The trajectory operator with hyperbolic limits #
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): there are symmetric S± and c > 0 with
c ‖v‖² ≤ ⟪S± (A± v), v⟫. By the inertia theorem this says exactly that A± are hyperbolic, with
no eigenvalue on the imaginary axis
(ContinuousLinearMap.hasLyapunovForm_iff_forall_eq_zero_of_apply_eq_neg_smul_of_apply_eq_smul).
This file proves that the trajectory operator
D_A u = u' + A u : W^{1,2}(ℝ, E) → L²(ℝ, E)
(TauCeti.WeakDerivStep.trajectoryOperator, for an L∞ class of A) is a Fredholm operator, with
index dim ker D_A - dim ker D_{-A†}. Neither symmetry of A nor any derivative of A is
assumed. A symmetric operator bounded below has the Lyapunov form S = A
(ContinuousLinearMap.hasLyapunovForm_of_isSymmetric), and a limit of symmetric operators is
symmetric (ContinuousLinearMap.hasLyapunovForm_of_tendsto_of_isSymmetric), so this applies in
particular to every continuous family of symmetric operators with invertible limits.
The estimate behind the theorem replaces the energy identity of the symmetric case by the
monotonicity of a quadratic form. For a C¹ family S(s) of symmetric operators and a compactly
supported C¹ path u, integrating the derivative of ⟪S u, u⟫ gives
∫ ⟪S (u' + A u), u⟫ = ∫ ⟪S A u, u⟫ - ½ ∫ ⟪S' u, u⟫
(TauCeti.integral_inner_apply_deriv_add_eq). Interpolating between S- and S+ across a
bounded interval [-R, R], the right side controls ∫ ‖u‖² up to an error supported in
[-R, R], while the left side is at most ‖S‖ ‖u' + A u‖ ‖u‖. This gives the estimate
‖u‖²_{W^{1,2}} ≤ C ‖D_A u‖²_{L²} + C ‖u‖²_{L²(-R, R)}
first on test paths (in TauCeti.Analysis.ODE.EnergyEstimate.Global) and then, by density, on
W^{1,2}(ℝ, E). Restriction to a bounded interval is compact, so Peetre's lemma gives a
closed range. The cokernel is the
orthogonal complement of the range, which lies in the kernel of the formal adjoint
-d/ds + A† (TauCeti.WeakDerivStep.orthogonal_range_trajectoryOperator_le_map_ker). That operator
is -D_{-A†}, where D_{-A†} = d/ds - A†, so it has the same kernel as D_{-A†}. Like the kernel
of D_A, that kernel is finite dimensional because E is
(TauCeti.WeakDerivStep.finiteDimensional_ker_trajectoryOperator); the Lyapunov forms are needed
only for the closed range.
Main declarations #
TauCeti.WeakDerivStep.exists_norm_sq_le_mul_norm_trajectoryOperator_sq_add_of_hasLyapunovForm: the estimate onW^{1,2}(ℝ, E).TauCeti.WeakDerivStep.isClosed_range_trajectoryOperator_of_hasLyapunovForm: the range ofD_Ais closed.TauCeti.WeakDerivStep.finiteDimensional_coker_trajectoryOperator_of_hasLyapunovForm: the cokernel ofD_Ais finite dimensional.TauCeti.WeakDerivStep.isFredholm_trajectoryOperator_of_hasLyapunovForm:D_Ais Fredholm.TauCeti.WeakDerivStep.isFredholm_trajectoryOperator_of_forall_eq_zero: the same conclusion when the limitsA±are hyperbolic, with no purely imaginary eigenvalue.TauCeti.WeakDerivStep.isFredholm_trajectoryOperator_iff: combined with the degenerate-limit results ofTauCeti.Analysis.ODE.TrajectoryOperator.DegenerateLimit, for a continuous family with symmetric limitsD_Ais Fredholm exactly whenA⁻andA⁺are injective.TauCeti.WeakDerivStep.index_trajectoryOperator_of_hasLyapunovForm: its index isdim ker D_A - dim ker D_{-A†}.
References #
- J. Robbin and D. Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc.
27 (1995), 1–33, §3 (the operator
d/ds + A(s)with hyperbolic limits). - M. Schwarz, Morse Homology, Birkhäuser, 1993, Chapter 2 (the Fredholm property of linearized trajectory operators).
- D. McDuff and D. Salamon, J-holomorphic Curves and Symplectic Topology, 2nd ed., Appendix A.1 (estimates with a compact error and the Fredholm criterion).
The trajectory operator on Sobolev paths #
The trajectory estimate for hyperbolic limits on Sobolev paths. Let A be a continuous
family of operators whose limits at ±∞ have Lyapunov forms. Then the trajectory operator
satisfies ‖u‖² ≤ C ‖D_A u‖² + C ‖u‖²_{L²(-R, R)} on W^{1,2}(ℝ, E). The coefficient of the
operator is any L∞ class AL of A.
The range of the trajectory operator is closed for hyperbolic limits. For a continuous
family of operators whose limits at ±∞ have Lyapunov forms, the image of u ↦ u' + A u from
W^{1,2}(ℝ, E) is a closed subspace of L²(ℝ, E).
The cokernel of the trajectory operator is finite dimensional for hyperbolic limits. For a
continuous family of operators whose limits at ±∞ have Lyapunov forms, the quotient of
L²(ℝ, E) by the range of u ↦ u' + A u is finite dimensional.
The trajectory operator with hyperbolic limits is Fredholm. For a continuous family A
of operators on a finite-dimensional real inner product space whose limits at ±∞ have Lyapunov
forms, u ↦ u' + A u is a Fredholm operator from W^{1,2}(ℝ, E) to L²(ℝ, E). Neither symmetry
nor differentiability of A is assumed.
The trajectory operator with hyperbolic limits is Fredholm, hyperbolicity stated through
eigenvalues. For a continuous family A of operators on a finite-dimensional real inner product
space whose limits at ±∞ have no purely imaginary eigenvalue (no ω and v, w, not both zero,
with A± v = -ω • w and A± w = ω • v), u ↦ u' + A u is a Fredholm operator from
W^{1,2}(ℝ, E) to L²(ℝ, E).
The trajectory operator is Fredholm exactly when its limits are invertible. Let A be a
continuous family of operators on a finite-dimensional real inner product space with symmetric
limits A± at ±∞. Then u ↦ u' + A u from W^{1,2}(ℝ, E) to L²(ℝ, E) is Fredholm if and
only if A⁻ and A⁺ are injective. Only the limits need to be symmetric. The coefficient of the
operator is any L∞ class AL of A.
The index of the trajectory operator with hyperbolic limits. Under the hypotheses of
TauCeti.WeakDerivStep.isFredholm_trajectoryOperator_of_hasLyapunovForm, the index of
D_A u = u' + A u is dim ker D_A - dim ker D_{-A†}, where BL is an L∞ class of the pointwise
adjoint A†: the cokernel of D_A is the kernel of the formal adjoint -d/ds + A†. Such a class
exists by TauCeti.WeakDerivStep.exists_ae_eq_adjoint; for a symmetric family, AL itself is
one.