2012/06/14 by Andrew Hassell, Hassell, Andrew, Peijie Lin +1 · 2 citations
Computer Science · Mathematics · #35J05 #42B37 #58J05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1206.2997
openalex publication_date 2012/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider Schroedinger operators on metric cones whose cross section is a closed Riemannian manifold (Y, h) of dimension d-1 ≥ 2. Thus the metric on the cone M = (0, ∞)r × Y is dr2 + r2 h. Let Δ be the Friedrichs Laplacian on M and V0 be a smooth function on Y, such that ΔY + V0 + (d-2)2/4 is a strictly positive operator on L2(Y), with lowest eigenvalue μ20 and second lowest eigenvalue μ21, with μ0, μ1 > 0. The operator we consider is H = Δ+ V0/r2, a Schrödinger operator with inverse square potential on M; notice that H is homogeneous of degree -2. We study the Riesz transform T = ∇ H-1/2 and determine the precise range of p for which T is bounded on Lp(M). This is achieved by making a precise analysis of the operator (H + 1)-1 and determining the complete asymptotics of its integral kernel. We prove that if V is not identically zero, then the range of p for Lp boundedness is d/ (min(1+d/2+μ0, d ) lt; p lt; d / (max(d/2-μ0, 0) ), while if V is identically zero, then the range is 1 lt; p lt; d / (max(d/2-μ1, 0 ). The result in the case V identically zero was first obtained in a paper by H.-Q. Li.