Existence and Mapping Properties of Wave Operator for the Schrödinger Equation with Singular Potential by Vladimir Georgiev and Angel Ivanov

We consider the Schrödinger equation in three dimensional space with small potential in the Lorentz space L3/2,∞ and we prove Strichartz-type estimates for the solution to this equation. Moreover, using the Cook’s method, we prove the existence of the wave operator. In the last section we prove equivalence between the homogeneous Sobolev spaces Ḣs and Ḣs V in the case 0 ≤ s < 3 2 . 1. Definitions and main results Consider the following Schrödinger equation with potential perturbation: i∂tu−∆u + V u = 0 , ∆ = ∂ x1 + ∂ x2 + ∂ x3 , (1) u(0, x) = u0(x), x ∈ R. (2) Here V = V (x) is a real-valued potential that satisfies the assumption ‖V ‖ L 3 2 ,∞) ≤ δ0, (3) where L are standard Lorentz spaces, L(p,∞) is the weak L space (see [1] for details). For δ0 > 0 sufficiently small one can define the bilinear form Q(u, v) = (∇u,∇v)L2(R3) + ∫ R3 V (x)u(x)v(x) d x (4) 2000 AMS Subject Classification. 35J10, 35P25, 35B45.