful discussion with us for the spectral flow.