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Locally constant invisible spaces are Laplacian invariant

PDF p. 11: Lemma 3.4 proof (ii)

Statement

Assume the Axiom of Choice and hypothesis (H): $n \geq 2$ , the nonempty compact obstacle $Θ \subset R^{n}$ has smooth boundary, its exterior $Ω = R^{n} ∖ Θ$ is connected, and $Θ$ is non-trapping in the generalized-ray sense. Choose $R > 0$ with $Θ \subset B_{R} (0)$ and fix a real-valued $φ \in C^{\infty} (R^{n}; [0, 1])$ such that $φ (x) = 0$ for $∣ x ∣ \leq R$ , $φ (x) > 0$ for $∣ x ∣ > R$ , and $φ (x) = 1$ for $∣ x ∣ \geq R + 1$ . Let $T_{0} > 0$ and $σ \in [- 2, 2)$ . Let $H_{D}^{σ} (Ω)$ be the Dirichlet spectral scale, $W (t) = e^{i t Δ_{D}}$ its evolution, and $N_{T}^{σ}$ the space of initial data whose observation $φ W (t) v$ vanishes in $H_{D}^{σ} (Ω)$ for every $t \in [0, T]$ , using this fixed $φ$ .

There exist $T_{0} / 2 \leq a < b \leq T_{0}$ such that $N_{T}^{σ}$ is the same finite-dimensional space $N$ for every $T \in [a, b]$ . For every $h \in [0, b - a]$ , $W (h) N \subseteq N$ . Identifying $N$ with its image in $H_{D}^{σ - 2} (Ω)$ under the canonical injective scale embedding, one has $Δ_{D} N \subseteq N$ , where $Δ_{D} : H_{D}^{σ} \to H_{D}^{σ - 2}$ is the bounded scale extension.

Statement references

Non-trapping obstacle

Control region and nested real cutoffs

Dirichlet spectral Sobolev scale and duality

Scale isometries and free Schrödinger evolution

Invisible initial-data spaces

Facts & Assumptions

[F1]

For the fixed $R$ and $φ$ in the statement, choose an auxiliary real cutoff $ψ \in C_{c}^{\infty} (R^{n}; [0, 1])$ equal to one on $B_{R + 2} (0)$ and supported in $B_{R + 3} (0)$ , as supplied by Control region and nested real cutoffs. The pair $φ, ψ$ satisfies the cutoff hypotheses of Finite dimensionality of invisible spaces. Thus under Choice and (H), every $N_{T}^{σ}$ formed with this fixed $φ$ is finite-dimensional for $T > 0$ and $σ \in [- 2, 2)$ . For $T^{'} \geq T$ , the definition gives $N_{T^{'}}^{σ} \subseteq N_{T}^{σ}$ . def-se-invisible-space

[F2]

The scale is the completed spectral scale of $I - Δ_{D}$ . Its canonical embedding $H_{D}^{σ} \to H_{D}^{σ - 2}$ is continuous and injective, and both levels are complex Hilbert spaces. Dirichlet spectral Sobolev scale and duality Spectral cutoff cores and unitary identifications of completed scales

[F3]

The operators $W (t)$ form a group on $H_{D}^{σ}$ . For every $v \in H_{D}^{σ}$ , $t \mapsto W (t) v$ is continuously differentiable in $H_{D}^{σ - 2}$ with derivative $i Δ_{D} W (t) v$ ; $Δ_{D}$ is the bounded scale extension specified in the statement. Scale isometries and free Schrödinger evolution

[F4]

A finite-dimensional complex linear subspace of a complex normed space is closed for the ambient norm. A finite-dimensional normed subspace is closed

[F5]

The geometry and non-trapping convention in (H) are those of Exterior and truncated obstacle domains and Non-trapping obstacle. Full Choice is retained as an assumption of the scale and finite-dimensionality suppliers. The Axiom of Choice

Proof

[F1] For the fixed $R$ and $φ$ in the statement, choose an auxiliary real cutoff $ψ \in C_{c}^{\infty} (R^{n}; [0, 1])$ equal to one on $B_{R + 2} (0)$ and supported in $B_{R + 3} (0)$ , as supplied by Control region and nested real cutoffs. The pair $φ, ψ$ satisfies the cutoff hypotheses of Finite dimensionality of invisible spaces. Thus under Choice and (H), every $N_{T}^{σ}$ formed with this fixed $φ$ is finite-dimensional for $T > 0$ and $σ \in [- 2, 2)$ . For $T^{'} \geq T$ , the definition gives $N_{T^{'}}^{σ} \subseteq N_{T}^{σ}$ . def-se-invisible-space

[F2] The scale is the completed spectral scale of $I - Δ_{D}$ . Its canonical embedding $H_{D}^{σ} \to H_{D}^{σ - 2}$ is continuous and injective, and both levels are complex Hilbert spaces. Dirichlet spectral Sobolev scale and duality Spectral cutoff cores and unitary identifications of completed scales

[F3] The operators $W (t)$ form a group on $H_{D}^{σ}$ . For every $v \in H_{D}^{σ}$ , $t \mapsto W (t) v$ is continuously differentiable in $H_{D}^{σ - 2}$ with derivative $i Δ_{D} W (t) v$ ; $Δ_{D}$ is the bounded scale extension specified in the statement. Scale isometries and free Schrödinger evolution

[F4] A finite-dimensional complex linear subspace of a complex normed space is closed for the ambient norm. A finite-dimensional normed subspace is closed

[F5] The geometry and non-trapping convention in (H) are those of Exterior and truncated obstacle domains and Non-trapping obstacle. Full Choice is retained as an assumption of the scale and finite-dimensionality suppliers. The Axiom of Choice

Given: Choice, (H), the fixed $R$ and $φ$ , $T_{0} > 0$ and $σ \in [- 2, 2)$ as in the statement.

Goal: Find a constant interval for the invisible spaces and prove the stated flow and Laplacian invariance.

1.1F1given

Each dimension $d (T) = \dim N_{T}^{σ}$ is a nonnegative integer. The nonempty set of these integers for $T \in [T_{0} / 2, T_{0})$ has a least element; choose $a$ in that interval attaining it and choose $b$ with $a < b < T_{0}$ . For every $T \in [a, b]$ , nesting gives $N_{T}^{σ} \subseteq N_{a}^{σ}$ and hence $d (T) \leq d (a)$ , while minimality gives $d (T) \geq d (a)$ . A subspace of a finite-dimensional space with the same dimension is the whole space. Thus all these spaces equal $N : = N_{a}^{σ}$ , which is finite-dimensional.

2.1F1F3step 1.1

Let $v \in N$ and $0 \leq h \leq b - a$ . Since $v \in N_{b}^{σ}$ , its observation vanishes at every time in $[0, b]$ . For $t \in [0, a]$ , the group law gives $φ W (t) W (h) v = φ W (t + h) v = 0$ , because $t + h \in [0, b]$ . Therefore $W (h) v \in N_{a}^{σ} = N$ .

3.1F2F3F4step 1.1step 2.1

Regard $N$ as a complex linear subspace of $H_{D}^{σ - 2}$ through the injective embedding. Its finite basis remains a basis of its image, so it is closed in that lower-scale norm. For $0 < h \leq b - a$ , both $W (h) v$ and $v$ belong to $N$ , and hence so does $(W (h) v - v) / h$ . Differentiability at zero in $H_{D}^{σ - 2}$ makes these quotients converge to $i Δ_{D} v$ . Closedness gives $i Δ_{D} v \in N$ .

4.1step 3.1∎

Since $N$ is a complex linear space, multiplication by $- i$ shows $Δ_{D} v \in N$ . As $v$ was arbitrary, $Δ_{D} N \subseteq N$ in the stated lower-scale identification. This and steps 1.1 and 2.1 prove the conclusion.

Source notes

The original article is cited for attribution of the constant-dimension and invariance argument in the proof of Lemma 3.4 on PDF page 11. The argument above uses the stated mathematical suppliers; the article's conclusion is not used as its own premise.

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2