Uniqueness properties of solutions to Schrödinger equations

Abstract

nX j=1 ∂ 2 u ∂x 2 (x) = 0, (harmonic functions) in the unit ball {x ∈ R n : |x| 2, since harmonic functions are still real analytic in {x ∈ R n : |x| < 1}. In fact, it is well-known that if P(x,D) is a linear elliptic differential operator with real analytic coefficients, andP(x,D)u = 0 in a open set ⊂ R n , then u is real analytic in . Hence, the (s.u.c.p.) also holds for such solutions. Through the work of Hadamard [28] on the uniqueness of the Cauchy problem (which is closely related to the strong unique continuation property discussed earlier) it became clear (for applications in nonlinear problems) that it would be desirable to establish the strong unique continuation property for operators whose coefficients are not necessarily real analytic, oreven C ∞ . The first results in this direction were found in the pioneering work of Carleman [9] (when ,