Details
| Original language | English |
|---|---|
| Pages (from-to) | 323-363 |
| Number of pages | 41 |
| Journal | Indagationes mathematicae |
| Volume | 32 |
| Issue number | 1 |
| Early online date | 6 Sept 2020 |
| Publication status | Published - Feb 2021 |
| Externally published | Yes |
Abstract
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.
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Indagationes mathematicae, Vol. 32, No. 1, 02.2021, p. 323-363.
Research output: Contribution to journal › Article › Research › peer review
}
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.
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.
UR - http://www.scopus.com/inward/record.url?scp=85091506751&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2002.01055
DO - 10.48550/arXiv.2002.01055
M3 - Article
AN - SCOPUS:85091506751
VL - 32
SP - 323
EP - 363
JO - Indagationes mathematicae
JF - Indagationes mathematicae
SN - 0019-3577
IS - 1
ER -