Dirichlet Laplacian and positive shift

PDF p. 6: §2, paragraph defining ΔD; PDF p. 8: opening paragraph

Assume the Axiom of Choice. Let $n\ge 2$, and let $\mathrm{\Theta }\subset {\mathbb{R}}^n$ be a nonempty compact set with smooth boundary and connected exterior $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$. Work in the complex Hilbert space $L^2(\mathrm{\Omega })$ of square-integrable functions modulo equality almost everywhere, with inner product $⟨u,v⟩={\int }_{\mathrm{\Omega }}u\bar{v}dx$.

For $k=1,2$, $H^k(\mathrm{\Omega })$ consists of the $L^2$ classes whose weak derivatives of order at most $k$ are in $L^2$; $H_0^1(\mathrm{\Omega })$ is the closure of $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ in the $H^1$ norm. Define the Dirichlet Laplacian ${\mathrm{\Delta }}_D$ by $D({\mathrm{\Delta }}_D)=H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$ and ${\mathrm{\Delta }}_{D}u={\sum }_{j=1}^n{\mathrm{\partial }}_j^{2}u$, where the derivatives are weak derivatives. Define $A=I-{\mathrm{\Delta }}_D$ on the same domain.

The operator ${\mathrm{\Delta }}_D$ is self-adjoint and nonpositive, and $A$ is self-adjoint with $A\ge I$ in the quadratic-form sense: $⟨Au,u⟩\ge \mathrm{\parallel }u{\mathrm{\parallel }}_2^2$ for $u\in D(A)$. Here self-adjoint means that the densely defined operator equals its Hilbert-space adjoint, including equality of domains. Let $E_A$ be the spectral projection-valued measure of $A$. For real $s$, write $A^s$ for the spectral operator associated with the Borel function $\lambda \mapsto \max (1,\lambda {)}^s$. For real $t$, write $e^{it{\mathrm{\Delta }}_D}$ for the bounded spectral operator associated with $\lambda \mapsto e^{it(1-\lambda )}$. These operators form the strongly continuous unitary Schrödinger group on $L^2(\mathrm{\Omega })$: for every $u\in L^2(\mathrm{\Omega })$ and $t_0\in \mathbb{R}$, ${\lim }_{t\to t_0}\mathrm{\parallel }{e}^{it{\mathrm{\Delta }}_D}u-e^{i{t}_0{\mathrm{\Delta }}_D}u{\mathrm{\parallel }}_2=0$.

Exterior and truncated obstacle domains

Complex integer-order Sobolev notation

Integer-order Sobolev spaces and their norms

Zero-boundary Sobolev space as a norm closure

The Axiom of Choice

Exterior Dirichlet realization, H² domain and form domain

Spectral theorem for unbounded self-adjoint operators (PVM form)

Unbounded Borel functional calculus: domains, products, spectral mapping

Strongly continuous one-parameter unitary group

A self-adjoint operator generates a strongly continuous unitary group

Spectral form domain and core of a semibounded operator

The geometry and smooth-boundary convention are those of Exterior and truncated obstacle domains. The complex weak Sobolev spaces and their norms are fixed by Complex integer-order Sobolev notation and Integer-order Sobolev spaces and their norms; Zero-boundary Sobolev space as a norm closure supplies the test-function closure. In particular, the definition of $H_0^1$ does not itself assert a trace characterization.

Under the stated Choice assumption, Exterior Dirichlet realization, H² domain and form domain applies to this smooth exterior and identifies the displayed domain and action with the self-adjoint Dirichlet realization. It supplies $A\ge I$ and the nonpositivity of ${\mathrm{\Delta }}_D$. Choice includes the countable-choice assumption used in the Sobolev definitions and is also assumed by the realization and spectral suppliers.

The exterior is nonempty and open, so $L^2(\mathrm{\Omega })$ is nonzero. The spectral theorem Spectral theorem for unbounded self-adjoint operators (PVM form) therefore supplies the unique spectral measure of $A$. The spectral calculus Unbounded Borel functional calculus: domains, products, spectral mapping defines the stated operators, with the domain of $A^s$ consisting of those $u$ for which $\int \max (1,\lambda {)}^{2s}d(E_A{)}_u(\lambda )$ is finite. Applying Spectral form domain and core of a semibounded operator with the lower bound $c=1$ shows that $E_A$ is carried on $[1,\mathrm{\infty })$. Thus the Borel multiplier $\max (1,\lambda {)}^s$ agrees with the usual positive power ${\lambda }^s$ on the spectral support. The bounded calculus in that supplier gives the group law and adjoint formula for the exponential functions; their product with their conjugates is one, so each $e^{it{\mathrm{\Delta }}_D}$ is unitary.

Applying math:lem-self-adjoint-operator-generates-a-strongly-continuous-unitary-group to the self-adjoint $A$ on $L^2(\mathrm{\Omega })$ under the stated Choice assumption gives strong continuity of $t\mapsto e^{itA}u$ for every $u\in L^2(\mathrm{\Omega })$. The bounded spectral calculus identifies $e^{it{\mathrm{\Delta }}_D}=e^{it}{e}^{-itA}$, because $e^{it(1-\lambda )}=e^{it}{e}^{-it\lambda }$. Thus for $u\in L^2(\mathrm{\Omega })$ and $t_0\in \mathbb{R}$, $\mathrm{\parallel }{e}^{it{\mathrm{\Delta }}_D}u-e^{i{t}_0{\mathrm{\Delta }}_D}u{\mathrm{\parallel }}_2\le \mathrm{\parallel }(e^{-itA}-e^{-i{t}_{0}A})u{\mathrm{\parallel }}_2+\mathrm{\mid }{e}^{it}-e^{i{t}_0}\mathrm{\mid }\mathrm{\parallel }u{\mathrm{\parallel }}_2⟶0(t\to t_0)\mathrm{.}$ Together with the group law and unitarity already established, this gives a strongly continuous one-parameter unitary group in the sense of Strongly continuous one-parameter unitary group.

The article is cited for its notation and the stated Dirichlet domain. The realization and spectral conclusions above use the independently supplied mathematical items, not the article as proof.