Covariant Schr\"odinger semigroups on Riemannian manifolds

TL;DR

This monograph develops a comprehensive theory of covariant Schr"odinger semigroups on sections of vector bundles over noncompact Riemannian manifolds, covering heat kernels, potentials, and applications in quantum mechanics.

Contribution

It introduces foundational and advanced results for covariant Schr"odinger semigroups on Riemannian manifolds, including heat kernel analysis and self-adjointness, from first principles.

Findings

Analysis of heat kernels on vector bundles

Criteria for essential self-adjointness of Schr"odinger operators

Applications to quantum mechanics and geometric analysis

Abstract

This monograph develops the theory of covariant Schr\"odinger semigroups acting on sections of vector bundles over noncompact Riemannian manifolds from scratch. Contents: I. Sobolev spaces on vector bundles II. Smooth heat kernels on vector bundles III. Basis differential operators in Riemannian manifolds IV. Some specific results for the minimal heat kernel V. Wiener measure and Brownian motion on Riemannian manifolds VI. Contractive Dynkin and Kato potentials VII. Foundations of covariant Schr\"odinger semigroups VIII. Compactness of $V(H^{\nabla}+1)^{-1}$ IX. $L^q$-properties of covariant Schr\"odinger semigroups X. Continuity properties of covariant Schr\"odinger semigroups XI. Integral kernels for covariant Schr\"odinger semigroups XII. Essential self-adjointness of covariant Schr\"odinger semigroups XIII. Smooth compactly supported sections as form core XIV. Applications (in quantum mechanics and geometric analysis)