Analytic semigroups approaching a Schrödinger group on real foliated metric manifolds

Abstract

On real metric manifolds admitting a co-dimension one foliation, sectorial operators are introduced that interpolate between the generalized Laplacian and the d'Alembertian. This is used to construct a one-parameter family of analytic semigroups that remains well-defined into the near Lorentzian regime. In the strict Lorentzian limit we identify a sense in which a well-defined Schrödinger evolution group arises. For the analytic semigroups we show in addition that: (i) they act as integral operators with kernels that are jointly smooth in the semigroup time and both spacetime arguments. (ii) the diagonal of the kernels admits an asymptotic expansion in (shifted) powers of the semigroup time whose coefficients are the Seeley-DeWitt coefficients evaluated on the complex metrics.

arXiv comment52 pages

References in corpus (10)

• Off-diagonal coefficients of the DeWitt-Schwinger and Hadamard representations of the Feynman propagator
• One-loop stress-tensor renormalization in curved background: the relation between $\zeta$-function and point-splitting approaches, and an improved point-splitting procedure
• Feynman Propagators on Static Spacetimes
• Proof of the symmetry of the off-diagonal Hadamard/Seeley-deWitt's coefficients in $C^{\infty }$ Lorentzian manifolds by a local Wick rotation
• An Evolution Equation Approach to the Klein-Gordon Operator on Curved Spacetime
• Feynman's i-epsilon prescription, almost real spacetimes, and acceptable complex spacetimes
• Strong Short Time Asymptotics and Convolution Approximation of the Heat Kernel
• The Regularized Hadamard Expansion
• A remark on the essential self-adjointness for Klein-Gordon type operators
• Wick rotation of the time variables for two-point functions on analytic backgrounds

Cited by in corpus (1)

• Propagators in curved spacetimes from operator theory