Xijun Hu,Yuwei Ou,Penghui Wang and Hao Zhu

1 School of Mathematics,Shandong University,Jinan,Shandong 250100,China

2 School of Mathematics(Zhuhai),Sun Yat-Sen University,Zhuhai,Guangdong 519082,China 3 Department of Mathematics,Nanjing University,Nanjing,Jiangsu 210093,China

Abstract.In this paper,we give a survey on the Hill-type formula and its applications.Moreover,we generalize the Hill-type formula for linear Hamiltonian systems and Sturm-Liouville systems with any self-adjoint boundary conditions,which include the standard Neumann,Dirichlet and periodic boundary conditions.The Hill-type formula connects the infinite determinant of the Hessian of the action functional with the determinant of matrices which depend on the monodromy matrix and boundary conditions.Further,based on the Hill-type formula,we derive the Krein-type trace formula.As applications,we give nontrivial estimations for the eigenvalue problem and the relative Morse index.

Key Words:Hill-type formula,trace formula,conditional Fredholm determinant,relative Morse index.

1 Introduction

The study of Hill-type formula begins with the original work of Hill[10]in 1877.In his study of the motion of lunar perigee,Hill considered the following equation:

where

is a realπ-periodic function.Letγ(t)be the fundamental solution of the associated first order system of(1.1),that is,

Suppose

are the eigenvalues of the monodromy matrixγ(π).In order to compute c,Hill obtained the following formula which connects the infinite determinant,corresponding to the differential operator,and the characteristic polynomial:

where the right hand side of(1.2)is the Fredholm determinant.We should point out that the right hand side of the original formula of Hill[10]is a determinant of an infinite matrix.In[10],Hill did not prove the convergence of the infinite determinant,and the convergence was proved by Poincar´e[24].The Hill-type formula for a periodic solution of Lagrangian system on manifold was given by Bolotin[2].In[3],Bolotin and Treschev studied the Hill-type formula for both continuous and discrete Lagrangian systems with Legendre convexity condition.For the periodic solution of ODE,the Hill-type formula was given by Denk[6].Please refer more works for Lagrangian systems at[5,7,16,21].As the beginning of a series of work,Hu and Wang[15]introduced the conditional Fredholm determinant and sucessfully generalized the Hill-type formula to Hamiltonian systems with S-periodic boundary conditions.Together with Ou,they obtained the Hill-type formula for Hamiltonian systems with Lagrangian boundary conditions[13].For the Sturm-Liouville systems,they derived the Hill-type formula with S-periodic boundary conditions in[12]and with Lagrangian boundary conditions in[17].By Taylor expansion of the parameterized Hill-type formula,they also derived the Krein-type trace formula[12,13,17].For the non-self-adjoint version of the Hill-type formula,we refer the readers to[16].

The goal of this paper is to derive the Hill-type formula and the Krein-type trace formula for Hamiltonian systems and Sturm-Liouville systems with any self-adjoint boundary conditions,which cover S-periodic boundary conditions and Lagrangian boundary conditions.The linear Hamiltonian system takes the form

where B,D∈C([0,T];S(2n)),and

Here,we denote by M(2n)and S(2n)the set of 2n×2n matrices and real symmetric matrices,respectively.B and D can be considered as bounded operators on H:=L2([0,T];C2n),defined by(Bx)(t)=B(t)x(t)and(D x)(t)=D(t)x(t).Letγλ(t)be the fundamental solution of(1.3),that is,

˙γλ(t)=Jn(B(t)+λD(t))γλ(t), γλ(0)=I2n.

It is well known that

γλ(t)∈Sp(2n):={M∈G L(R2n)|MTJnM=Jn}

for any t∈[0,T].

The self-adjoint boundary condition can be described by Lagrangian subspaces.More precisely,the standard symplectic structureωn(x,y)on C2nis defined by

ωn(x,y)=〈Jnx,y〉,

where〈·,·〉is the standard Hermitian inner product.A Lagrangian subspace V of(C2n,ωn)is an isotropic subspace of dimension n,that is,for any x,y∈V,ωn(x,y)=0.Denote by L a g(C2n,ωn)the set of Lagrangian subspaces of C2n.It is well known that L a g(C2n,ωn)is homeomorphic to the unitary group U(n).Let

(V,Ω):=(C2n⊕C2n,−ωn⊕ωn),

which is 4n-dimensional symplectic space. As above,we denote the set of 2ndimensional Lagrangian subspaces by L a g(V,Ω).Then any self-adjoint boundary condition can be written as

whereΛ∈L a g(V,Ω).

We will explain some important boundary conditions.Let

SpC(2n):={M∈G L(C2n)|M∗JnM=Jn}

We define the operator

with domain

It is well known that forλ∈ρ(A),the resolvent(A−λ)−1is not a trace class operator,but a Hilbert-Schmidt operator.Assume that A−B is non-degenerate throughout this paper,then we set

which will be written as F(B,D)for short without confusion.Throughout this paper,we use I to denote the identity on a Hilbert space.Since F(B,D)is not necessarily a trace class operator,the Fredholm determinant det(I−F(B,D))is not well-defined.Instead we will use the definition of conditional Fredholm determinant,which was introduced in[15],see Section 2 for details.Another definition of the infinite dimensional determinant based on zeta function is referred to[9,26].

Recall that(XT,YT)Tis a frame ofΛin(1.4).Then we have the following Hill-type formula for Hamiltonian system(1.3)–(1.4).

Theorem 1.1.Assume that A−B is non-degenerate,then

where the left hand side is the conditional Fredholm determinant,and the right hand side is independent of the choice of the frame(XT,YT)T.

Remark 1.2.Assume that A−B is non-degenerate.Let S∈Sp(2n)∩O(2n).Then the following Hill-type formula for S-periodic boundary conditions was obtained in[15]:

Let Z0,Z1be frames ofΛ0,Λ1.It is evident thatγλ(T)Z0is a frame ofγλ(T)Λ0and(γλ(T)Z0,Z1)is a 2n×2n matrix.Then the following Hill-type formula for the real Lagrangian boundary conditions was obtained in[13]:

Here,the left hand side of(1.9)–(1.10)is the conditional Fredholm determinant,and the right hand side of(1.10)is independent of the choice of the frames Z0,Z1.

It is worthy to point out that the Hill-type formula(1.10)is also true for the complex Lagrangian boundary conditions,and the proof is similar as Theorem 1.1 in[13].The trace formula can be derived from the Hill-type formula.Set

Let Gj=P−1MjX and F=F(B,D)for simplicity.Then we get the following Krein-type trace formula.

Theorem 1.2.Under the above notations,we have for m∈N,

In particular,

Since Fmis a trace class operator for m≥2,we have

where the algebraic multiplicity ofλjis counted.

The original work of the trace formula is due to Krein[19,20]in 1950’s.In fact,Krein considered the following system

where{λj}are the eigenvalues for the system(1.13),and

Moreover,Krein gave an interesting stability criterion:

Next,we consider the Sturm-Liouville system

where P,Q∈W1,2([0,T];M(n)),R,R1∈C([0,T];M(n)),P(t)is invertible and P(t),R(t),R1(t)∈S(n)for t∈[0,T].Let

Then as in(1.4),any self-adjoint boundary condition can be written as

where

By the Legendre transformation,(1.16)corresponds to the linear Hamiltonian system

where

We define the operator

on L2([0,T];Cn)with domain

It is well-known that A is a self-adjoint operator.We assume that A is non-degenerate.Then R1A−1is a trace class operator and the classical Fredholm determinant det(I+λR1A−1)can be defined.We still useγλ(t)to denote the fundamental solutions of(1.18)with the initial dataγλ(0)=I2n.Now,we give the Hill-type formula for Sturm-Liouville system(1.16)–(1.17).

Theorem 1.3.Assume that A is non-degenerate,then

Similarily,we have the trace formula.

Theorem 1.4.Assume that A is non-degenerate,then for m∈N,

Finally,we give some applications of the trace formula.As an application,we will give some estimations on the non-degeneracy of the linear system.It is well-known that the system preserves the non-degeneracy under small perturbations.A natural question will arise:can we give an upper bound for the perturbation such that,under the smaller perturbation,the systems preserve the non-degeneracy?By the trace formula,we can answer this question partly.Details can be found in Section 5.As another application,the trace formula could be used to estimate the relative Morse index for Hamiltonian systems and the Morse index for Lagrangian systems.It is well-known that the relative Morse index(or Morse index)is equal to the Maslov-type index for the path of symplectic matrices and the Maslov-type index is a successful tool in judging the linear stability[14,22].We will not discuss the stability in the present paper,and interested readers could find the details in[12,13].

We would like to point out that there are some other interesting applications of the Hill-type formula given by Portaluri and Wu[25],which relates to the spectral flow and degree theory.

The paper is organized as follows.Section 2 is devoted to preliminaries on conditional Fredholm determinant and conditional trace.In Section 3,the Hill-type formula and the Krein-type trace formula are proved for Hamiltonian systems with any selfadjoint boundary conditions.These two type’s formula are proved for Sturm-Liouville systems with any self-adjoint boundary conditions in Section 4.Some estimations of relative Morse index and stability criteria are given in Section 5.

2 Conditional Fredholm determinant and conditional trace

In this section,we introduce some preliminary results,which include properties of the conditional Fredholm determinant and conditional trace developed in[13,15].

The original idea of the conditional Fredholm determinant comes from[15].For the classical theory of Fredholm determinant,if C is a trace class operator on a Hilbert space H,then the Fredholm determinant det(I+αC)is well-defined and it is an entire function onα.In the study of Hamiltonian system,however,A−1is not necessarily a trace class operator but a Hilbert-Schmidt operator,where A is assumed to be invertible.This makes the traditional Fredholm determinant not well-defined here.To break this barrier,the conditional Fredholm determinant was introduced in[15]for a class of Hilbert-Schmidt operator satisfying the following condition.

Definition 2.1.Let P={Pk}be a sequence of orthogonal projections on the Hilbert space H such that

1)R an ge(Pk)⊂Ran ge(Pm)for k≤m,

2)Pkconverges to I in strong operator topology.

A Hilbert-Schmidt operator F is called to have the”trace finite condition”associated to P if the limit limk→∞Tr(PkF Pk)exists and is finite.In this case,we still use the notation Tr(F):=limk→∞Tr(PkF Pk),which is called the”conditional trace”of F associated to P.

Denote by J(P)the set of Hilbert-Schmidt operators with trace finite condition associated to P.It is easy to see that J(P)is a linear space and

J1⊂J(P)⊂J2,

where J1and J2are the ideals of trace class operators and Hilbert-Schmidt operators,respectively.In general,J(P)is not an ideal of the algebra of bounded linear operators.

For F∈J2,the regularized Fredholm determinant can be defined(see,for example,[27]):

det2(I+F)=det((I+F)e−F).

Denote by‖F‖2the Hilbert-Schmidt norm for F∈J2.Then for any sequence of finite rank operators{Fk}such that‖Fk−F‖2→0 as k→∞,we have

Therefore,if F∈J2has the trace finite condition associated to{Pk},then setting Fk=PkF Pk,we have

which means

is convergent.To simplify the notation,we have the following definition.

Definition 2.2.If F∈J2has the trace finite condition associated to{Pk},then we define the conditional Fredholm determinant

where Fk=PkF Pk.

The conditional Fredholm determinant shares many properties with the classical Fredholm determinant.For example,in[12,15],by using Montel’s Theorem,we have

Lemma 2.1.If F∈J2has the trace finite condition,then the function det(I+αF)is an entire function on the variableα.

The next lemma collects some basic properties of the determinant.

Lemma 2.2(Theorem 3.5 in[15]).If D,F∈J2have the trace finite condition,then

2)det(I+D)det(I+F)=det(I+D+F+D F),

3)det(I+D)/=0 if and only if I+D is invertible,

and only ifλ0is a zero point of det(I+αD)of order k.

Following[13],we may change a symplectic basis and assume that the Lagrangian frames of V0,V1are

respectively,where for−π/2<θj≤π/2,

C(θ)=di a g(cos(θ1),···,cos(θn)), S(θ)=di a g(sin(θ1),···,sin(θn)).

Then

with the corresponding eigenfunctions

where ejis the standard j-th basis of C2n.In the following lemma.Let PNbe the projections from H to span{ej,k:1≤j≤n,|k|≤N}.

Lemma 2.3(Proposition 2.2 in[13]).

3 Hill-type formula and Krein-type trace formula for Hamiltonian systems

In this section,we prove the Hill-type formula for Hamiltonian systems with any selfadjoint boundary condition.By Taylor expansion of the parameterized Hill-type formula,we derive the Krein-type trace formula.

3.1 Hill-type formula for Hamiltonian systems with any self-adjoint boundary condition

where Xij,Yij∈M(n).Then by(1.5)we have

Let

with the boundary condition

where we identify the subspacesΛ0andΛTwith their frames.To confirm that(3.3)is a complex Lagrangian boundary condition,we need to check thatΛ0andΛTare Lagrangian subspaces in(C4n,ω2n).Note that the basis here isω2n,and thus we will verify that

It is evident thatΛ0is Lagrangian.Direct computation implies

Then by(3.1),we have

which verifies that(3.3)is a complex Lagrangian boundary condition.Let

and define the operator

with domain

ThenˆA is a self-adjoint operator and has compact resolvent.Then by the discussion in(2.2),we have

The corresponding eigenfunctions are denoted by

Recall that

Then the Hamiltonian system(3.2)–(3.3)can be rewritten as

with the boundary condition

Moreover,we have the following conclusion.

Following[22],denote

for

and mij,nij∈M(n).Let

Then by Lemma 2.3 ii),we have

exists and is finite.Hence,the conditional Fredholm determinant det(I−D(A−B)−1)is well defined associated to the projections{PN}.Denote

the fundamental solutions of(1.3)with the initial dataγλ(0)=I2n.Then I2n◇γλ(t)is the fundamental solutions of(3.2)with the initial data I2n◇γλ(0)=I4n.Now,we are in a position to prove Theorem 1.1:the Hill-type formula for Hamiltonian system(1.3)–(1.4).Proof of Theorem 1.1.Since A−B is non-degenerate,we have z(t)≡0 is the only solution of(1.3)–(1.4)forλ=0.By Lemma 3.1,

For k=0,1,we have

Then we prove that

Assume that(3.9)is true.By(3.6)we have

Then(1.8)follows from(3.10),(3.8)and(3.7).

It rests to prove(3.9).By Lemma 3.1,

Let

Then v2is a generalized eigenfunction ofλ∈σ(D(A−B)−1)satisfying

(D(A−B)−1−λ)v2=v1

by induction we can construct a corresponding generalized eigenfunction vi+1ofλ∈σ(D(A−B)−1)satisfying

(D(A−B)−1−λ)vi+1=vi

and vice versa.

This proves(3.9).

3.2 Krein-type trace formula for Hamiltonian systems with any self-adjoint boundary condition

In this subsection,we derive the Krein-type trace formula for Hamiltonian systems with any self-adjoint boundary condition.The idea is similar as Section 2 in[12].First,we study the Taylor expansion of the parameterized conditional Fredholm determinant and the linearly parameterized monodromy matrices,separately.Then we obtain the Kreintype trace formula by comparing the coefficients of the expansions in the parameterized Hill-type formula.

Recall that F=F(B,D)and the conditional Fredholm determinant det(I−αF)is well-defined associated to{PN},which is given in Subsection 3.1.Then by Theorem 2.6 in[16]we have

Next,we consider the expansion of the parameterized monodromy matrices.Let

M(α)=Y−γα(T)X, f(α)=det M(α)det M(0)−1and P=γ0(T)−1Y−X.

Then

f(α)=det(γ0(T)−1(Y−γα(T)X))det(P−1)

=det(I2n−P−1(γ0(T)−1γα(T)−I2n)X).

Following Subsection 2.2 in[12],we have

where Mjis defined in(1.11).Then

where Gj=P−1MjX.Similar to(2.6)in[12],we have f(α)=eg(α),where

Now,we give the proof of Theorem 1.2.

Proof of Theorem 1.2.By Theorem 1.1,we get the parameterized Hill-type formula det(I−αF)=f(α).Then the proof is done by comparing the coefficients of(3.11)and(3.12).

4 Hill-type formula and Krein-type trace formula for the Sturm-Liouville system with any self-adjoint boundary condition

In this section,we prove the Hill-type formula and the Krein-type trace formula for the Sturm-Liouville system(1.16)–(1.17).Recall that

is defined on L2([0,T];Cn)with domain EΛ.If A is invertible,then R1A−1∈J1.Thus,the classical Fredholm determinant det(I+λR1A−1)can be defined.We need the following lemma,whereγλ(t)is the fundamental solutions of(1.18)with the initial data γλ(0)=I2n,and(XT,YT)Tis a frame ofΛin(1.17).

Lemma 4.1.

i)det(I+λ0R1A−1)=0 if and only if det(Y−γλ0(T)X)=0.

ii)Suppose R1>0.Then the order of det(I+λR1A−1)at a zero pointλ0is the same as that of det(Y−γλ(T)X)atλ0.

By Theorem 3.5(c)in[27]again,k0=τ1(−1/λ0,R1A−1).

Since

we have R1A−1is diagonable and thus−1/λ0is semi-simple,which means τ1(−1/λ0,R1A−1)=τ2(−1/λ0,R1A−1).

Now,we are ready to prove Theorem 1.3:the Hill-type formula for Sturm-Liouville system(1.16)–(1.17).Proof of Theorem 1.3.Since R1A−1∈J1,by Theorem 3.7 in[27]we have

First,we prove(1.20)for R1>0.Following the proof of Lemma 2.3 in[17],we have γλ(T)is an entire function onλand

‖γλ(T)‖≤C0|λ|1/2exp(C|λ|1/2).

Then

g(λ):=det(Y−γλ(T)X)

is an entire function and it is not hard to check that for anyε>0,there exists Cεsuch that

|g(λ)|≤Cεexp(ε|λ|).

By Lemma 4.1,all the zeros(counting orders)of g(λ)are{λj}.Moreover,∑j|λj|−1<∞due to R1A−1∈J1.Since A is non-degenerate,g(0)/=0.By Lemma 3.6 in[27],we have

and in particular,

which proves(1.20)for R1>0.In the general case,we chooseα0∈R such that R1−α0>0 and A+α0is non-degenerate.We rewriteγλ(T)byγλ(T,R1)to indicate its dependence on R1.Following(2.5)in[17],we have

where we used(4.2).This proves(1.20)in the general case.

Next,we give the proof of Theorem 1.4.

Proof of Theorem 1.4.Since R1A−1∈J1,we have by(5.12)in[27]that

Let

f(λ)=det(Y−γλ(T)X)·det(Y−γ0(T)X)−1.

Similar to(3.12),we have f(λ)=eh(λ),where

Here Gj=P−1MjX,MjandˆD(t)are defined in(1.11),D(t)is defined in(1.19).By Theorem 1.3,we get the parameterized Hill-type formula det(I+λR1A−1)=f(λ).Comparing the coefficients of(4.3)and(4.4),we have

This complete the proof.

Moreover,we have

Tr(R1A−1)k=Tr(Fk(B,D)), k∈N,

where B and D are defined in(1.18).

When the Hamiltonian system comes from the Legendre transformation of Sturm-Liouville system,the operator F(B,D)∈J∞,and thus,

(4.5)does not hold for general Hamiltonian systems,and a counterexample could be found in Subsection 3.3 of[13].

Finally,we give an example to illustrate how to get infinite identity from the trace formula.Consider the eigenvalue problem

with the boundary conditions

It is well known that the k-th eigenvalueλkis the k-th positive solution of the next transcendental equation

However,λkcan only be solved numerically.As an application of the trace formula,we have the following equality,which itself is interesting:

Obviously,forθ=0,(4.9)gives the well known identity

and forθ=π/2,(4.9)gives the identity

5 Applications

The relative Morse index for linear Hamiltonian system is equal to the Maslov-type index for the corresponding fundamental solutions,and the Maslov-type index is a very useful tool in studying the multiplicity and stability of periodic solution in Hamiltonian systems[22].In this section,we give the relation of conditional Fredholm determinant and relative Morse index,also we estimate the relative Morse index by the trace formula.Therefore the trace formula could be used to judge the linear stability via the Maslov-type index.

5.1 Relation with the relative Morse index and Maslov-type index

A simple way to understand the relative Morse index I(A−B,A−B−D)is from the viewpoint of spectral flow.For reader’s convenience,we first give a brief review of the spectral flow.The spectral flow was introduced by Atiyah,Patodi and Singer[1]in their study of index theory on manifolds with boundary.Let{A(θ),θ∈[0,1]}be a continuous path of self-adjoint Fredholm operators on a Hilbert space H.Roughly speaking,the spectral flow of path{A(θ),θ∈[0,1]}counts the net change in the number of negative eigenvalues of A(θ)as θ goes from 0 to 1,where the enumeration follows from the rule that each negative eigenvalue crossing to the positive axis contributes+1 and each positive eigenvalue crossing to the negative axis contributes−1,and for each crossing,the multiplicity of eigenvalue is counted.

Our main interests in this paper are Hamiltonian dynamics.Given a Hamiltonian system

In general,suppose

Bs(t)=B(s,t)∈C([0,1]×[0,T];S(2n)).

There is another natural topological characterization for each orbit of the system,namely its Maslov index[4].Here we use the Maslov index theory of the complex Lagrangian subspaces,the details of which can be found in[29].Let(C2n,ωn)be the complex symplectic vector space.Recall that a complex subspace V is Lagrangian ifωn|V=0 and dimCV=n.Let V±=ker(iJ∓I2n).Then any Lagrangian subspace can be expressed as Gr(U)={(x,Ux)|x∈V+},where U:V+→V−is some unitary matrix,and the converse is also true.This shows that the(complex)Lagrangian Grassmannian Lag(C2n)is homeomorphic(isomorphic)to the unitary group U(n),which we denote by F:Lag(C2n)→U(n).For V1,V2∈Lag(C2n),it is obvious that

dim(V1∩V2)=dim ker(F(V2)−1F(V1)−In).

For any fixed U∈U(n),let

ΣU={U0∈U(n)|det(U−1U0−In)=0)}

be the singular cycle of U.For any U0∈ΣU,the path eitU0(|t|<ε)is transversal toΣU.Let U(t)(t∈[a,b])be any path in U(n).Forε>0 small enough,e−εiU(a)and e−εiU(b)are not in the singular cycle of U,and the intersection number[e−εiU(t):ΣU]is well defined.For a path of complex Lagrangian subspaces V(t)andΛ∈L a g(C2n),we define the Maslov index to be

µ(Λ,V(t)):=[e−εiF(V(t)):ΣF(Λ)].

For the fundamental solutionγ(t),t∈[0,T]of the linearized Hamiltonian system along the solution z(t),the Maslov index of z is defined to be

µ(z)=µ(Λ,Gr(γ(t))),

whereΛ∈L a g(V,Ω)is the boundary condition.

Theorem 5.1.In[14],the authors showed that for each orbit of the Hamiltonian system,its relative Morse index is equal to its Maslov index under the Lagrangian boundary conditions,that is,

The Maslov-type index for symplectic paths is a powerful tool in studying the stability problem.We give its relation to the Maslov index,and the details can be found in[22,23].As usual,consider paths in Sp(2n):

PT(2n)={γ∈C([0,T];Sp(2n))|γ(0)=I2n}.

For anyω∈U andγ∈PT(2n),we define

the intersection number withεto be a small positive number,where

is the codimensional one hypersurface in Sp(2n)with

Then we have the following theorem,which is from[23].

Theorem 5.2.For anyγ∈PT(2n),we have

i1(γ)+n=µ(△,Gr(γ(t))),

iω(γ)=µ(Gr(ω),G r(γ(t))), ω∈U{1},

where△is the diagonal G r(I2n)and G r(ω)=Gr(ωI2n).

Now,we briefly review the Morse index theorem of Sturm-Liouville systems with any self-adjoint boundary condition,and the details can be found in[18].Consider the Sturm-Liouville system(1.16).We define the Morse index of Sturm-Liouville operator by m−(A)which is the number of total negative eigenvalues of A,and set m0(A)=ker(A).Then we have the following Morse index theorem from[18],which gives the relation between the Morse index of A and the Maslov index of the corresponding Hamiltonian system.

Theorem 5.3.For the Sturm-Liouville system(1.16)under the boundary condition(1.17),we have

µ(Λ,Gr(γ(t)),t∈[0,T])−m−(A)=n−i(Gr(I2n),Λ,ΛD),

where i(Gr(I2n),Λ,ΛD)is the Duistermaat triple index.

ΛD={(z(0),z(T))∈C4n|y(0)=y(T)=0},

{(z(0),z(T))∈C4n|x(0)=x(T)=0}.

We list several important examples to compute i(Gr(I2n),Λ,ΛD).

1.The first is from Theorem 1.2 in[14].Let V be any subspace ofΛNand the boundary condition be given byΛV=J V⊥⊕V,where J=−Jn⊕Jn.Then

i(Gr(I2n),Λ,ΛD)=n−dim(V⊥∩G r(−In)).

There are two important cases:

(i)dim(V⊥∩Gr(−In))=dim(M−I2n),∀V=Gr(M),M∈G L(Cn).Let V1and V2be two subspaces of Cn.Then

(ii)dim(V⊥∩Gr(−In))=dim(V⊥1)∩V⊥2,∀V=V1⊕V2.As special cases,we have

a.Dirichlet boundary condition:i(Gr(I2n),ΛD,ΛD)=0.

b.Neumann boundary condition:i(G r(I2n),ΛN,ΛD)=n.

c.Periodic boundary condition:i(Gr(I2n),G(I2n),ΛD)=0.

2.Separated boundary conditions.For this case,we have

For the special case y(0)=Asx(0),y(T)=Aex(T),we have

i(Gr(I2n),Λs⊕Λe,ΛD)=n+m+(As−Ae),

where m+(A)is the total number of positive eigenvalues of operator A.

5.2 Estimation of the relative Morse index and the stability criteria

In this subsection,we consider the relation of conditional Fredholm determinant and relative Morse index,also we could estimate the relative Morse index by the Krein-type trace formula.As in[12,15],we have the following theorem.

Theorem 5.4.Assume A−B and A−B−D are non-degenerate,then det(I−F(B,D))>0(<0)if and only if I(A−B,A−B−D)is even(odd).

In[12],the trace of Fk(B,D)was used to get the nontrivial estimation of relative Morse index.Although we dealt with operators in the S-periodic case in[12],it is totally same for the Lagrangian boundary conditions.The following theorem is from[12].

Proposition 5.1.Suppose A−B is non-degenerate and D>0.Then for k∈N,

whereυ(A−B−D)=dimker(A−B−D).

Proposition 5.2.Suppose D>0.Then

I(A−B,A−B+D)=I(A−B,A−B−D)+υ(A−B−D)=0.

Theorem 5.5.Suppose A−B is non-degenerate.Assume that there exist D1,D2∈B(2n)such that D10.If there exists k∈2N such that Tr Fk(B,Dj)<1 for j=1,2,then A−B−D is non-degenerate,and moreover,I(A−B,A−B−D)=0.

Theorem 5.6.Suppose A−B is non-degenerate and D1≤D≤D2,where D1<0,D2>0.

Let

m−=inf{[Tr(F(B,D1)k)],k∈2N} and m+=inf{[Tr(F(B,D2)k)],k∈2N}.

Then

−m−≤I(A−B,A−B−D)≤m+.

Connected with the trace formula(1.12),we can give an estimation of relative Morse index by the trace of matrices.As a corollary of Theorem 5.5,we have

Let e(M)be the total number of eigenvalues of M on U.Then a simple but useful stability criterion is

Then we have following propositions.

For the applications of the trace formula to stability estimation in the planar threebody problem,we refer the readers to[12,13]for the details.

Acknowledgements

The first author is partially supported by NSFC(Nos.12071255 and 11790271)and National Key R&D Program of China(2020YFA0713300).The second authors is partially supported by NSFC(No.11801583).The third author is Partially supported by NSFC(Nos.11471189,and 11871308).