The Essential Self-adjointness of the Wave Operator on Radiative Spacetimes

Abstract We prove the essential self-adjointness of the d’Alembertian $$\square _g$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mo>□</mml:mo> <mml:mi>g</mml:mi> </mml:msub> </mml:math> , allowing a larger class of spacetimes than previously considered, including those that arise from perturbing Minkowski spacetime by gravitational radiation. We emphasize the fact, proven by Taira in related settings, that all tempered distributions u satisfying $$\square _g u = \lambda u +f$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo>□</mml:mo> <mml:mi>g</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> for $$\lambda \in \mathbb {C}\backslash \mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>C</mml:mi> <mml:mo>\</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> and f Schwartz are Schwartz. The proof is fully microlocal and relatively quick given the “de,sc-” machinery recently developed by the third author.