Kato’s inequality and form boundedness of Kato potentials on arbitrary Riemannian manifolds
Copyright policy )Kato’s inequality and form boundedness of Kato potentials on arbitrary Riemannian manifolds
Let M M be a Riemannian manifold and let E → M E\to M be a Hermitian vector bundle with a Hermitian covariant derivative ∇ \nabla . Furthermore, let H ( 0 ) H(0) denote the Friedrichs extension of ∇ ∗ ∇ / 2 \nabla ^*\nabla /2 and let V : M → E n d ( E ) V:M\to \mathrm {End}(E) be a potential. We prove that if V V has a decomposition of the form V = V 1 − V 2 V=V_1-V_2 with V j ≥ 0 V_j\geq 0 , V 1 V_1 locally integrable and | V 2 | \left | V_2 \right | in the Kato class of M M , then one can define the form sum H ( V ) := H ( 0 ) ∔ V H(V):=H(0)\dotplus V in Γ L 2 ( M , E ) \Gamma _{\mathsf {L}^2}(M,E) without any further assumptions on M M . Applications to quantum physics are discussed.
Mathematics - Differential Geometry, Friedrichs extensions, Riemannian manifold, Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.), FOS: Physical sciences, Heat and other parabolic equation methods for PDEs on manifolds, Applications of stochastic analysis (to PDEs, etc.), Methods of global Riemannian geometry, including PDE methods; curvature restrictions, Mathematical Physics (math-ph), Linear symmetric and selfadjoint operators (unbounded), Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, Differential Geometry (math.DG), quantum physics, FOS: Mathematics, Kato class, Spectral Theory (math.SP), Mathematical Physics, geodesics, Analysis of PDEs (math.AP)
Mathematics - Differential Geometry, Friedrichs extensions, Riemannian manifold, Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.), FOS: Physical sciences, Heat and other parabolic equation methods for PDEs on manifolds, Applications of stochastic analysis (to PDEs, etc.), Methods of global Riemannian geometry, including PDE methods; curvature restrictions, Mathematical Physics (math-ph), Linear symmetric and selfadjoint operators (unbounded), Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, Differential Geometry (math.DG), quantum physics, FOS: Mathematics, Kato class, Spectral Theory (math.SP), Mathematical Physics, geodesics, Analysis of PDEs (math.AP)
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