2023/03/30 by Andrés Larraín-Hubach, Jacob N. Shapiro, Larraín-Hubach, Andrés +1
Computer Science · Mathematics · #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.2303.17722
openalex publication_date 2023/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove an explicit weighted estimate for the semiclassical Schrödinger operator P = - h2 ∂2x + V(x;h) on L2(ℝ), with V(x;h) a finite signed measure, and where h >0 is the semiclassical parameter. The proof is a one dimensional instance of the spherical energy method, which has been used to prove Carleman estimates in higher dimensions and in more complicated geometries. The novelty of our result is that the potential need not be absolutely continuous with respect to Lebesgue measure. Two consequences of the weighted estimate are the absence of positive eigenvalues for P, and a limiting absorption resolvent estimate with sharp h-dependence. The resolvent estimate implies exponential time-decay of the local energy for solutions to the corresponding wave equation with a compactly supported measure potential, provided there are no negative eigenvalues and no zero resonance, and provided the initial data have compact support.