Decay estimates for Schrödinger heat semigroup with inverse square potential in Lorentz spaces II

2020/12/15 by Kazuhiro Ishige, Ishige, Kazuhiro, Yujiro Tateishi +1
Mathematics · #Advanced Mathematical Physics Problems #Spectral Theory in Mathematical Physics #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2012.08064

Abstract

Let H:=-Δ+V be a nonnegative Schrödinger operator on L²(\bf R^N), where N≥ 2 and V is a radially symmetric inverse square potential. Let ‖∇^αe^-tH‖_{(L^p,σ→ L^q,θ)} be the operator norm of ∇^αe^-tH from the Lorentz space L^p,σ(\bf R^N) to L^q,θ(\bf R^N), where α∈\0,1,2,…\. We establish both of upper and lower decay estimates of ‖∇^αe^-tH‖_{(L^p,σ→ L^q,θ)} and study sharp decay estimates of ‖∇^αe^-tH‖_{(L^p,σ→ L^q,θ)}. Furthermore, we characterize the Laplace operator -Δ from the view point of the decay of ‖∇^αe^-tH‖_{(L^p,σ→ L^q,θ)}.

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