TY - JOUR

T1 - Global Propagator for the Massless Dirac Operator and Spectral Asymptotics

AU - Capoferri, Matteo

AU - Vassiliev, Dmitri

N1 - Funding Information:
We are grateful to Yiannis Petridis for providing useful references. Furthermore, we would like to thank an anonymous referee for insightful comments, in particular, for suggesting the argument in Remark 3.9.
Publisher Copyright:
© 2022, The Author(s).

PY - 2022/9

Y1 - 2022/9

N2 - We construct the propagator of the massless Dirac operator W on a closed Riemannian 3-manifold as the sum of two invariantly defined oscillatory integrals, global in space and in time, with distinguished complex-valued phase functions. The two oscillatory integrals—the positive and the negative propagators—correspond to positive and negative eigenvalues of W, respectively. This enables us to provide a global invariant definition of the full symbols of the propagators (scalar matrix-functions on the cotangent bundle), a closed formula for the principal symbols and an algorithm for the explicit calculation of all their homogeneous components. Furthermore, we obtain small time expansions for principal and subprincipal symbols of the propagators in terms of geometric invariants. Lastly, we use our results to compute the third local Weyl coefficients in the asymptotic expansion of the eigenvalue counting functions of W.

AB - We construct the propagator of the massless Dirac operator W on a closed Riemannian 3-manifold as the sum of two invariantly defined oscillatory integrals, global in space and in time, with distinguished complex-valued phase functions. The two oscillatory integrals—the positive and the negative propagators—correspond to positive and negative eigenvalues of W, respectively. This enables us to provide a global invariant definition of the full symbols of the propagators (scalar matrix-functions on the cotangent bundle), a closed formula for the principal symbols and an algorithm for the explicit calculation of all their homogeneous components. Furthermore, we obtain small time expansions for principal and subprincipal symbols of the propagators in terms of geometric invariants. Lastly, we use our results to compute the third local Weyl coefficients in the asymptotic expansion of the eigenvalue counting functions of W.

KW - Dirac operator

KW - Global Fourier integral operators

KW - Hyperbolic propagators

KW - Weyl coefficients

UR - http://www.scopus.com/inward/record.url?scp=85135708770&partnerID=8YFLogxK

U2 - 10.1007/s00020-022-02708-1

DO - 10.1007/s00020-022-02708-1

M3 - Article

AN - SCOPUS:85135708770

VL - 94

JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

SN - 0378-620X

IS - 3

M1 - 30

ER -