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
SN - 0378-620X
VL - 94
JO - Integral Equations and Operator Theory
JF - Integral Equations and Operator Theory
IS - 3
M1 - 30
ER -