Continuity Properties of Integral Kernels Associated with Schrödinger Operators on Manifolds

2004/10/31 by Jochen Bruening, Jochen Brüning, V. A. Geyler +2
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Bounded function #Diagonal #Fourier integral operator #Geometry #Heat equation #Heat kernel #Hölder condition #Integral equation #Kernel (algebra) #Mathematical analysis #Mathematics #Numerical methods in inverse problems #Operator (biology) #Pure mathematics #Scalar (mathematics) #Singular integral #Spectral Theory in Mathematical Physics #math-ph #math.FA #math.MP #msc:47B25 #msc:47B34 #msc:47D08

paper · pdf · doi:10.1007/s00023-006-0322-z

published as Ann. Henri Poincare 8 (2007) 781-816 · 38 pages, major revision; to appear in Annales Henri Poincare (2007)

arxiv created 2006/07/19 · openalex publication_date 2007/06/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

For Schroedinger operators (including those with magnetic fields) with singular (locally integrable) scalar potentials on manifolds of bounded geometry, we study continuity properties of some related integral kernels: the heat kernel, the Green function, and also kernels of some other functions of the operator. In particular, we show the joint continuity of the heat kernel and the continuity of the Green function outside the diagonal. The proof makes intensive use of the Lippmann-Schwinger equation.

Citations

• Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds
• Continuous integral kernels for unbounded Schrödinger semigroups and their spectral projections
• On-diagonal singularities of the Green functions for Schrödinger operators
• Higher Angular Momenta and Long Range Interaction in Resonance Reactions
• A Panoramic View of Riemannian Geometry