The index of the trajectory operator #
Let A : ℝ → End(E) be a continuous family of operators on a finite-dimensional real inner
product space E, with limits A± at ±∞ that have Lyapunov forms
(ContinuousLinearMap.HasLyapunovForm), so that the trajectory operator
D_A u = u' + A u : W^{1,2}(ℝ, E) → L²(ℝ, E) is Fredholm
(TauCeti.WeakDerivStep.isFredholm_trajectoryOperator_of_hasLyapunovForm). Invertible symmetric
limits have Lyapunov forms (ContinuousLinearMap.hasLyapunovForm_of_isSymmetric). This file
proves:
- the index of
D_Adepends only on the limitsA±: two such families with the same limits give operators of the same index; - the index vanishes when the two limits agree and are invertible and symmetric;
- the index is the dimension of the space of square-integrable solutions of
v' = -A vminus that ofv' = A v, and this holds for every continuous symmetricAfor whichD_Ahas closed range; - for invertible symmetric limits, the index is
n⁻(A⁻) - n⁻(A⁺), wheren⁻(B)is the number of negative eigenvalues ofBcounted with multiplicity, that is, the dimension of the sum of its eigenspaces at negative eigenvalues; - for arbitrary limits with symmetric Lyapunov forms
S±, the index isn⁻(S⁻) - n⁻(S⁺). In particular two symmetric Lyapunov forms of the same operator have the same negative inertia.
For the first statement, the affine family (1 - t) A + t B has the same limits as A and B for
every real t, so each of its trajectory operators is Fredholm. The trajectory operator depends
continuously on its L∞ coefficient (TauCeti.WeakDerivStep.lipschitzWith_trajectoryOperator),
so the index is constant along this family. For the second, a constant coefficient A₀ with a
spectral gap gives an injective D_{A₀} and an injective formal adjoint D_{-A₀}, by the
coercive estimate TauCeti.WeakDerivStep.ker_trajectoryOperator_eq_bot, so D_{A₀} is bijective
and has index zero; a family with equal limits A₀ has the index of the constant family.
The first statement is the invariance of the index under deformations of the coefficient family
that fix its limits; the second is its value when the limits agree. Together they reduce the
computation of the index for arbitrary limits to that of one family for each pair of limits. The
third statement is the form that computation takes: the kernel of D_A is the space of
square-integrable solutions of v' + A v = 0 (TauCeti.WeakDerivStep.kerTrajectoryOperatorEquiv),
and the cokernel is the kernel of the formal adjoint D_{-A}.
For the fourth statement we use the model family ψ(-s) A⁻ + ψ(s) A⁺, where ψ is Mathlib's
Real.smoothTransition. On [0, ∞) it is ψ(s) A⁺, a scalar multiple of one symmetric operator,
so a solution of v' = -ψ(s) A⁺ v multiplies the coordinate of v along an eigenvector of A⁺
with eigenvalue λ by exp (-λ ∫₀ˢ ψ). It is square integrable on (0, ∞) exactly when v 0
lies in the positive spectral subspace W of A⁺. Likewise it is square integrable on (-∞, 0)
exactly when v 0 lies in the negative spectral subspace U of A⁻. So the kernel of the model
operator is W ⊓ U, and the same argument for -A identifies its cokernel with Wᗮ ⊓ Uᗮ,
because the positive and negative spectral subspaces of an invertible symmetric operator are
orthogonal complements. Hence the index is dim (W ⊓ U) - dim (W ⊔ U)ᗮ = dim W + dim U - dim E,
which is n⁻(A⁻) - n⁻(A⁺).
For the fifth statement, the index is also constant along (1 - t) A + t B when the limits move,
as long as (1 - t) A± + t B± keeps a Lyapunov form for t ∈ [0, 1]. A symmetric Lyapunov form
S of A is a Lyapunov form of every (1 - t) A + t S with 0 ≤ t ≤ 1
(ContinuousLinearMap.hasLyapunovForm_one_sub_smul_add_smul), so the limits A± can be deformed to
the invertible symmetric operators S±, where the fourth statement applies. For a constant family,
computing the index with the forms S and S' at the two ends shows n⁻(S) = n⁻(S').
Main declarations #
TauCeti.WeakDerivStep.index_trajectoryOperator_eq_of_tendsto: families with the same limits have trajectory operators of the same index.TauCeti.WeakDerivStep.bijective_trajectoryOperator_const: a constant coefficient with a spectral gap gives a bijective trajectory operator.TauCeti.WeakDerivStep.index_trajectoryOperator_eq_zero_of_tendsto: a family with equal limits at both ends gives index zero.TauCeti.WeakDerivStep.index_trajectoryOperator_eq_finrank_sub_finrank_of_isClosed_range: when the range ofD_Ais closed, the index ofD_Ais the dimension of the square-integrable solutions ofv' = -A vminus that ofv' = A v.TauCeti.WeakDerivStep.index_trajectoryOperator_eq_finrank_sub_finrank: the same formula for a continuous symmetric family with invertible limits.TauCeti.WeakDerivStep.index_trajectoryOperator_eq_finrank_pointSpectralSubspace_sub_finrank: the index is the number of negative eigenvalues ofA⁻minus that ofA⁺.TauCeti.WeakDerivStep.index_trajectoryOperator_eq_of_forall_hasLyapunovForm: the index is constant along an affine deformation whose limits keep Lyapunov forms.index_trajectoryOperator_eq_finrank_pointSpectralSubspace_sub_finrank_of_forall_le_inner, inTauCeti.WeakDerivStep: for limits with symmetric Lyapunov formsS±, the index isn⁻(S⁻) - n⁻(S⁺).ContinuousLinearMap.finrank_pointSpectralSubspace_Iio_eq_of_forall_le_inner: two symmetric Lyapunov forms of one operator have the same number of negative eigenvalues.
References #
- D. McDuff and D. Salamon, J-holomorphic Curves and Symplectic Topology, 2nd ed., Appendix A.1 (stability of the Fredholm index).
- J. Robbin and D. Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc.
27 (1995), 1--33 (the index of
d/ds + A(s)as a spectral flow, and its homotopy invariance). - A. Ostrowski and H. Schneider, Some theorems on the inertia of general matrices, J. Math. Anal. Appl. 4 (1962), 72–84 (the inertia of Lyapunov forms).
- M. Schwarz, Morse Homology, Chapter 2 (the index of linearized trajectory operators).
Homotopy invariance of the index through hyperbolic limits. Let A and B be continuous
families of operators on a finite-dimensional real inner product space with limits A± and B±
at ±∞, such that (1 - t) A± + t B± has a Lyapunov form for every t ∈ [0, 1]. Then the
trajectory operators u ↦ u' + A u and u ↦ u' + B u from W^{1,2}(ℝ, E) to L²(ℝ, E) have
the same Fredholm index. Unlike TauCeti.WeakDerivStep.index_trajectoryOperator_eq_of_tendsto,
the limits may move.
The index of the trajectory operator depends only on the limits. Let A and B be
continuous families of operators on a finite-dimensional real inner product space with the same
limits A± at ±∞, which have Lyapunov forms. Then the trajectory operators u ↦ u' + A u and
u ↦ u' + B u from W^{1,2}(ℝ, E) to L²(ℝ, E) have the same Fredholm index.
A constant invertible coefficient gives a bijective trajectory operator. For a symmetric
operator A₀ on a finite-dimensional real inner product space with α ‖v‖ ≤ ‖A₀ v‖ for some
α > 0, the trajectory operator u ↦ u' + A₀ u from W^{1,2}(ℝ, E) to L²(ℝ, E) is bijective.
Both it and its formal adjoint -d/ds + A₀ are injective, and its range is closed.
Equal limits give index zero. Let A be a continuous family of operators on a
finite-dimensional real inner product space with the same invertible symmetric limit A₀ at -∞
and at +∞. Then the trajectory operator u ↦ u' + A u from W^{1,2}(ℝ, E) to L²(ℝ, E) has
index zero.
The index of a trajectory operator with closed range counts square-integrable solutions.
For a continuous family A of symmetric operators on a finite-dimensional real inner product
space E such that the range of D_A u = u' + A u from W^{1,2}(ℝ, E) to L²(ℝ, E) is closed,
the index of D_A is the dimension of the space of square-integrable solutions of v' = -A v
minus the dimension of the space of square-integrable solutions of v' = A v.
The index of the trajectory operator counts square-integrable solutions. For a continuous
family A of symmetric operators on a finite-dimensional real inner product space with invertible
limits at both ends, the index of D_A u = u' + A u from W^{1,2}(ℝ, E) to L²(ℝ, E) is the
dimension of the space of square-integrable solutions of v' = -A v minus the dimension of the
space of square-integrable solutions of v' = A v. This specializes
TauCeti.WeakDerivStep.index_trajectoryOperator_eq_finrank_sub_finrank_of_isClosed_range.
The index as a difference of Morse indices #
The index of the trajectory operator is a difference of Morse indices. Let A be a
continuous family of operators on a finite-dimensional real inner product space with invertible
symmetric limits A± at ±∞. Then the index of the
trajectory operator u ↦ u' + A u from W^{1,2}(ℝ, E) to L²(ℝ, E) is the number of negative
eigenvalues of A⁻ minus the number of negative eigenvalues of A⁺, counted with multiplicity:
the dimensions of the sums of their eigenspaces at negative eigenvalues.
The index of the trajectory operator with hyperbolic limits is a difference of inertia
indices of Lyapunov forms. Let A be a continuous family of operators on a finite-dimensional
real inner product space whose limits A± at ±∞ have symmetric Lyapunov forms S±:
c± ‖v‖² ≤ ⟪S± (A± v), v⟫ with c± > 0. Then the index of the trajectory operator
u ↦ u' + A u from W^{1,2}(ℝ, E) to L²(ℝ, E) is the number of negative eigenvalues of S⁻
minus the number of negative eigenvalues of S⁺, counted with multiplicity. Neither A nor its
limits need to be symmetric; for invertible symmetric limits one may take S± = A±, which gives
TauCeti.WeakDerivStep.index_trajectoryOperator_eq_finrank_pointSpectralSubspace_sub_finrank.
The negative inertia of a Lyapunov form depends only on the operator. If S and S' are
symmetric Lyapunov forms of the same operator A on a finite-dimensional real inner product space,
c ‖v‖² ≤ ⟪S (A v), v⟫ and c' ‖v‖² ≤ ⟪S' (A v), v⟫ with c, c' > 0, then S and S' have the
same number of negative eigenvalues, counted with multiplicity. By the inertia theorem of Ostrowski
and Schneider, which is not formalized here, this number is the dimension of the sum of the
generalized eigenspaces of A at eigenvalues of negative real part.