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Heat Kernel Estimates for Schr\"odinger Operators with Decay at Infinity on Parabolic Manifolds

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arxiv 2501.04221 v2 pith:SJVAMEGA submitted 2025-01-08 math.AP math.PR

Anthony Graves-McCleary , Laurent Saloff-Coste This is my paper · ORCID
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keywords odingerschrdeltaestimatesgiveheatinfinitykernel
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abstract

We give estimates for positive solutions for the Schr\"odinger equation $(\Delta_\mu+W)u=0$ on a wide class of parabolic weighted manifolds $(M, d\mu)$ when $W$ decays to zero at infinity faster than quadratically. These can be combined with results of Grigor'yan to give matching upper and lower bounds for the heat kernel of the corresponding Schr\"odinger operator $\Delta_\mu+W$. In particular, this appears to complement known results for Schr\"odinger operators on $\mathbf{R}^2$.

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  1. Relaxed uniqueness conditions for the parabolic Schrodinger equation on Riemannian manifolds

    math.AP 2025-05 conditional novelty 6.0 of 10

    When a potential grows at infinity, uniqueness for the parabolic Schrödinger equation holds under a weaker integral condition on the solution, with the improvement set by the decay rate of positive stationary solutions.

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