Covariant Schrdinger Semigroups on Riemannian Manifolds [Hardcover]

This monograph discusses covariant Schr?dinger operators and their heat semigroups on noncompact Riemannian manifolds and aims to fill a gap in the literature, given the fact that the existing literature on Schr?dinger operators has mainly focused on scalar Schr?dinger operators on Euclidean spaces so far. In particular, the book studies operators that act on sections of vector bundles. In addition, these operators are allowed to have unbounded potential terms, possibly with strong local singularities.?

The results presented here provide the first systematic study of such operators that is sufficiently general to simultaneously treat?the natural operators from quantum mechanics, such as magnetic Schr?dinger operators with singular electric potentials, and those from geometry, such as squares of Dirac operators that have smooth but endomorphism-valued and possibly unbounded potentials.

The book is largely self-contained, making it accessible for graduate and postgraduate students alike. Since it also includes unpublished findings and new proofs of recently published results, it will also be interesting for researchers from geometric analysis, stochastic analysis, spectral theory, and mathematical physics..

Sobolev spaces on vector bundles.- Smooth heat kernels on vector bundles.- Basis differential operators on Riemannian manifolds.- Some specific results for the minimal heat kernel.- Wiener measure and Brownian motion on Riemannian manifolds.- Contractive Dynkin potentials and Kato potentials.- Foundations of covariant Schr?dinger semigroups.- Compactness of resolvents for covariant Schr?dinger operators.- L^p properties of covariant Schr?dinger semigroups.- Continuity properties of covariant Schr?dinger semigroups.- Integral kernels for covariant Schr?dinger semigroup.- Essential self-adjointness of covariant Schr?dinger semigroups.- Form cores.- Applications.