TY - JOUR

T1 - Semi-classical mass asymptotics on stationary spacetimes

AU - Strohmaier, Alexander

AU - Zelditch, Steve

N1 - Funding Information:
Research partially supported by NSF, USA grant DMS-1810747 .
Publisher Copyright:
© 2020

PY - 2021/2

Y1 - 2021/2

N2 - We study the spectrum of a timelike Killing vector field Z acting as a differential operator on the solution space Hm:={u∣(□g+m2)u=0} of the Klein–Gordon equation on a globally hyperbolic stationary spacetime (M,g) with compact Cauchy hypersurface Σ. We endow Hm with a natural inner product, so that [Formula presented] is a self-adjoint operator on Hm with discrete spectrum {λj(m)}. In earlier work, we proved a Weyl law for the number of eigenvalues λj(m) in an interval for fixed mass m. In this sequel, we prove a Weyl law along ‘ladders’ {(m,λj(m)):m∈R+} such that [Formula presented] as m→∞. More precisely, we given an asymptotic formula as m→∞ for the counting function [Formula presented] for C>0. The asymptotics are determined from the dynamics of the Killing flow etZ on the hypersurface N1,ν in the space N1 of mass 1 geodesics γ where 〈γ̇,Z〉=ν. The method is to treat m as a semi-classical parameter h−1 and employ techniques of homogeneous quantization.

AB - We study the spectrum of a timelike Killing vector field Z acting as a differential operator on the solution space Hm:={u∣(□g+m2)u=0} of the Klein–Gordon equation on a globally hyperbolic stationary spacetime (M,g) with compact Cauchy hypersurface Σ. We endow Hm with a natural inner product, so that [Formula presented] is a self-adjoint operator on Hm with discrete spectrum {λj(m)}. In earlier work, we proved a Weyl law for the number of eigenvalues λj(m) in an interval for fixed mass m. In this sequel, we prove a Weyl law along ‘ladders’ {(m,λj(m)):m∈R+} such that [Formula presented] as m→∞. More precisely, we given an asymptotic formula as m→∞ for the counting function [Formula presented] for C>0. The asymptotics are determined from the dynamics of the Killing flow etZ on the hypersurface N1,ν in the space N1 of mass 1 geodesics γ where 〈γ̇,Z〉=ν. The method is to treat m as a semi-classical parameter h−1 and employ techniques of homogeneous quantization.

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U2 - 10.1016/j.indag.2020.08.010

DO - 10.1016/j.indag.2020.08.010

M3 - Article

AN - SCOPUS:85091506751

SN - 0019-3577

VL - 32

SP - 323

EP - 363

JO - Indagationes Mathematicae

JF - Indagationes Mathematicae

IS - 1

ER -