[Paper Review] Covariant Schr\"odinger semigroups on noncompact Riemannian manifolds

Batu Güneysu|arXiv (Cornell University)|Jan 4, 2018

advanced mathematical theories3 citations

TL;DR

This monograph establishes a comprehensive theory of covariant Schrödinger semigroups on vector bundles over noncompact Riemannian manifolds, using Sobolev spaces, heat kernel analysis, and stochastic processes via Wiener measure and Brownian motion. The key contribution is the rigorous foundation for the essential self-adjointness, compactness, and $L^q$-boundedness of these semigroups under general potential conditions, enabling applications in 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 $V(H^{ abla}+1)^{-1}$ 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)

Motivation & Objective

• To develop a rigorous functional analytic framework for covariant Schrödinger operators on vector bundles over noncompact Riemannian manifolds.
• To establish the existence and regularity of heat kernels and their connection to stochastic processes on manifolds.
• To characterize the essential self-adjointness and compactness properties of Schrödinger operators under general potential conditions.
• To provide a form core of smooth, compactly supported sections for the associated quadratic forms.
• To extend applications to quantum mechanics and geometric analysis through the theory of covariant semigroups.

Proposed method

• Construction of Sobolev spaces of sections on vector bundles over noncompact Riemannian manifolds using the Levi-Civita connection and Riemannian volume measure.
• Analysis of smooth heat kernels on vector bundles via parametrix methods and estimates on curvature and geometry.
• Use of Wiener measure and Brownian motion on manifolds to represent semigroups and derive probabilistic representations.
• Application of Dynkin and Kato potential classes to control singularities and ensure contractivity and regularity.
• Derivation of integral kernels for covariant Schrödinger semigroups using heat kernel asymptotics and perturbation theory.
• Proof of essential self-adjointness via the form core property and spectral theory on noncompact domains.

Experimental results

Research questions

• RQ1 Under what conditions is the covariant Schrödinger operator on a vector bundle over a noncompact Riemannian manifold essentially self-adjoint?
• RQ2 How do contractive Dynkin and Kato potentials influence the $L^q$-boundedness and continuity of the associated semigroups?
• RQ3 What is the precise relationship between the heat kernel on the manifold and the integral kernel of the covariant Schrödinger semigroup?
• RQ4 Can smooth, compactly supported sections serve as a form core for the Schrödinger operator in the noncompact setting?
• RQ5 How do geometric properties of the manifold, such as curvature and volume growth, affect the spectral and regularity properties of the semigroup?

Key findings

• The covariant Schrödinger operator $H^{\nabla} + V$ is essentially self-adjoint on smooth, compactly supported sections under suitable potential conditions, including Kato and Dynkin classes.
• The operator $V(H^{\nabla}+1)^{-1}$ is compact if $V$ is in the Kato class, implying discrete spectrum under appropriate geometric constraints.
• The Schrödinger semigroup $e^{-t(H^{\nabla}+V)}$ is bounded on $L^q$ for all $q \in [1, \infty]$ under $V$ in the Kato class, with uniform bounds depending on curvature and volume growth.
• The semigroup admits an integral kernel that is jointly continuous and dominated by the heat kernel, ensuring strong Feller properties.
• Smooth, compactly supported sections form a form core for the Schrödinger operator, enabling approximation arguments in the associated quadratic form domain.
• The theory provides a rigorous foundation for quantum mechanical models on curved spaces and geometric analysis on noncompact manifolds, including spectral theory and heat kernel estimates.