Operators with Periodic Hamiltonian Flows in Domains with the Boundary

TL;DR

This paper derives sharp spectral asymptotics for operators with periodic Hamiltonian flows in bounded domains, addressing complexities introduced by boundary reflections and refractions.

Contribution

It extends spectral asymptotics to operators with complex boundary interactions and non-Weyl corrections, even for non-second-order operators.

Findings

Derived sharp spectral asymptotics with non-Weyl correction

Analyzed Hamiltonian flow branching at boundaries

Addressed periodicity in complex boundary conditions

Abstract

We consider operators in the domains with the boundaries and derive sharp spectral asymptotics (containing non-Weyl correction) in the case when Hamiltonian flow is periodic. Even if operator is scalar but not second order (or even second-order but there is an inner boundary with both refraction and reflection present) Hamiltonian flow is branching on the boundary and the notion of periodicity becomes more complicated.