Equivalence of classical and quantum completeness for real principal type operators on the circle

2020/04/16 by Taira, Kouichi · 3 citations
#35S05 #81Q10 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2004.07547

Abstract

In this article, we prove that the completeness of the Hamilton flow and essential self-adjointness are equivalent for real principal type operators on the circle. Moreover, we study spectral properties of these operators. The proof is based on the construction of eigenfunctions with non-real eigenvalues which is well-known in scattering theory. Moreover, the relationship between scattering theory and the essential self-adjointness is explained.

Cited by

• Long-time behavior of resonant time-dependent perturbations of periodic transport equations on ℝ
• Feynman spectral action of the wave operator on asymptotically de Sitter spaces
• Self-adjoint extensions of singular Sturm-Liouville operators on graphs and Weyl's law

Related

• The virial theorem and the method of multipliers in spectral theory
• Essential self-adjointness of real principal type operators
• Essential self-adjointness of semi-bounded operators
• Completeness Relation in Renormalized Quantum Systems
• Asymptotic completeness in dissipative scattering theory