Unique continuation of Schrödinger-type equations for ∂ II

2024/06/15 by Yifei Pan, Yuan Zhang, Pan, Yifei +1
Mathematics · #Spectral Theory in Mathematical Physics #Numerical methods in inverse problems #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.2406.10749

Abstract

In this paper, we extend our earlier unique continuation results \citePZ2 for the Schrödinger-type inequality |∂ u| ≤ V|u| on a domain in \mathbb Cⁿ by removing the smoothness assumption on solutions u = (u₁, …, u_N). More specifically, we establish the unique continuation property for W_loc^1,1 solutions when the potential V∈ L_loc^p , p>2n; and for W_loc^1,2n+ε solutions when V∈ L_loc²ⁿ with N=1 or n = 2. Although the unique continuation property fails in general if V∈ L_loc^p, p<2n, we show that the property still holds for W_loc^1,1 solutions when V is a small constant multiple of (1)/(|z|).

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