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adjoint approximating sequence assertions boundary conditions boundary values closure compact support compactness theorem completes the proof cone consequence Corollary corresponding cp(x cp(y Cq(R D(AQ data with compact decomposition define denotes dense distribution sense divergence free divergence theorem Doctor of Philosophy equations in free eventually outgoing free space functions with compact gonal graph norm group U(t Hence infinitely differentiable initial data initial value problem iu)m Lax and Phillips Lemma lies in D(A lies in Mq Lipschitz continuous maximal non-positive boundary Maxwell's equations obstacle obtain orthogonal outgoing and incoming previous lemma Proof of Theorem properties prove Lemma rn^x satisfy scattering theory skew-adjoint smooth functions square integrable subspaces Suppose tempered distribution translation representation unitary operators unperturbed group UQ(t value problem vanishes identically wave equation