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Sturm theory with applications in geometry and classical mechanics

Classical Sturm non-oscillation and comparison theorems as well as the Sturm theorem on zeros for solutions of second order differential equations have a natural symplectic version, since they describe the rotation of a line in the phase plane of the equation. In the higher dimensional symplectic version of these theorems, lines are replaced by Lagrangian subspaces and intersections with a given line are replaced by non-transversality instants with a distinguished Lagrangian subspace. Thus the symplectic Sturm theorems describe some properties of the Maslov index. Starting from the celebrated paper of Arnol’d on symplectic Sturm theory for optical Hamiltonians, we provide a generalization of his results to general Hamiltonians. We finally apply these results for detecting some geometrical information about the distribution of conjugate and focal points on semi-Riemannian manifolds and for studying the geometrical properties of the solutions space of singular Lagrangian systems arising in Celestial Mechanics.


Introduction
Symplectic Sturm theory has a lot of predecessor, like Morse, Lidskii, Bott, Edwards, Givental who proved the Lagrangian nonoscillation of the Picard-Fuchs equation for hyperelliptic integrals. The classical Sturm theorems on oscillation, non-oscillation, alternation and comparison for a second-order ordinary differential equation have a symplectic nature. They, in fact, describe the rotation of a straight line through the origin of the phase plane of the equation. A line through the origin is a special 1-dimensional subspace of the phase plane: it is, in fact a Lagrangian subspace.
Starting from this observation, as clearly observed and described by Arnol'd in [Arn86], the higher-dimensional symplectic generalization of the Sturm theory has been obtained by replacing lines by Lagrangian subspaces and instants of intersections between lines, by instants of nontransversality. Such instants in the terminology of Arnol'd has been termed moments of verticality. Thus, in higher dimension, the rotation of a straight line through the origin has been replaced by the evolution of a Lagrangian subspace through the phase flow of the linear Hamiltonian system in the phase space. The phase flow defines, in this way, a curve of Lagrangian subspaces and moments of verticality correspond to the intersection instants between this curve and a hypersurface (with singularities) in the Lagrangian Grassmannian manifold, called (in the Arnol'd terminology), the train of a distinguished Lagrangian subspace. Such a train is a transversally oriented variety and by using such an orientation, it is possible to define an integer-valued intersection index, insure that the Lagrangian function is non-negative. This is a pretty important information and gives deep insight on the spectral analytic properties of the problem. In fact, up to a shifting constant (discussed in Section 1) that is bounded by the number of degrees of freedom, the Maslov index coincides with the Morse index. Now, under the signature assumptions on the kinetic and potential energy, it follows that the Morse index is zero and hence the the Maslov index is bounded by the number of degrees of freedom. This, however, is not the end of the story, since the bound on the Maslov index doesn't imply, in general, a bound on the total number of crossing instants. However, in the case of plus curve, it does. This is why in the theorem the Maslov intersection index is considered with respect to the Dirichlet Lagrangian (and in fact such a Hamiltonian is Dirichlet optical, being Legendre convex).
An extremely useful result in applications is Theorem 4.1: a generalized version of the Sturm comparison theorem. In this case, on the contrary, is not important to work with plus Lagrangian curves. This fact, has been already recognized by the third author in [MPP07]. Loosely speaking, the monotonicity between Hamiltonian vector fields implies an inequality on the Maslov index and if the Hamiltonian system is induced by a second order Lagrangian system C 2 -convex in the velocity, this implies an inequality on the Morse indices. From a technical viewpoint the proof of this result is essentially based upon the homotopy invariance of the Maslov index. An essential ingredient in the proof is provided by a spectral flow formula for paths of unbounded self-adjoint firs order (Fredholm) operators with dense domain in L 2 .
Finally in the last section we provide some applications essentially in differential topology and classical mechanics. More precisely, we prove some interesting new estimates about the conjugate and focal points along geodesics on semi-Riemannian manifolds, improving the estimates provided by authors in [JP09,Section 4]. We stress on the fact that classical comparison theorems for conjugate and focal points in Riemannian manifolds and more generally on Lorenzian manifolds but for timelike geodesics, requires curvature assumptions or Morse index arguments. On general semi-Riemannian manifolds having non-trivial signature, the curvature is never bounded and the index form has always infinite Morse index and co-index. The second application we provide is based upon an application of the Sturm comparison theorem to the Kepler problem in the plane with fixed (negative) energy.
Considerable effort has been focused on improving the readability of the manuscript and on explaining the main ideas and involved techniques. Notation For the sake of the reader, we introduce some notation that we shall use henceforth without further reference throughout the paper.
Id V or in shorthand notation just Id denotes the identity; -For T ∈ L (V, W ), we define the pull-back of C ∈ B(W ) through the map T as -(V, ω) denotes a 2n-dimensional (real) symplectic vector space and J denotes a complex structure on V ; Sp(V, ω) the symplectic group ; sp(V, ω) denotes the symplectic Lie algebra. GL(V ) denotes the general linear group. The symplectic group of (R 2n , ω) is denoted by Sp(2n) and its Lie algebra simply by sp(2n). We refer to a matrix in sp(2n) as the set of Hamiltonian matrices.
Lagrangian action functional with the Maslov-type index of z x corresponding to x through the Legendre transform. Our basic references are [Dui76,APS08,HS09]. Let T R n ∼ = R n ⊕ R n be the tangent space of R n endowed with coordinates (q, v). Given T > 0 and the Lagrangian function L ∈ C 2 ([0, T ] × T R n , R), we assume that the following two assumptions hold L is exactly quadratic in the velocities v meaning that the function L(t, q, v) is a polynomial of degree at most 2 with respect to v.
Under the assumption (L1) the Legendre transform defined by Remark 1.1. The assumption (L2) is in order to guarantee that the action functional is twice Frechét differentiable. It is well-known, in fact, that the smoothness assumption on the Lagrangian is in general not enough. The growth condition required in (L2) is related to the regularity of the Nemitski operators. For further details we refer to [PWY19] and references therein.
We denote by H := W 1,2 ([0, T ], R n ) be the space of paths having Sobolev regularity W 1,2 and we define the Lagrangian action functional A : H → R as follows Let Z ⊂ R n ⊕ R n be a linear subspace and let us consider the linear subspace Notation 1.2. In what follows we shall denote by A Z the restriction of the action A onto H Z ; thus in symbols we have A Z := A HZ .
It is well-know that critical points of the functional A on H Z are weak (in the Sobolev sense) solutions of the following boundary value problem where Z ⊥ denotes the orthogonal complement of Z in T * R n and up to standard elliptic regularity arguments, classical (i.e. smooth) solutions.
Remark 1.3. We observe, in fact, that there is an identification of Z × Z ⊥ and the conormal subspace of Z, namely N * (Z) in T * R n . For further details, we refer the interested reader to [APS08].
We assume that x ∈ H Z is a classical solution of the boundary value problem given in Equation (1.1). We observe that, by assumption (L2) the functional A is twice Fréchet differentiable on H. Being the evaluation map from H Z into H a smooth submersion, also the restriction A Z is twice Fréchet differentiable and by this we get that d 2 A Z (x) coincides with D 2 A Z (x).
By computing the second variation of A Z at x we get and finally R(t) := ∂ qq L t, x(t), x ′ (t) .
Now, by linearizing the ODE given in Equation (1.1) at x, we finally get the (linear) Morse-Sturm boundary value problem defined as follows We observe that u is a weak (in the Sobolev sense) solution of the boundary value problem given in Equation (1.2) if and only if u ∈ ker I. Moreover, by elliptic bootstrap it follows that u is a smooth solution.
Denoting by J 0 the (standard) complex structure namely the automorphism J 0 : T * R n → T * R n defined by J 0 (p, q) = (−q, p) whose associated matrix is given by Notation 1.4. In what follows, T * R n is endowed with a coordinate system z = (p, q), where p = (p 1 , . . . , p n ) ∈ R n and q = (q 1 , . . . , q n ) ∈ R n . we shall refer to q as configuration variables and to p as the momentum variables.
By setting z(t) := P (t)u ′ (t) + Q(t)u(t), u(t) T , the Morse-Sturm equation reduces to the following (first order) Hamiltonian system in the standard symplectic space We now define the double standard symplectic space (R 2n ⊕ R 2n , −ω 0 ⊕ ω 0 ) and we introduce the matrix J 0 := diag(−J 0 , J 0 ) where diag( * , * ) denotes the 2 × 2 diagonal block matrix. In this way, the subspace L Z given by is thus Lagrangian.
Notation 1.5. The following notation will be used throughout the paper. If x is a solution of (1.1) we denote by z x the corresponding function defined by (1.5) Definition 1.6. Let x be a critical point of A. We denote by ι Z (x) the Morse index of x namely Let z x be defined in Equation (1.5). We define the Maslov index of z x as the integer given by where ψ denotes the fundamental solution of the Hamiltonian system given in Equation (1.3).
Theorem 1.7. Under the previous notation and if assumptions (L1) & (L2) are fulfilled the functional A : H Z → R is of regularity class C 2 . If x is a critical point of A Z , then ι Z (x) is finite. Moreover there exists a non-negative integer c(Z) ∈ {0, . . . , n} such that the following equality holds Proof. For the proof of this result we refer the reader to [HS09, Theorem 3.4 & Theorem 2.5].
Remark 1.8. The integer c(Z) depend upon the boundary conditions. However the authors in [HS09, Section 3], computed c(Z) in some interesting cases.
• (Periodic) Z := ∆ ⊂ R n ⊕ R n (where ∆ denotes the graph of the identity in R n ) and We observe that in the case of separate boundary conditions, i.e. Z = Z 1 ⊕ Z 2 , then we get that [HS09, Equation (3.28)] for further details). Remark 1.9. It is not surprising that in the Dirichlet case and in the Neumann we get the n and 0. In fact the Morse index of a critical point x ∈ H of the action A get its largest possible value with respect to Neumann boundary conditions and the smallest possible value with respect to Dirichlet boundary conditions.
The last result of this section provides a bound on the Maslov index of z x when x is a minimizer. Proof. Being x minimizer, it follow that ι Z (x) = 0 and by Theorem 1.7, we get that The conclusion now follows from the fact that c(Z) ∈ {0, . . . , n}.
A direct consequence of Proposition 1.10 in the case of natural Lagrangian, namely Lagrangian of the form where as usually K(v) and V (q) denote respectively the kinetic and the potential function, is the following result.
Corollary 1.11. Let L be a C 2 -natural Lagrangian having a C 2 -concave potential energy and let x ∈ H Z be a critical point of A Z . Then Proof. Being L(t, q, v) = K(v) − V (q), we get that the Lagrangian function L is C 2 -convex. Let x ∈ H be a critical point of A. By the C 2 -convexity of the Lagrangian, we get that ι(x) = 0 on H and in particular ι Z (x) = 0 for every Z ⊂ R n ⊕ R n . By Theorem 1.7 ι LZ (z x ) = c(Z), and the conclusion now follows by using Proposition 1.10.
Remark 1.12. A common Z, often occurring in the applications, is represented by Z := Z 1 ⊕ (0) where Z 1 is a linear subspace of R n . This subspace directly appears in the classical Sturm nonoscillation theorem [Arn86, Section 1].

Sturm Theory and symplectic geometry
The aim of this section is to provide a generalization of the Sturm Alternation and Comparison Theorems proved by Arnol'd in [Arn86] in the case of optical Hamiltonian. The abstract idea behind these results relies on a careful estimates of the Hörmander (four-fold) index which is used for comparing and estimating the difference of the Maslov indices with respect to two different Lagrangian subspaces. Our basic reference for this section is [ZWZ18, Section 3] and references therein. We stress on the fact that, even in the (classical) case of optical Hamiltonians, we provide new and sharper estimates. For the sake of the reader, we refer to Section A for the main definitions and properties of the intersection indices as well as for the basic properties of the Lagrangian Grassmannian Λ(V, ω) of the symplectic space (V, ω).

A generalization of Sturm Alternation Theorem
In the 2n-dimensional symplectic space (V, ω), let us consider λ ∈ C 0 [a, b], Λ(V, ω) and µ 1 , µ 2 ∈ Λ(V, ω). We now define the two non-negative integers k 1 , k 2 given by and we let k := max{k 1 , k 2 }. We are in position to state and to prove the first main result of this section.
Theorem 2.1. Under the previous notation, the following inequality holds: Proof. The proof of this result is a consequence of Proposition A.14, Equation (A.12) and Remark A.11. First of all, we start to observe that For i = 1, 2, we denote by π ǫi (resp. π δi ) the projection onto the symplectic reduction mod ǫ i (resp. δ i ). So, we have , it follows that Q ǫi (resp. Q δi ) are quadratic forms on n − dim ǫ i (resp. n − dim δ i ) vector space. So, the inertia indices are integers between 0 and n − dim ǫ i (resp. n − dim δ i ). In conclusion, we get that By using these inequalities together with Equation (2.1), we get that Putting the inequalities given in Formula 2.1 all together, we get where k = max{k 1 , k 2 }. This concludes the proof.
Remark 2.2. Loosely speaking, by Theorem 2.1, we can conclude that the smaller is the difference of a Lagrangian path with respect to two Lagrangian subspaces the higher is the intersection between them.
Corollary 2.3. Under the notation of Theorem 2.1 and assuming that λ In particular, if λ is a closed path, then we get that Proof. We observe that By this, we get that n − k n − k 1 is less or equal than n − dim I. This concludes the proof of the first claim. The second claim readily follows by observing that for loops of Lagrangian subspaces, we have dim I = n.
Remark 2.4. It is worth noticing that in the case of Lagrangian loops, the ι CLM -index is actually independent on the vertex of the train. This property was already pointed out by Arnol'd in his celebrated paper [Arn67].
Corollary 2.5. Under notation of Theorem 2.1 and if µ 1 ∩ λ(a) = µ 1 ∩ λ(b) = ∅, then we have Proof. We observe that By this, we get that n − k n − k 2 is less or equal than n − dim J.
Remark 2.6. We observe that if the four Lagrangians λ(a), λ(b), µ 1 , µ 2 are mutually transversal, then k = 0. Thus in this case the modulus of the difference of the Maslov indices computed with respect to two (distinguished) Lagrangian is bounded by n.
Remark 2.7. We observe that Corollary 2.3 and Corollary 2.5 are well-known. More precisely Corollary 2.3 agrees with [JP09, Corollary 3.4] and Corollary 2.5 corresponds to [JP09, Proposition 3.3]. As by-product of the previous arguments we get that the inequalities proved by authors in aforementioned paper were not sharp. It is worth noticing that the proof provided by authors is completely different from the one given in the present paper and it mainly relies on a careful estimate of the inertial indices of symmetric bilinear forms obtained by using the atlas of the Lagrangian Grassamannian and its transition functions.
Thus, we have Proof. By taking into account the symplectic invariance of the ι CLM -index, we get and The proof now immediately follows by theorem 2.1.
By restricting Theorem 2.1 to curves of Lagrangian subspaces induced by the evolution of a fixed Lagrangian under the phase flow of a linear Hamiltonian system we get a generalization of the Sturm Alternation Theorem proved by Arnol'd in [Arn86]. More precisely, let us consider the linear where k := max{k 1 , k 2 } and Remark 2.10. We stress on the fact that in the aforementioned paper, Arnol'd proved the Alternation Theorem for the class of quadratic Hamiltonian functions that are optical with respect to the two distinguished Lagrangian subspaces L 1 and L 2 . In the classical formulation, author provides a bound on the difference of non-transversality moments of the evolution of a Lagrangian path with respect to two distinguished Lagrangian subspaces.

Iteration inequalities for periodic boundary conditions
In this section we provide some simple estimates on the Conley-Zehnder index ofwhich can be obtained directly from Theorem 2.1. Given a symplectic space (V, ω), we consider the direct sum V 2 := V ⊕ V , endowed with the symplectic form ω 2 := −ω ⊕ ω, defined as follows for all v 1 , v 2 , w 1 , w 2 ∈ V and we recall that Remark 2.12. We observe that the Conley-Zehnder index was originally defined for symplectic paths having non-degenerate final endpoint meaning that Gr ψ(b) ∩ ∆ = {0}. We emphasize that, for curves having degenerate endpoints with respect to ∆ there are several conventions for how the endpoints contribute to the Maslov index. For other different choices we refer the interested reader to [RS93, LZ00, DDP08] and references therein.

Proof. The proof of this result follows by [RS93, Theorem 3.2] and Equation (A.7)
Theorem 2.14.
Before proving this result, we observe that the maximal dimension of the isotropic subspace ǫ is an even number less or equal than 2n. This is for instance the case in which P = Id.
Proof. By invoking Lemma 2.13, we start to observe that and by Definition 2.11, we know that ι CZ (ψ(t); t ∈ [0, T ]) = ι CLM ∆, Gr ψ(t); t ∈ [0, T ] . Summing up, we get where in the last equality we used Lemma A.13, (I). We observe that ι Gr P, ∆, L ⊕ L 0 is equal to the extended coindex of a quadratic form on a Lagrangian subspace of the reduced space V ǫ := ǫ ω /ǫ (see Equations (A.12)). Thus the sum of all inertial indices is bounded from above by 1/2 dim V ǫ which is equal to 2n − dim ǫ.
Remark 2.15. For an explicit computation of the term ι L ⊕ L, ∆, Gr (P ) , we refer the interested reader to [Por08,FK14] and references therein.
Definition 2.16. Given L ∈ Λ(V, ω), we term the L-Maslov index the integer given by As direct consequence of Theorem 2.14 and Definition 2.16 we get the following.
Lemma 2.17. Under notation of Theorem 2.14, the following inequality holds: Proof. The proof of the first inequality in Equation (2.3) comes directly by Theorem 2.14. The second inequality follows by observing that W ⊇ (L ⊕ L) ∩ ∆ and thus dim W n.
Typically in concrete applications, one is faced with the problem of estimating the difference of the ι CLM -indices of two different Lagrangian curves with respect to a distinguished Lagrangian subspace. These Lagrangian curves are nothing but the evolution under the phase flow of two distinguished Lagrangians.
We set Then for any ω ∈ U, let us consider the hypersurface in Sp(2n) defined as As proved by Long (cf. [Lon02] and references therein), for any M ∈ Sp(2n) 0 ω , we define a co- , we define the concatenation of the two paths as For any n ∈ N, we define the following special path ξ n ∈ P T (2n) as follows where ⋄ denotes the diamond product of matrices. (Cf. [Lon02] for the definition).
Proof. For the proof of this result, we refer the interested reader to [LZ00, Corollary 2.1].
Given L ∈ Λ(V, ω) and ψ ∈ P T (2n), we define the continuous curve ℓ m : By the affine scale invariance of the Maslov index, for any given L ∈ Λ(n), we get By taking into account the additivity property of the Maslov index under concatenations of paths and Lemma 2.13, we infer In particular, if L is P -invariant (namely P L ⊆ L), then we have Proposition 2.20. Let ψ ∈ P T (2n) and m ∈ N. Then Proof. By invoking the Bott type iteration formula given in Equation (2.4), Definition 2.11 and Lemma 2.19, we get For every ω ∈ U, using Lemma A.13, we have Summing up, we finally get Now, for every root of unit ω i , by using analogous arguments as given in the proof of Theorem 2.1, we get that the triple index ι Gr P, ∆, ∆ ωi is equal to the extended coindex of a quadratic form Furthermore, use (A.10), we have ι Gr P, ∆, ∆ ω ≥ dim(∆ ω ∩ Gr (P )). It follows that This concludes the proof.
Remark 2.21. For an analogous estimate, we refer the interested reader to [DDP08, Corollary 3.7, Equation (12)]. We remark that the estimate provided in Proposition 2.20 coincides with the one proved by authors in [Lon02,Equation (19), Theorem 3, pag.213] with completely different methods once observed that ι CZ (ψ(y), t ∈ [0, T ] = i 1 (ψ) + n where i 1 is the index appearing in the aforementioned book of Long.

Optical Hamiltonian and Lagrangian plus curves
This section is devoted to discuss a monotonicity property of the crossing forms for a path of Lagrangian subspaces with respect to a distinguished Lagrangian subspace L 0 ; such a property is usually termed L 0 -positive (respectively L 0 -negative) or L 0 -plus (respectively L 0 -minus) property.
We start with the following definition.
] is a crossing instant, we define the multiplicity of the crossing instant t 0 , the positive integer Remark 3.2. We observe that an analogous definition holds for L 0 -minus curves just by replacing plus by minus.
Remark 3.3. We stress on the fact that the plus condition strongly depends on the train Σ(L 0 ). In fact, as we shall see later, a curve of Lagrangian subspaces could be a plus curve with respect to a train but not with respect to another (or even worse with respect to any other).
Thus for L 0 -plus curves we get the following result.
) be a L 0 -plus curve. Then we have: Proof. We observe that if ℓ is a L 0 -plus curve then Since ℓ is a plus curve, each crossing instant is non-degenerate and in particular isolated. So, on a compact interval are in finite number. We conclude the proof using Equation (A.5).
In this paragraph we provide sufficient conditions on the Hamiltonian function in order the lifted Hamiltonian flow at the Lagrangian Grassmannian level is a plus curve with respect to a distinguished Lagrangian subspace.
On the symplectic space (R 2n , ω 0 ), let H : [0, T ] × R 2n → R be a (smooth) Hamiltonian and let us consider the first order Hamiltonian system given by (ω 0 and J 0 have been introduced at page 6). By linearizing Equation (3.1) along a solution z 0 , we get the system We denote by ψ the fundamental solution of the Hamiltonian system given at Equation (3.2).
Remark 3.5. We observe that if H is quadratic and t-independent, the linear Hamiltonian vector field in Equation (3.2) is t-independent, i.e. B(t) = B. In this particular case, we get ψ(t) = exp(tJ 0 S).
Definition 3.6. Let L 0 , L ∈ Λ(n) and let ℓ : Some important special classes of L 0 -optical Hamiltonians where L 0 is the Dirichlet (resp. Neumann) Lagrangian is represented by Hamiltonian having some convexity properties with respect to the momentum (resp. configuration) variables.
Proposition 3.7. Let H : R 2n → R be a C 2 -convex Hamiltonian and let z 0 be a solution of the Hamiltonian system given in Equation (3.1). Then we get that H with respect to the 1. momentum variables is L D -optical 2. configuration variables is L N -optical.
Proof. We prove only the first statement, being the second completely analogous. Given L ∈ Λ(n), let us consider the Lagrangian curve pointwise defined by ℓ(t) := ψ(t)L. Let t 0 be a crossing instant for ℓ with respect to the Dirichlet Lagrangian L D . By using Equation (A.9) and Equation (3.3), we get that Since H is C 2 convex in the p-variables, it follows that the crossing form Γ given in Equation (3.4) is positive definite. The conclusion now follows by the arbitrarily of t 0 .
Corollary 3.8. Let H : R 2n → R be a C 2 -strictly convex Hamiltonian function and let z 0 be a solution of the Hamiltonian system given in Equation (3.1). Then H is L 0 -optical with respect to every L 0 ∈ Λ(n).
Proof. In fact, since H is C 2 -strictly convex, this in particular implies that B(t) = D 2 H t, z 0 (t) is positive definive and hence every restriction is positive definite. The conclusion now follows directly by using once again Equation (A.9).
Remark 3.9. We consider the Hessian of H along a solution z 0 of the Hamiltonian system given in Equation (3.1), given by Equation (3.3) and we observe that in terms of the block matrices entering in the Hessian of H, the condition for H to be C 2 -strictly convex is equivalent to The equivalence readily follows by the characterization of positive definiteness of a block matrices in terms of the Schur's complement. Thus, in general, if the Lagrangian L given in Definition (3.6) is not in a special position with respect to L D and L N , the opticality property strongly depends upon the all blocks appearing in the Hessian of H.
We are now in position to prove the Sturm non-oscillation theorem.
. Let ψ be the fundamental solution of the linearized system given in Equation Proof. Let x be the critical point (with Dirichlet boundary conditions) of the action functional corresponding to the solution z 0 . Then the Morse index of x is 0, since the (natural) Lagrangian L corresponding to the Hamiltonian H is C 2 convex. In particular by Theorem 1.7, we have Here Z = (0) ⊕ (0), L Z = L D , and by taking into account Remark 1.8 we get that c(Z) = n. Then ι LZ (z 0 ) = n and by Definition 1.6 we have Note that L D ∩ (ψ(0)L D ) = n and the Hamiltonian is L D -optical . By lemma 3.4, we have From Definition A.9 and Proposition A.14 we get By [HWY18, Equation (1.17)], we have where the last equality follows by [ZWZ18, Corollary 3.14]. By equation (A.10) and (3.6), we have We get By this inequality and by Equation (3.5), we get that The thesis follows by observing that in the case of positive curves, it holds that Remark 3.11. It is worth noticing that, in fact Now, since the natural Hamiltonian is C 2 Legendre convex, as direct consequence of Proposition 3.7, we get that the curve t → ℓ 0 (t) is L D -plus and by using Lemma 3.4, the local contribution to the ι CLM -index is through the multiplicity. This concludes the proof.
Remark 3.12. By using the suggestive original Arnol'd language, the Sturm non-oscillation theorem given in Theorem 3.10 could be rephrased by stating that Nonoscillation Theorem. If the potential energy is nonpositive, then the number of moments of verticality does not exceed the number n of degrees of freedom.
The non-positivity of the potential energy implies that the quadratic Lagrangian is strictly positive and hence the Morse index of associated Lagrangian action functional vanished identically.
Theorem 3.13. [Sturm Alternation Theorem for plus-curves] Under the above notation, the following holds: where k := max{k 1 , k 2 } and Proof. The idea of the proof is similar wit h theorem 2.9 but it needs more precise estimate. Note Then by theorem A.14, we get By using Equation (A.10) and Equation (3.7), we get that Moreover, for arbitrary Lagrangian subspaces α, β, γ, we have Then by (3.8) it follows that (3.10) By using Equation (3.9) and Equation (3.10), we get the thesis arguing precisely as given in Theorem 2.9 .
Remark 3.14. We observe that the estimates provided in Theorem 3.13 is, in general, sharper than the one proved by Arnol'd for which the difference was bounded by n. • if ν L 2 , [α, β]) > n − k, then there is at least one crossing instant of ℓ with L 1 ; • if ν L 1 , [α, β]) > n − k, then there is at least one crossing instant of ℓ with L 2 .
Proof. The proof follows immediately by using triangular inequality and Theorem 3.13.

Sturm comparison principles
In this section we provide some new comparison principles as well as a generalization of the classical Sturm comparison principle. Our first result is a generalization of the comparison principle which was proved by third named author in [Off00, Section 5].
Remark 4.2. Before proving this result, we observe that assumption 2. corresponds to require that the triple index is as large as possible. In fact, by assumption 1. the term dim(L 1 ∩ L 2 ∩ L 3 ) drops down. This assumption, somehow replaces the condition on Q(L 1 , L 2 ; L 3 ) to be positive definite in this (maybe degenerate) situation.
A direct consequence of the Theorem 4.1, we get the following result which is in the form appearing in [Off00, Theorem 5.1].
Remark 4.4. Corollary 4.3 provides a generalization of [Off00, Theorem 5.1] which was proved for paths of symplectic matrices arising as fundamental solutions of Hamiltonian systems. Moreover we removed the Legendre convexity condition as well as the transversality condition between the Lagrangian subspaces L 1 and L 2 , which, in concrete applications such a conditions are pretty difficult to be checked.
Theorem 4.5. Under the notation of Theorem 4.1, we assume that Proof. We start to observe that by assumption 3. and assumption 1. we get that ι CLM (L 3 , ℓ 2 (t); t ∈ [a, b]) k.
The last result of this section is a generalized version of the Sturm comparison theorem proved by Arnol'd in the case of optical Hamiltonians. The proof of this result is essentially based on spectral flow techniques and for the sake of the reader we refer to Appendix B for the basic definitions, notation and properties. Now, for i = 1, 2 let us consider the Hamiltonians H i : [0, T ] ⊕ R 2n → R and the induced Hamiltonian systems z ′ (t) = J 0 ∇H i t, z(t) . (4.1) By linearizing Equation (4.1) at a common equilibrium point z 0 , we get where B i (t) = D 2 H i (t, z 0 (t)). For i = 1, 2, we denote by ψ i the fundamental solution of the corresponding linearized Hamiltonian system (4.2). For s ∈ [0, 1], we define the two-parameter family of symmetric matrices as follows Given L ∈ Λ(2n), we denote by D(T, L) the subspace of W 1,2 paths defined by and we define the two parameter family of first order linear operators: It is well-known that for every (s, r) ∈ [0, 1] ⊕ [0, 1], the linear operator A (s,r) is unbounded selfadjoint in L 2 with dense domain D(T, L). We also observe that being the domain independent on (s, r) the linear operator A (s,r) : Theorem 4.6. (First Comparison theorem) Let L ∈ Λ(2n) and under the notation above, we assume Then we get sf(A 2 ) sf(A 1 ) where A 1 := A (0,r) and A 2 := A (1,r) .
Before proving the result, we observe that the assumption (C1) guarantees that the curve s → A (s,r) is a plus-curve.
Proof. The proof of this result is based upon the homotopy invariance of the spectral flow. Let us consider the two parameter family of operators A (s,r) defined above, and we observe that, as direct consequence of the homotopy invariance (since the rectangle R is contractible), we get that We now observe that the first term sf A (s,0) , s ∈ [0, 1] = 0. This follows by the fact that A (s,0) is a fixed operator. Let us now consider the second term in the right-hand side of Equation (4.4), namely sf A (s,1) , s ∈ [0, 1] . By Lemma B.5 we can assume that for δ > 0 sufficiently small the path A δ s := A (s,1) + δ Id where Id denotes the identity on L 2 , has only regular crossings. So, by the homotopy invariance of the spectral flow we get that In order to relate the spectral flow for a path of Hamiltonian operators with the Maslov index of the induced Lagrangian curve, we need to use a spectral flow formula.
Let us now consider the path s → L s of unbounded Hamiltonian operators that are selfadjoint in L 2 and defined on the domain D(T, L) given in Equation (4.3) where s → E s (t) is a C 1 path of symmetric matrices such that E 0 (t) = 0 2n and E 1 (t) = E(t), where we denoted by 0 2n the 2n ⊕ 2n zero matrix. where ψ denotes the solution of Remark 4.8. The basic idea behind the proof of Proposition 4.7 is to perturb the path s → L s in order to get regular crossing (which it is possible as consequence of the fixed endpoints homotopy invariance). Once this has been done, for concluding, it is enough to prove that the local contribution at each crossing instant to the spectral flow is the opposite of the local contribution to the Maslov index. This can be achieved by comparing the crossing forms as in [HS09, Lemma 2.4] and to prove that the crossing instants for the path s → L s are the same as the crossing instants of the path s → Gr ψ s and at each crossing s 0 the kernel dimension of the operator L s0 is equal to the dim(L ∩ Gr ψ s0 ). The conclusion follows once again by using the homotopy properties of the ι CLM -index and the spectral flow.
Theorem 4.9. (Second Comparison theorem) Under the notation above, we assume Then we get Proof. The proof readily follows by Theorem 4.6 and Proposition 4.7.
As direct consequence of Theorem 4.5 we get the following useful result.
Proof. The proof follows as direct application of Theorem 4.9, in the case in which L = ∆ and of [KOP19, Equation (3.8)].
Remark 4.11. An analogous of Corollary 4.10 already appears in [Arn86, Corollary 2 (Oscillation Theorem]. In this result, however, author estimates from below the moments of verticality, namely the Maslov index with respect to the Dirichlet Lagrangian. We also observe that the opposite inequality appearing in Corollary 4.10 with respect to the aforementioned Arnol'd result is due essentially to the fact that in that paper author considered Lagrangian paths ending in the vertex of the train, whereas we are considering Lagrangian paths starting at the vertex of the train.
We close this section with a comparison theorem for Morse-Sturm systems. For i = 1, 2, let us consider the natural quadratic Hamiltonians H i : R 2n → R of the form Let Z ⊂ R n ⊕ R n be a linear subspace, L Z ∈ Λ(2n) be the Lagrangian subspace defined by Equation (1.4) and, for i = 1, 2, we denote by ι Z (B i ) the Morse-index of the index form of the Morse-Sturm system corresponding to B i .

Some applications in geometry and classical mechanics
The aim of this final section is to give some applications in differential geometry and in classical mechanics. Inspired by [JP09] from which we borrow some notation, in Subsection 5.1 we shall prove some comparison results between the conjugate and focal points along a geodesic on semi-Riemannian manifold. In Subsection 5.2 some applications to the planar Kepler problem where provided.

Comparison Theorems in semi-Riemannian geometry
Let (M, g) be semi-Riemannian n-dimensional manifold, and let D be the covariant derivative of the Levi-Civita connection of the metric tensor g. We denote by R the Riemannian curvature tensor, chosen according to the following sign convention R(ξ, η) := [D ξ , D η ] − D [ξ,η] . Given a geodesic γ : [a, b] → M the Jacobi (deviation) equation along γ is given by The Jacobi equation is a linear second order differential equation whose flow Φ defines a family of isomorphisms . It is easy to check that in this way we get a smooth curve ℓ : [a, b] → Λ(V, ω). We set L 0 := ℓ(a) = L a 0 . 1 Now, consider a smooth connected submanifold P of M , with γ(a) ∈ P and γ ′ (a) ∈ T γ(a) P ⊥ (where ⊥ is the orthogonal with respect to g) and we assume that the restriction of g to T γ(a) P is non-degenerate. (This condition is always true if M is either Riemannian or Lorentzian and γ is timelike). Let S be the second fundamental form of P at γ(a) in the normal direction γ ′ (a), seen as a g-symmetric operator S : T γ(a) P → T γ(a) P . An instant t 0 ∈ (a, b] is P -focal if there exists a nonzero P -Jacobi field vanishing at t 0 . The multiplicity of a mechanical P -focal instant is the multiplicity of the P -Jacobi fields vanishing at t 0 . To every submanifold P of M , we associate a Lagrangian subspace L P ⊂ V defined by Proposition 5.2. Under the previous notation, the following inequality holds Remark 5.3. The last inequality appearing in Proposition 5.2 coincide with that one proved by authors in [JP09, Proposition 4.3].
As direct consequence of the triangular inequality and Proposition 5.2, we get the following.
Corollary 5.4. Under the notation of Proposition 5.2, we get that, for any interval The last result of this paragraph is quite useful in the applications. Loosely speaking, claims that the absence of conjugate (respectively focal instants gives an upper bound on the number of focal (respectively conjugate) instants Proposition 5.5. If γ has no conjugate instant, then Proof. If γ has no conjugate instants, then ι CLM L 0 , ℓ(t); t ∈ [a, b] = 0. The result directly follows by applying Proposition5.2. Similarly for the second claim.
Let now consider two smooth connected submanifold P, Q of M , with γ(a) ∈ P ∩ Q and γ ′ (a) ∈ T γ(a) P ⊥ ∩ T γ(a) Q ⊥ (where ⊥ is the orthogonal with respect to g) and we assume that the restriction of g to T γ(a) P and to T γ(a) Q are non-degenerate. We set where S P and S Q denote the shape operators of P and Q, respectively.
Proposition 5.6. Let L be either L P or L Q . Then we have and d := max{dim P, dim Q}.

Simple Mechanical systems and mechanical focal points
This final section is devoted to study the so-called P -kinetic focal and conjugate points in the case of simple mechanical systems and to derive some interesting estimates relating the qualitative and variational behavior of orbits in some singular Lagrangian systems.
In this paragraph we stall by recalling some well-known facts and to fix our notation. The main references are [Sma70a,Sma70b,Pin75] and references therein.
Definition 5.7. Let (M, g) be a finite dimensional Riemannian manifold and V : M → R be a smooth function. The triple (M, g, V ) is called a simple mechanical system. The manifold M is called the configuration space and its tangent bundle T M is usually called the state space. A point in T M is a state of the mechanical system which gives the position and the velocity. The kinetic energy K of the simple mechanical system is the function The smooth function V is called the potential energy (function) of the system and finally the total energy function Notation 5.8. Everywhere in the paper we shall denote by V the potential energy and by U the potential function and we recall that V = −U .
Example 5.9. (The n-body problem) Consider n point masses particles (bodies) with masses m 1 , . . . , m n ∈ R + moving in the d-dimensional Euclidean space E d . So the positions of the bodies is described by the vector q = (q 1 , . . . , q n ) ∈ (E d ) n . The kinetic energy is Clearly the kinetic energy is induced by the Riemannian metric cdot, The n-bodies moves under the influence of the Newtonian potential energy defined by The function V is singular at the collision set defined by ∆ := (q 1 , . . . , q n ) ∈ (E d ) n q i = q j for some i = j .
Then V is a smooth function on M := (E d ) n \∆ thus defining a simple dynamical system (M, K, V ).
Definition 5.10. A physical path (orbit, trajaectory) of a simple mechanical system (M, g, V ) is a smooth path γ in M satisfying the Newton Equation where D/dt denotes the covariant derivative relative of the Levi-Civita connection D of the Riemannian metric g and where ∇ g denotes the gradient defined by g.
Remark 5.11. If V = 0 then the physical path are just geodesics of the Riemannian manifold. Moreover if g is the Euclidean metric, then the left-hand side of Equation (5.3) reduces to γ ′′ and the gradient ∇ g appearing in the right-hands side of that equation is the usual gradient.
By the conservation law of the total energy function along a physical path and since in the Riemannian world the kinetic energy is non-negative 2 a physical path of total energy h ∈ R must lie in the set where M denotes the topological closure of the set usually called the h-configuration space or the Hill's region. If h is a regular value of V , then M is a smooth manifold with boundary The Jacobi metric g corresponding to the value h of a simple mechanical system (M, g, V ) is given by Remark 5.12. We observe that g defines a honest Riemannian metric on M which degenerate on ∂M .
The next result, which relates the physical paths of energy h and the geodesics on the Hill's region with respect to the Jacobi metric, goes back to Jacobi.
Proposition 5.13. (Jacobi) The physical paths of (M, g, V ) of total energy h are, up to time re-parametrization, geodesics of the Riemannian manifold (M, g).
We now consider the configuration space M to be the Euclidean plane E 2 endowed with a polar coordinate system (r, θ). Take the origin to be the center of central force so that the potential energy V of the problem depends only upon r (thus is θ independent). We assume that the particle has mass m = 1 so that the kinetic energy is K(q, v) = v 2 /2 for all v ∈ E 2 . The Jacobi metric of this simple mechanical system in polar coordinates is given by The mechanical Gaussian curvature can be easily computed (cfr. [Pin75, Proposition 2.1]) and it is given by Assuming that h is a regular value of V meaning that V ′ = 0 on the boundary ring then by continuity it readily follows the following result.
Lemma 5.14. [Pin75, Proposition 2.1 & Proposition 2.2] Suppose h is a regular value of V and that the boundary ring ∂M = ∅. Then there is an annulus region of the boundary ∂M on which the mechanical Gaussian curvature is positive. Moreover K(q) → +∞ as q → ∂M .

The planar Kepler problem
In polar coordinates the Jacobi metric for the planar Kepler problem is Remark 5.15. As recently observed by Montgomery in [Mon18, Section 4], in the particular case of zero energy h = 0 it reduces to g 0 = 2 dr 2 r + dθ 2 and by setting ρ = 2r 1/2 it can be written as follows which is the metric of cone over a circle of radius 1/2.
In the standard planar Kepler problem, the mechanical Gaussian curvature is In particular we get that In the two dimensional case the mechanical Jacobi field, reduces to where s denotes the Jacobi arc-length. Since |K| |h|/4, and as a direct consequence of Proposition 4.12, we get the following.
Theorem 5.16. Let γ be a Keplerian ellipse. Then the first conjugate point occurs at Jacobi distance less than 2 π |h| .
Proof. In fact, since |K(s)| |h| 4 , by setting R 1 (s) = |K(s)| and R 2 (s) := |h|Id and by using Proposition 4.12, we get that the associated block diagonal matrices B 1 and B 2 are ordered, meaning that pointwise we have B 1 (s) B 2 (s) for every s ∈ [0, 1]. Thus, by invoking once again Proposition 4.12 and Theorem 1.7, we have Since crossing instants (or a verticality moments) correspond to conjugate points. (Cfr. [MPP05] and references therein for further details), the result follows once observed that |K| |h|/4 and |h|/4 is the Gaussian curvature of the sphere of radius 2/ |h|. This concludes the proof.

A A symplectic excursion on the Maslov index
The purpose of this Section is to provide the basic definitions, properties and symplectic preliminaries used in the paper. We recall the basic definition, the main properties of the intersection number for curves of Lagrangian subspaces with respect to a distinguished one and we fix our notation. Our basic references are [RS93, CLM94, LZ00, MPP05, MPP07, HS09, BJP14, BJP16, PWY19].

A.1 Symplectic preliminaries and the Lagrangian Grassmannian
A finite dimensional (real) symplectic vector space, is a pair (V, ω), where V is a (real, even dimensional) vector space, and ω : V × V → R is an antisymmetric non-degenerate bilinear form on V . A complex structure on the real vector space V is an automorphism J : V → V such that J 2 = −Id. With such a structure V becomes a complex vector space. We denote by Sp(V, ω) the symplectic group of (V, ω) which is the closed Lie subgroup of the general linear group GL(V ) consisting of all isomorphisms that preserve ω. The Lie algebra sp(V, ω) of Sp(V, ω) consists of all endomorphisms X : V → V such that ω(X·, ·) is a symmetric bilinear form on V , i.e. ω(Xv, w) = ω(Xw, v), for all v, w ∈ V . Here and throughout, unless different stated, (V, ω) denotes a 2n-dimensional (real) symplectic space.
We start by recalling some classical definition and notation that we will use throughout the paper. First of all, a (linear) subspace I ⊂ V is termed isotropic if the restriction of ω on I vanishes identically. Now, given an isotropic subspace I of the symplectic Euclidean space (V, ·, · , ω), we shall identify the quotient space I ω /I with the orthogonal complement V I of I in I ω and we call V I the symplectic reduction of V modulo I. Thus, by definition: Notice that if I is isotropic, also JI is isotropic. Moreover V I = V JI . This follows from Equation (A.1) and the orthogonality relations between ω and ⊥ . Moreover We observe that V I is a symplectic space since V I ∩ V ω I = {0}. Thus, we get the symplectic decomposition of V : V = V I ⊕ V ⊥ I . A special class of isotropic subspaces is played by the so-called Lagrangian subspaces. More precisely, a maximal (with respect to the inclusion) isotropic subspace of (V, ω) is termed a Lagrangian subspace. We denote by Λ(V, ω) (or in shorthand notation by Λ) the collection of all Lagrangian subspaces of V . So, if (V, ω) is a 2n-dimensional (real) symplectic space, a Lagrangian subspace of V is an n-dimensional subspace L ⊂ V such that L = L ω where L ω denotes the symplectic orthogonal . We denote by Λ = Λ(V, ω) the Lagrangian Grassmannian of (V, ω), namely the set of all Lagrangian subspaces of (V, ω); thus Λ(V, ω) : Notation A.1. Here and throughout the Lagrangian Grassmannian of the standard symplectic space will be denoted by Λ(n). Moreover, we set and we shall refer to L D as the Dirichlet (or horizontal) Lagrangian subspace whilst to L N as the Neumann (vertical) Lagrangian subspace.
In this subsection we recall some basic facts on the differentiable structure of Λ(V, ω). We start to observe that Λ(V, ω) has the structure of a compact real-analytic submanifold of the Grassmannian of all n-dimensional subspaces of V . Moreover the dimension of Λ(V, ω) is 1 2 n(n+1) and an atlas on Λ is given as follows.
Definition A.2. We term Maslov cycle with vertex at L 0 or train with vertex L 0 (by using Arnol'd terminology [Arn86, Section 2]), the algebraic (stratified) variety defined by The top-stratum Λ 1 (L 0 ) is co-oriented meaning that it has a transverse orientation. To be more precise, for each L ∈ Λ 1 (L 0 ), the path of Lagrangian subspaces (−δ, δ) → e tJ L cross Λ 1 (L 0 ) transversally, and as t increases the path points to the transverse direction. Thus the Maslov cycle is two-sidedly embedded in Λ(V, ω) and, based on the topological properties of the Lagrangian Grassmannian manifold, it is possible to define a fixed endpoints homotopy invariant ι CLM -which is a generalization of the classical notion of Maslov index for paths of Lagrangian subspaces.

A.2 On the CLM-index: definition and computation
Our basic references for this subsection are the beautiful papers [RS93,CLM94,LZ00].
We For further reference we refer the interested reader to [CLM94] and references therein. Following authors in [LZ00, Section 3], and references therein, let us now introduce the notion of crossing form that gives an efficient way for computing the intersection indices in the Lagrangian Grassmannian context.
Let ℓ be a C 1 -curve of Lagrangian subspaces such that ℓ(0) = L andl(0) = L. Now, if W is a fixed Lagrangian subspace transversal to L. For v ∈ L and small enough t, let w(t) ∈ W be such that v + w(t) ∈ ℓ(t). Then the form is independent on the choice of W .
Remark A.6. For the sake of comparison with the results proven in [JP09] we remark that ι CLM (L 0 , ℓ 2 ) can be defined by using the Seifert Van Kampen theorem for groupoids as the unique Z-valued homomorphism that it is locally defined as difference of the coindices as in [JP09, Equation (2-3)]. It is worth noticing that in that respect the local chart we are considering here is the opposite of the one considered in that paper.
A particular interesting situation which often occurs in the applications is the one in which ℓ(t) := ψ(t)L where ψ ∈ C 1 [a, b], Sp(2n) . Usually, in fact, such a ψ is nothing but the fundamental solution of a linear Hamiltonian system.
In this situation, in fact, as direct consequence of Equation (A.3) and Equation (A.6), we get that for such a path Assuming that ψ is the fundamental solution of the linear Hamiltonian system where t → B(t) is a path of symmetric matrices, then by Equation (A.8), we get that Example A.7. In this example we compute the crossing form with respect to the Dirichlet and Neumann Lagrangian for a special curve of Lagrangian subspaces in the symplectic space (R 2n , ω 0 ) by using the fact that for any L ∈ Λ(V, ω), the map δ L : Sp(V, ω) → Λ(V, ω) defined by δ L (A) := AL is a real-analytic fibration. Let L 0 be either the Dirichlet or the Neumann Lagrangian, ℓ : [a, b] → Λ(n) be a smooth curve having a crossing instant with Σ(L 0 ) at the instant t 0 ∈ (a, b). First case: L 0 = L D . We assume that ℓ(t 0 ) is transverse to L N (otherwise it is enough to consider a different Lagrangian decomposition). By the local description of the atlas of the Lagrangian Grassmannian, ℓ(t 0 ) is a graph of a (symmetric) linear map A : R n → R n , namely ℓ(t 0 ) = { (p, q) ∈ R n × R n | q = Ap } and hence There exists ε > 0 sufficiently small and ψ : (t 0 − ε, t 0 + ε) → Sp(2n) with ψ(t 0 ) = Id such that ℓ(t) = ψ(t)ℓ(t 0 ). With respect to the Lagrangian decomposition L D ⊕ L N = R 2n we can write ψ(t) in the block form as follows By an immediate computation, it follows that the crossing form is given by where p ∈ ker A is the unique vector in R n such that ξ = (p, 0). Second case: L 0 = L N . We assume that ℓ(t 0 ) is transverse to L D ; thus in this case, we can assume that ℓ(t 0 ) = { (p, q) ∈ R n × R n | p = Bq } and hence Under the above notation, it follows that the crossing form is given by where q ∈ ker B is the unique vector in R n such that η = (0, q).
Remark A.8. Before closing this section, one more comment on the Maslov intersection index defined by author in the quoted paper. We observe that, for a general Lagrangian path, the (intersection) Maslov index defined by Arnol'd in [Arn86, Section 2] (namely ι Ar ) differ from ι CLM because of the contribution of the endpoints. In the aforementioned paper, author only considered paths of Lagrangian subspaces such that the starting point doesn't belong to the train of a distinguished Lagrangian L 0 whereas the final endpoint coincides with the vertex. However, if we restrict on this particular class of Lagrangian paths and assuming that the Hamiltonian defining these paths through the lifting to the Lagrangian Grassmannian is L 0 -optical, then we have ι CLM (L 0 , ℓ(t); t ∈ [0, T ]) = ι Ar (L 0 , ℓ(t), t ∈ [0, T ]) − n where ι Ar denotes the Maslov index defined in [Arn86, Section 2]. This fact easily follows by observing that the local contribution given by the endpoints to the ι CLM index is through the coindex at the final point and the index of the starting point. We also observe that the Lagrangian paths defined by the evolution of a Lagrangian subspace under the phase flow, have in general, degenerate starting point. Thus, in order to fit with the class of Lagrangian paths defined by Arnol'd it is natural to parametrize the paths in the opposite direction. However, since the contribution at the end points is different, in the definition of ι CLMindex such a re-parametrization changes the Maslov index not only for a sign changing but also for a correction term which depends upon the endpoints. This fact is pretty much put on evidence in the Sturm-type comparison theorems.
We close this section by recalling some useful properties of the ι CLM -index.

A.3 On the triple and Hörmander index
A crucial ingredient which somehow measure the difference of the relative Maslov index with respect to two different Lagrangian subspaces is given by the Hörmader index. Such an index is also related to the difference of the triple index and to its interesting generalization provided recently by the last author and his co-authors in [ZWZ18]. For, we start with the following definition of the Hörmander index. Then the Hörmander index is the integer given by Compare [ZWZ18,Equation (17), pag. 736] once observing that we observe that ι CLM (λ, µ) corresponds to Mas{µ, λ} in the notation of [ZWZ18].
Properties of the Hörmander index. We briefly recall some well-useful properties of the Hörmander index.
The Hörmander index is computable as difference of two indices each one involving three different Lagrangian subspaces. This index is defined in terms of the local chart representation of the atlas of the Lagrangian Grassmannian manifold, given in Equation (A.2).
Authors in [ZWZ18, Lemma 3.2] give a useful property for calculating such a quadratic form.
We observe that if (α, β) is a Lagrangian decomposition of (V, ω) and β ∩ γ = {0} then π reduces to the identity and both terms dim(α ∩ γ) and dim(α ∩ β ∩ γ) drop down. In this way the triple index is nothing different from the the quadratic form Q defining the local chart of the Proof. For the proof, we refer the interested reader to [ZWZ18, Corollary 3.16].
The next result, which is the main result of [ZWZ18], allows to reduce the computation of the Hörmander index to the computation of the triple index.
We are now in position to introduce the spectral flow. Given a C 1 -path L : [a, b] → BF sa (W, H), the spectral flow of L counts the net number of eigenvalues crossing 0. where we denoted byL t0 the derivative of L with respect to the parameter t ∈ [a, b] at the point t 0 . A crossing is called regular, if Γ(L, t 0 ) is non-degenerate. If t 0 is a crossing instant for L, we refer to m(t 0 ) the dimension of ker L t0 . Remark B.3. It is worth noticing that regular crossings are isolated, and hence, on a compact interval are in a finite number.
In the case of regular curve (namely a curve having only regular crossings) we introduce the following Definition. We recall the following well-known result.
Definition B.6. The C 1 -path L : [a, b] ∋ t → L t ∈ BF sa (H) is termed positive or plus path, if at each crossing instant t * the crossing form Γ(L, t * ) is positive definite.
Remark B.7. We observe that in the case of a positive path, each crossing is regular and in particular the total number of crossing instants on a compact interval is finite. Moreover the local contribution at each crossing to the spectral flow is given by the dimension of the intersection. Thus given a positive path L, the spectral flow is given by Remark B.10. We observe that a direct proof of Equation (B.2) can be easily conceived as direct consequence of the homotopy properties of BF sa + (H). Remark B.11. We observe that the definition of spectral flow for bounded selfadjoint Fredholm operators given in Definition B.4 is slightly different from the standard definition given in literature in which only continuity is required on the regularity of the path. (For further details, we refer the interested reader to [RS95,Wat15] and references therein). Actually Definition B.4 represents an efficient way for computing the spectral flow even if it requires more regularity as well as a transversality assumption (the regularity of each crossing instant). However, it is worth to mentioning that, the spectral flow is a fixed endpoints homotopy invariant and for admissible paths (meaning for paths having invertible endpoints) is a free homotopy invariant. By density arguments, we observe that a C 1 -path always exists in any fixed endpoints homotopy class of the original path.
Remark B.12. It is worth noting, as already observed by author in [Wat15], that the spectral flow can be defined in the more general case of continuous paths of closed unbounded selfadjoint Fredholm operators that are continuous with respect to the (metric) gap-topology. However in the special case in which the domain of the operators is fixed, then the closed path of unbounded selfadjoint Fredholm operators can be regarded as a continuous path in BF sa (W, H). Moreover this path is also continuous with respect to the aforementioned gap-metric topology.
The advantage to regard the paths in BF sa (W, H) is that the theory is straightforward as in the bounded case and, clearly, it is sufficient for the applications studied in the present manuscript.