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Forward and backward real-linear propagators

PDF p. 17: §4 opening paragraph, definition of U(t)

Statement

Assume the Axiom of Choice and hypotheses (H) and (W). Thus $Ω$ is the smooth connected exterior of the compact nontrapping obstacle, $T > 0$ , $λ \in R ∖ {0}$ , and $w \in C ([0, T]; H_{D}^{2} (Ω) \cap W^{2, \infty} (Ω))$ is the reference solution specified by (W). Put $V_{1} = 2 λ ∣ w ∣^{2}$ and $V_{2} = λ w^{2}$ . For every $θ \in [- 2, 2]$ , both homogeneous systems $i z_{t} + Δ_{D} z + V_{1} z + V_{2} \overline{z} = 0$ and $i v_{t} + Δ_{D} v + V_{1} v - V_{2} \overline{v} = 0$ are well posed forward and backward on $[0, T]$ in the mild class $C ([0, T]; H_{D}^{θ} (Ω))$ . Specifically, for every initial time $r$ and datum in this space there is a unique solution satisfying the corresponding free-flow Duhamel equation.

Let $S (t, r)$ be the evolution for the first system and let $U (t)$ map the datum at time zero to the solution of the second. These maps are bounded invertible real-linear maps on $H_{D}^{θ} (Ω)$ ; their inverses are the respective reverse-time evolutions. There is a constant $C_{θ} \geq 1$ , independent of $t \in [0, T]$ and the datum, such that $C_{θ}^{- 1} ∥ v_{0} ∥_{H_{D}^{θ}} \leq ∥ U (t) v_{0} ∥_{H_{D}^{θ}} \leq C_{θ} ∥ v_{0} ∥_{H_{D}^{θ}}$ . The Dirichlet condition is encoded by the realization and spectral scale; no classical trace is asserted for arbitrary negative-order data.

Statement references

Exterior and truncated obstacle domains

Reference trajectory hypothesis W

Dirichlet spectral Sobolev scale and duality

Linearized equation and its real-duality adjoint

Continuous and mixed time-space function spaces

The Axiom of Choice

Facts & Assumptions

[F1]

Under Choice, (H), and (W), the Dirichlet spectral space $E = H_{D}^{θ} (Ω)$ is a complex Banach space, and $W (t) = e^{i t Δ_{D}}$ is a strongly continuous unitary group on it Exterior and truncated obstacle domains Reference trajectory hypothesis W Dirichlet spectral Sobolev scale and duality Scale isometries and free Schrödinger evolution The Axiom of Choice.

[F2]

For $j = 1, 2$ , the maps $t \mapsto M_{V_{j} (t)}$ are continuous in operator norm on $E$ , uniformly bounded on $[0, T]$ ; complex conjugation is a bounded real-linear operation on $E$ . At negative orders the multipliers are their pivot-dual extensions Time-dependent potential and conjugation bounds on the scale. The two displayed systems and their signs are those of the linearized pair Linearized equation and its real-duality adjoint.

[F3]

For a strongly continuous bounded group on a complex Banach space regarded as real, a norm-continuous bounded real-linear perturbation has a unique two-parameter evolution solving its oriented Volterra equation. The evolution is real-linear, obeys the cocycle and inverse identities, and has the exponential norm bound stated in Bounded real-linear perturbations preserve solution estimates. Its Countable Choice hypothesis follows from the assumed Axiom of Choice The Axiom of Choice.

Proof

[F1] Under Choice, (H), and (W), the Dirichlet spectral space $E = H_{D}^{θ} (Ω)$ is a complex Banach space, and $W (t) = e^{i t Δ_{D}}$ is a strongly continuous unitary group on it Exterior and truncated obstacle domains Reference trajectory hypothesis W Dirichlet spectral Sobolev scale and duality Scale isometries and free Schrödinger evolution The Axiom of Choice.

[F2] For $j = 1, 2$ , the maps $t \mapsto M_{V_{j} (t)}$ are continuous in operator norm on $E$ , uniformly bounded on $[0, T]$ ; complex conjugation is a bounded real-linear operation on $E$ . At negative orders the multipliers are their pivot-dual extensions Time-dependent potential and conjugation bounds on the scale. The two displayed systems and their signs are those of the linearized pair Linearized equation and its real-duality adjoint.

[F3] For a strongly continuous bounded group on a complex Banach space regarded as real, a norm-continuous bounded real-linear perturbation has a unique two-parameter evolution solving its oriented Volterra equation. The evolution is real-linear, obeys the cocycle and inverse identities, and has the exponential norm bound stated in Bounded real-linear perturbations preserve solution estimates. Its Countable Choice hypothesis follows from the assumed Axiom of Choice The Axiom of Choice.

Given: The hypotheses, $θ \in [- 2, 2]$ , an initial time $r \in [0, T]$ , and a datum $x \in E$ .

Goal: Construct the two evolutions and prove their inverse and uniform norm bounds.

1.1F1F2

Define $P_{+} (t) u = i (V_{1} (t) u + V_{2} (t) \overline{u})$ and $P_{-} (t) u = i (V_{1} (t) u - V_{2} (t) \overline{u})$ . By the multiplier and conjugation bounds, these are bounded real-linear operators on $E$ and are norm-continuous in $t$ ; multiplication by $i$ is bounded on the complex Banach space $E$ . Their operator norms are bounded by a common finite constant $K$ on $[0, T]$ .

2.1F1F2F3step 1.1

Apply the real evolution result separately to $W$ and $P_{+}$ and to $W$ and $P_{-}$ . It gives unique solutions of $u (t) = W (t - r) x + \int_{r}^{t} W (t - ρ) P_{\pm} (ρ) u (ρ) d ρ$ , where the Bochner integral is oriented when $t < r$ . These are precisely the mild formulations of the two stated systems, with the Dirichlet realization encoded in $W$ and $E$ . Their evolution maps are real-linear and bounded, satisfy the cocycle law, and have inverse given by reversing the two time arguments. This proves forward and backward well-posedness in the stated mild class.

3.1F1F3step 1.1step 2.1∎

Since $W$ is unitary, the bound in [F3] gives $∥ S_{\pm} (t, r) ∥ \leq e^{K ∣ t - r ∣}$ for both evolutions. Write $U (t) = S_{-} (t, 0)$ . The inverse identity gives $∥ U (t)^{- 1} ∥ = ∥ S_{-} (0, t) ∥ \leq e^{K T}$ as well as $∥ U (t) ∥ \leq e^{K T}$ . Applying the inverse bound to $U (t) v_{0}$ yields the lower inequality, and the upper inequality follows directly. Thus $C_{θ} = e^{K T}$ proves the asserted uniform equivalence and all conclusions.

Source notes

The article's PDF page 17 attributes the forward and backward well-posedness and the notation $U (t)$ ; page 20 uses $S (t, r)$ for the linearized propagator. The proof here applies the stated multiplier, free-flow and real-linear evolution suppliers. The article citation is attribution, not a proof premise.

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2