Unique continuation for Schrödinger operators in dimension three or less
Eric T. Sawyer · Annales de l’institut Fourier · 1984
We show that the differential inequality | Δ u | ≤ v | u | has the unique continuation property relative to the Sobolev space H l o c 2 , 1 ( Ω ) , Ω ⊂ R n , n ≤ 3 , if v satisfies the condition ( K n loc ) lim r → 0 sup x ∈ K ∫ | x - y | < r | x - y | 2 - n v ( y ) d y = 0 for all compact K ⊂ Ω , where if n = 2 , we replace | x - y | 2 - n by - log | x - y | . This resolves a conjecture of B. Simon on unique continuation for Schrödinger operators, H = - Δ + v , in the case n ≤ 3 . The proof uses Carleman’s approach together with the following pointwise inequality valid for all N = 0 , 1 , 2 , ... and any u ∈ H c 2 , 1 ( R