2024/09/04 by Aparajita Dasgupta, Dasgupta, Aparajita, Lalit Mohan +3
Mathematics · #22E30 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Primary 46F05 #Secondary 58J40 #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2409.03027
openalex publication_date 2024/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This article investigates the wave equation for the Schrödinger operator on ℝn, denoted as H0:=-Δ+V, where Δ is the standard Laplacian and V is a complex-valued multiplication operator. We prove that the operator H0, with Re(V)≥ 0 and Re(V)(x)→∞ as |x|→∞, has a purely discrete spectrum under certain conditions. In the spirit of Colombini, De Giorgi, and Spagnolo, we also prove that the Cauchy problem with regular coefficients is well-posed in the associated Sobolev spaces, and when the propagation speed is Hölder continuous (or more regular), it is well-posed in Gevrey spaces. Furthermore, we prove that it is very weakly well-posed when the coefficients possess a distributional singularity.