1 Introduction

The Hörmander index in the finite-dimensional case

Yuting ZHOU, Li WU, Chaofeng ZHU

1 Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China

2 Department of Mathematics, Shandong University, Jinan, 250100, China

Higher Education Press and Springer-Verlag Berlin Heidelberg 2017

Abstract In this paper, we calculate Hörmander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system on compact symplectic manifold with symplectic trivial tangent bundle and given autonomous Hamiltonian system on regular compact energy hypersurface of symplectic manifold with symplectic trivial tangent bundle.

footnotetext: Received December 14, 2016; accepted January 18, 2017
     Corresponding author: Chaofeng ZHU, E-mail: zhucf@nankai.edu.cn

Keywords Maslov index, Hörmander index, Maslov-type index, symplectic reduction
MSC 53D12, 58J30

1 Introduction

Let be a symplectic vector space. Let be four Lagrangian subspaces of . The Hörmander index has been introduced by L. Hörmander [Ho71, Sect. 3.3] in the finite-dimensional case when hold for , who also gave the explicit formula to calculate it. The notion was generalized by B. Booss and K. Furutani [BoFu99, Proposition 2.1] in the finite-dimensional case and [BoFu99, Definition5.2] in the strong symplectic Hilbert case when are two Fredholm pairs of Lagrangian subspaces of V and is finite-dimensional. Recently, B. Booss and the third author [BoZh14, Definition 3.4.4] generalized the notion to the symplectic Banach case.

The splitting number [Lo02, Denition 9.1.4] is a special case of the Hörmander index. It turns out that the study of the Hörmander index in the full generality is very important in the study of Hamiltonian systems (see [Lo02] for the applications of the splitting numbers and [LoZhZh06, LiZh14] for the study of multiplicity of the brake orbits).

In [Go09], M. de Gosson gave a very elegant definition of the Hörmander index in the finite-dimensional case in great generality. His definition differs slightly from ours. By admitting half-integer indices, it yields more simple proofs, but may be more difficult to be used in concrete applications in Morse theory.

We calculate the Hörmander index in the finite-dimensional case and get the following main result.

Theorem 1.1.

Let be a complex symplectic vector space of dimension . Let be four Lagrangian subspaces of . Denote by the triple index defined by [Du76, (2.16)] (see Corollary 3.12 below). Then we have

(1)
(2)

Our main result does not require the transversal conditions . We use the result to get some new iteration inequalities of Maslov-type index. Then we use the inequalities to prove almost existence of mean indices for given complete autonomous Hamiltonian system on compact symplectic manifold with symplectic trivial tangent bundle and given autonomous Hamiltonian system on regular compact energy hypersurface of symplectic manifold with symplectic trivial tangent bundle.

The paper is organized as follows. In , we review the historical literatures and introduce our main result. In , we review the notions of Maslov index and Maslov-type index in the finite-dimensional case. In , we study the Hörmander index in the finite-dimensional case and prove Theorem 1.1. In , we prove some iteration inequalities of Maslov-type index. In , we prove almost existence of mean indices for given complete autonomous Hamiltonian system on compact symplectic manifold with symplectic trivial tangent bundle and given autonomous Hamiltonian system on regular compact energy hypersurface of symplectic manifold with symplectic trivial tangent bundle.

In this paper we denote the sets of natural, integral, real, complex numbers, the unit circle in the complex plane, the set of all linear operators on V, the set of all invertible linear transformations on and the set of all self-adjoint operators on Hilbert space , by , , , , , , and respectively. We denote by the complex dimension of a complex linear space . We denote by the set of all linear maps between vector spaces and . For a linear operator , we denote by the graph of . For two maps and , we denote by the composite map defined by for each . Without further explanation, the coefficient field is in the rest of this paper.

2 The Maslov index and the Maslov-type index

In this section we review the definition of the Maslov index and the Maslov-type index in the finite-dimensional case. Firstly we recall the basic concepts and properties of symplectic vector space.

Definition 2.1.

Let be a complex vector space.

  1. A mapping

    is called a symplectic form on , if it is a non-degenerate skew-Hermitian form. Then we call a complex symplectic vector space.

  2. Let and be two finite-dimensional symplectic vector spaces. A linear map is called symplectic, if is invertible and for each . We denote by the set of all such symplectic linear maps , and for the finite-dimensional symplectic vector space . We denote by and if there is no confusion.

  3. Let be a linear subspace of . The annihilator of is defined by

  4. A linear subspace of is called symplectic, isotropic, co-isotropic, or Lagrangian if

    respectively.

  5. The Lagrangian Grassmannian consists of all Lagrangian subspaces of . We write if there is no confusion.

If , the space is path-connected. It is nonempty if and only if the signature .

The following definition of the Maslov index is taken from [BoZh14, §2.2].

Let be a -dimensional complex vector space. Let be an inner product on . Then there is an operator such that is self-adjoint and for all .

Denote by the positive (negative) eigenspace of . Given a path , of pairs of Lagrangian subspaces of , let be generators for , i.e., and (see [BoZh13, Proposition 2]). Then the family is a continuous family of unitary operators on Hilbert space .

Note that the eigenvalues of are on the unit circle . Recall that each map in can be lifted to a map in . By [Ka95, Theorem II.5.2], there are continuous functions such that the eigenvalues of the operator for each (counting algebraic multiplicity) have the form

Denote by the integer part of and . Define

(3)
Definition 2.2.

We define the Maslov index of the path by

(4)
(5)

By definition, is an integer which does not depend on the choices of the arguments . By [BoZh13, Proposition 6], it does not depend on the particular choice of the inner product.

Remark 2.3.

Let be a real symplectic -dimensional symplectic space. Let be in and , be a path in such that . Denote by the Maslov-Arnol’d index defined by [Du76, (2.8)]. By [BoZh14, Proposition 3.27], we have

We recall some concepts in [Du76, Zh06] for the calculation of the Maslov index.

Let be a finite-dimensional symplectic vector space. Let , be a path in differentiable at . Define the form on by

(6)

where , and . It is well-known that the Hermitian form is independent on the choice of with .

Definition 2.4.

Let be a finite-dimensional symplectic vector space. Let , be a curve of pairs of Lagrangian subspaces of . For , the crossing form on is defined by

(7)

where . A crossing is a time such that . A crossing is called regular if is non-degenerate.

Proposition 2.5.

([Du76, Lemma 2.5], [Zh06, Proposition 4.1]) Let be a finite-dimensional symplectic vector space. Let , be a curve of pairs of Lagrangian subspaces of with only regular crossings. Then the crossings are finite, and we have

(8)

where we denote by , the Morse positive index, the nullity, and the Morse negative index of an Hermitian form respectively.

For each , we define

(9)
Definition 2.6.

(cf. [Zh06, Definition 4.6]) Let , be two finite-dimensional symplectic vector space. Then is a symplectic vector space. Let . Let , be a path in . The Maslov-type index is defined to be . If , we define . If and , we denote by , , and .

The following lemma gives the tangent vector of given symplectic path.

Lemma 2.7.

(cf. [Du76, Lemma 3.1]) Let , be two finite-dimensional symplectic vector space. Then is a symplectic vector space. Let , be a path in . Then for each , we have

(10)

where we denote by .

The Maslov-type indices have the following properties.

Lemma 2.8.

(cf. [Lo02, Theorem 6.1.8]) Let be a finite-dimensional symplectic vector space. Let . Let be a symplectic path. Then there exists a neighborhood of in such that, for each , there holds that

(11)
Proof.

Let be a convex neighborhood of in , the Lie algebra of such that the map is a diffeomorphism. Set

Then for each , there exists and , such that , . Thus we have a homotopy by . Denote by the path , . By the definition of the Maslov-type index we can choose small enough such that for each . By the homotopy invariance and path additivity of the Maslov-type index we have . The inequality (11) then follows. ∎

Lemma 2.9.

(cf. [Zh06, Lemma 4.4]) Let , be finite-dimensional symplectic vector spaces. Let be a Lagrangian subspace of . Let , be symplectic paths. Then we have

(12)

where .

3 Calculation of the Hörmander index

In this section we study the form and the triple index . Then we express the Hömander index via the triple index.

3.1 The form

Let be a complex symplectic vector space with three isotropic subspaces . Define on by

(13)

for all , where , , .

Remark 3.1.

Assume that are Lagrangian subspaces of and . Then the form defined by [Du76, (2.3)] is here.

With the above notions, we have , and . It follows that . So does not depend on the choices of and and the form is a well-defined Hermitian form on . Moreover, the last two equalities in (13) hold, i.e., , and we have

(14)

The following lemma is well-known in the non-degenerate case.

Lemma 3.2.

Let be a complex symplectic vector space with three isotropic subspaces . Denote by , and .
(a) Let , where , , . Then we have .
(b) We have .

Proof.

(a) Note that . So we have

Similarly we have .

(b) Let be a linear subspace of such that with a base . Let , be such that . Then . Let be the -linear span of . By (a), for each we have , where and . In this case . So are linearly independent, and . Hence . Since is arbitrarily chosen, we have . Similarly we have . It follows that . Similarly we have and . ∎

Here we give the kernel of the form .

Lemma 3.3.

Let be a complex symplectic vector space with three isotropic subspaces . Then we have

(15)
(16)
(17)
(18)

In particular, holds if , or , or .

Proof.

Since , we have

Similarly we have . So (15) holds.

Since , by (13) we have

So (16) follows. Similarly we get (18). By (15) and (16) we get (17).

If , by (17) we have

Similarly, by (16) and (18), holds if , or . ∎

Corollary 3.4.

Let be a finite-dimensional complex symplectic vector space with three Lagrangian subspaces . Then we have .

We now study when the form is zero if

Lemma 3.5.

Let be an Abelian group with three subgroups . Then the following three conditions are equivalent:

  • ,

  • ,

  • .

Proof.

By the symmetry of the statement, we only need to prove that (i)(ii).

Assume that (i) holds. Clearly we have . Let . Then there exist and such that . So . By (i), there exist and such that . So and . ∎

Lemma 3.6.

Let be a vector space with three finite-dimensional linear subspaces . Then we have

(19)

The equality in (19) holds if and only if .

Proof.

We have . Since , we have

So (19) holds. The equality in (19) holds if and only if , if and only if . ∎

Corollary 3.7.

Let be a complex symplectic vector space of dimension with three Lagrangian subspaces . Then we have

(20)

The equality in (20) holds if and only if , if and only if .

Proof.

Since are Lagrangian subspaces of and , we have and

By Lemma 3.6, the inequality (20) holds, and the equality in (20) holds if and only if . By Corollary 3.4, holds if and only if . ∎

If is an isotropic subspace of such that , defines a symplectic form on . Moreover, for each isotropic subspace , the image of under the canonical homomorphism: is an isotropic subspace of .

Let be three isotropic subspaces of . Assume that . Then we have , and

So we have , and (cf. [Du76, (2.11)])

(21)

here and .

Lemma 3.8.

Let be a finite-dimensional complex symplectic vector space. Let , be a path in differentiable at . Let be a Lagrangian subspace of . Assume that is positive definite. Then there exists an such that for , we have , and

(22)
Proof.

By Proposition 2.5, there exists an such that holds for each . Set and . By the proof of [BoZh14, Corollary 1.3.4], there is a such that and . Since is positive definite, there exists an such that for , the form is positive (negative) definite. Let be in . By (21), the form is positively (negatively) definite. Note that . By (21) again, the form is positive (negative) definite. By Lemma 3.2, (22) holds. ∎

3.2 The triple index and the Hörmander index

Let be a complex symplectic vector space of dimension . Let be four Lagrangian subspaces of .

Definition 3.9.

([BoZh14, Definition 3.4.4]) Assume that there are continuous paths and , of Lagrangian subspaces of such that , , , . Then the Hörmander index is defined by

(23)
(24)

By [BoZh14, Proposition 2.3.1.b,f], we have

(25)
(27)
Lemma 3.10.

Let , be a Lagrangian path of complex symplectic vector space . Let be such that . Then we have

(28)
(29)
Proof.

Let , be a path of linear maps such that, the form , is Hermitian for each , , . Consider a special path . By [BoZh14, Lemma 2.3.2], we have

The equality (29) follows from (28). ∎

Corollary 3.11.

([Ho71, (3.3.5),(3.3.7)]), [Du76, (2.10),(2.13)]) Let be Lagrangian subspaces of complex symplectic vector space . Assume that , . Then we have

(30)
(31)
Proof.

By (27), (25), (27), Lemma 3.10, Corollary 3.4 and (14) we have