A Schrödinger equation

Abstract

Schrödinger equations are of great interest for both the physicist and the mathematician. A lot of papers have been written on them. We only want to treat one example in these lectures assuming G = ℝ³ and perturbing the Δ operator with a potential V. One usually distinguishes between ‘short-range’ potentials and ‘long-range’ potentials. Roughly speaking V is short-range if V ∈ ℒ ^loc₂ and |V(x)| = O (|x|^−1−ε) for |x| → ∞ and ε > 0. On the other hand the Coulomb potential V(x) = −q |x| ⁻¹, q > 0, is long-range.