Invisible initial-data spaces

PDF p. 11: Step 3, definition of NσT and nesting statements

Assume the Axiom of Choice. Let $\mathrm{\Omega }$ be the smooth exterior domain, $H_D^{\sigma }(\mathrm{\Omega })$ its Dirichlet spectral scale, $W(t)=e^{it{\mathrm{\Delta }}_D}$ the corresponding evolution, and $\phi$ the real observation cutoff. For $T>0$ and $-2\le \sigma \le 2$, define the invisible initial-data space by $N_T^{\sigma }=\{v_0\in H_D^{\sigma }(\mathrm{\Omega }):\phi W(t)v_0=0\text{ in }{H}_D^{\sigma }(\mathrm{\Omega })\text{ for every }t\in [0,T]\}\mathrm{.}$ For negative $\sigma$, multiplication is the bounded extension defined through the pivot antidual pairing. The space $N_T^{\sigma }$ is a closed complex linear subspace of $H_D^{\sigma }(\mathrm{\Omega })$. If $T^{\mathrm{\prime }}\ge T$, then $N_{T^{\mathrm{\prime }}}^{\sigma }\subseteq N_T^{\sigma }$; if $-2\le \sigma \le {\sigma }^{\mathrm{\prime }}\le 2$, then $N_T^{{\sigma }^{\mathrm{\prime }}}\subseteq N_T^{\sigma }$ under the canonical scale embedding.

Dirichlet spectral Sobolev scale and duality

Scale isometries and free Schrödinger evolution

Control region and nested real cutoffs

Endpoint multiplier bounds extended to negative spectral orders

Spectral cutoff cores and unitary identifications of completed scales

The Axiom of Choice

Linear subspace of a vector space

The scale and its pivot antidual convention are specified by Dirichlet spectral Sobolev scale and duality, and the assumed Axiom of Choice is the hypothesis used by the scale and flow suppliers The Axiom of Choice. The cutoff from Control region and nested real cutoffs is smooth and constant outside a ball, so it and its derivatives through order two are bounded. The multiplier result Endpoint multiplier bounds extended to negative spectral orders therefore gives a bounded complex-linear multiplication operator on every $H_D^{\sigma }$ in the stated range, including the pivot-antidual extension at negative orders. The evolution Scale isometries and free Schrödinger evolution is bounded and complex-linear on each such space. Consequently $\phi W(t)$ is a bounded complex-linear operator for each fixed $t$, and $N_T^{\sigma }$ is the intersection over $t\in [0,T]$ of its closed kernels. It is therefore a closed complex linear subspace. The longer-time inclusion follows directly because $[0,T]\subseteq [0,T^{\mathrm{\prime }}]$.

For ${\sigma }^{\mathrm{\prime }}\ge \sigma$, Spectral cutoff cores and unitary identifications of completed scales gives a continuous injective embedding from the higher to the lower scale, extending the identity on the common spectral core. The evolution is consistent across the scale by Scale isometries and free Schrödinger evolution. The multiplier actions are also consistent: they agree with ordinary multiplication on the common core, and boundedness on both levels and density of the core in the higher level extend that agreement to all of $H_D^{{\sigma }^{\mathrm{\prime }}}$. Thus an element whose observation vanishes at the higher level also has zero observation at the lower level, proving the asserted index nesting.

The article is cited for the definition and the two nesting assertions on PDF page 11. The justification uses the independent scale, flow and multiplier suppliers listed above; the article is attribution rather than proof of these properties.