The kernel of the trajectory operator consists of classical solutions #
Let A : ℝ → End(F) be a continuous family of bounded operators on a real Banach space F. An
element u of the kernel of the trajectory operator D_A u = u' + A u on the weak-derivative
graph over Lᵖ(ℝ, F) is a weak solution of u' = -A u. Since A u is locally integrable, u is
almost everywhere a constant plus a primitive of -A u; that primitive is continuous, so along it
the integrand -A u is continuous, and the primitive is a classical C¹ solution. Conversely,
every classical solution v of v' + A v = 0 with v ∈ Lᵖ comes from the kernel of D_A: its
derivative -A v is again in Lᵖ because A is essentially bounded.
For p = 2 this identifies the kernel of D_A with the space
TauCeti.squareIntegrableSolutions A of initial values v 0 of square-integrable solutions of
v' + A v = 0, by evaluating the continuous representative at time 0; that this is injective is
uniqueness of solutions of linear equations. Neither symmetry of A, finite dimensionality of F
nor any Fredholm property of D_A is used. The resulting formula for the index of D_A as a
difference of dimensions of spaces of square-integrable solutions is
TauCeti.WeakDerivStep.index_trajectoryOperator_eq_finrank_sub_finrank_of_isClosed_range.
Main declarations #
TauCeti.squareIntegrableSolutions: the initial values of the square-integrable solutions ofv' + A(s) v = 0.TauCeti.squareIntegrableSolutions_neg_smul_id_eq_bot: a scalar equationv' = b vhas no nonzero square-integrable solution when its scalar solutiongis not square integrable.TauCeti.hasDerivAt_exp_smul_sub_smul_id: multiplication byexp wturns solutions ofv' + A v = 0into solutions ofu' + (A - w') u = 0.TauCeti.squareIntegrableSolutions_sub_smul_id_le: for a bounded-below weightw, the square-integrable solutions ofu' + (A - w') u = 0have initial values among those ofv' + A v = 0.TauCeti.WeakDerivStep.exists_hasDerivAt_of_trajectoryOperator_eq_zero: an element of the kernel ofD_Ahas a classical solution ofv' = -A vas representative.TauCeti.WeakDerivStep.exists_trajectoryOperator_ae_eq_of_hasDerivAt: anLᵖclassical solution ofv' = g - A v, withg ∈ Lᵖ, is the value of an elementuwithD_A u = g.TauCeti.WeakDerivStep.exists_trajectoryOperator_eq_zero_of_hasDerivAt: anLᵖclassical solution ofv' = -A vis the value of an element of the kernel ofD_A.TauCeti.WeakDerivStep.kerTrajectoryOperatorEquiv: the kernel ofD_AonW^{1,2}(ℝ, F)is linearly equivalent toTauCeti.squareIntegrableSolutions A.TauCeti.WeakDerivStep.finrank_ker_trajectoryOperator: the kernel ofD_AonW^{1,2}(ℝ, F)has the dimension ofTauCeti.squareIntegrableSolutions A.TauCeti.WeakDerivStep.finiteDimensional_ker_trajectoryOperator: that kernel is finite dimensional whenFis.TauCeti.WeakDerivStep.ker_trajectoryOperator_eq_bot_iff:D_Ais injective onW^{1,2}(ℝ, F)exactly whenv' + A v = 0has no nonzero square-integrable solution.
References #
- M. Schwarz, Morse Homology, Birkhäuser, 1993, Chapter 2 (the kernel and cokernel of the linearized trajectory operator as spaces of decaying solutions).
- J. Robbin and D. Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc. 27 (1995), 1--33.
The initial values at time 0 of the square-integrable solutions v : ℝ → F of the linear
equation v' + A(s) v = 0 on the whole line. For the coefficient of a trajectory operator
D_A = d/ds + A, this is a model of the kernel of D_A
(TauCeti.WeakDerivStep.kerTrajectoryOperatorEquiv).
Instances For
A vector lies in TauCeti.squareIntegrableSolutions A when it is the value at time 0 of a
square-integrable solution of v' + A v = 0.
If the scalar equation g' = b g has a solution with g 0 = 1 that is not square
integrable, then v' = b v has no nonzero square-integrable solution in F: every solution is
g(s) • v 0.
Conjugation by an exponential weight. If u' = -T u at s and w has derivative w'
at s, then exp w • u solves the shifted equation (exp w • u)' = -(T - w' • id) (exp w • u) at
s.
Weighted solutions are square-integrable solutions. Let w have derivative w' and be
bounded below. If u is a square-integrable solution of u' + (A - w') u = 0, then
v = exp (w 0 - w) • u is a square-integrable solution of v' + A v = 0 with the same initial
value. In terms of the trajectory operator, the kernel of D_A on the weighted space
{u | exp w • u ∈ W^{1,2}} is contained in its kernel on W^{1,2}.
Elements of the kernel are classical solutions. If A is continuous and u lies in the
kernel of the trajectory operator D_A u = u' + A u on the weak-derivative graph over Lᵖ(ℝ, F),
then the value of u agrees almost everywhere with a classical solution v of v' = -A v.
Classical solutions give elements of the weak-derivative graph. If v is a classical
solution of v' = g - A v with v and g in Lᵖ(ℝ, F), then v is the value of an element u
of the weak-derivative graph over Lᵖ(ℝ, F) with D_A u = g. No continuity of A is needed: the
derivative g - A v lies in Lᵖ because A is essentially bounded.
Classical solutions lie in the kernel. If v is a classical solution of v' = -A v that
lies in Lᵖ(ℝ, F), then v is the value of an element u of the weak-derivative graph over
Lᵖ(ℝ, F) with D_A u = 0.
The kernel of the trajectory operator is the space of square-integrable solutions. For a
continuous family A of bounded operators on a real Banach space F, with L∞ class AL,
evaluating the continuous representative at time 0 is a linear equivalence from the kernel of
D_A u = u' + A u on W^{1,2}(ℝ, F) to the initial values of the square-integrable solutions of
v' + A v = 0.
Instances For
The equivalence kerTrajectoryOperatorEquiv evaluates at time 0 any continuous
representative of the value of a kernel element.
The kernel of D_A u = u' + A u on W^{1,2}(ℝ, F) has the dimension of the space of
square-integrable solutions of v' + A v = 0.
The kernel of the trajectory operator D_A u = u' + A u on W^{1,2}(ℝ, F) is finite
dimensional when F is: it embeds into the initial values of the solutions of v' + A v = 0.
Injectivity of the trajectory operator. For a continuous family A of bounded operators on
a real Banach space F, the trajectory operator D_A u = u' + A u on W^{1,2}(ℝ, F) is injective
exactly when v' + A v = 0 has no nonzero square-integrable solution.