Abstract
We study the essential self-adjointness for real principal type differential operators. Unlike the elliptic case, we need geometric conditions even for operators on the Euclidean space with asymptotically constant coefficients, and we prove the essential self-adjointness under the null non-trapping condition.
Highlights
In this paper, we consider formally self-adjoint real principal type operator P = Op(p) on the Euclidean space Rn with n 1, where Op(·) denotes the Weyl quantization
A typical example is the Klein-Gordon operator with variable coefficients, and the propagation of singularities plays an essential role in the proof of the essential self-adjointness
The essential self-adjointness for Klein–Gordon operators on scattering Lorentzian manifolds is proved by Vasy [Vas20] under the same null non-trapping condition
Summary
The study of essential self-adjointness has a long history but mostly on operators of elliptic type (see [RS80, Chapter X and reference therein]). The essential self-adjointness for Klein–Gordon operators on scattering Lorentzian manifolds is proved by Vasy [Vas20] under the same null non-trapping condition. We had independently found a proof of the essential self-adjointness using different method for compactly supported perturbations (we discuss the basic idea in Appendix C). In Subsection 3.2, we show the local smoothness implies an weighted Sobolev estimate, which is sufficient for the proof of the essential self-adjointness. In Appendix C, we give a simplified proof of the essential self-adjointness for the compactly supported perturbation case. In this case the relatively involved argument of Subsection 3.2 is not necessarily
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
