Perturbed Schrödinger equation with singular potential and initial data

Abstract

We consider linear Schrödinger equation perturbed by delta distribution with singular potential and the initial data. Due to the singularities appearing in the equation, we introduce two kinds of approximations: the parameter's approximation for potential and the initial data given by mollifiers of different growth and the approximation for the Green function for Schrödinger equation with regularized derivatives. These approximations reduce the perturbed Schrödinger equation to the family of singular integral equations. We prove the existence-uniqueness theorems in Colombeau space Gp,q([0, T) × Rn), 1 ≤ p, q ≤ ∞ employing novel stability estimates (w.r.) to singular perturbations for ε → 0, which imply the statements in the framework of Colombeau generalized functions. In particular, we prove the existence-uniqueness result in Gs,g([0, T) × Rn) and G2,2([0, T) × Rn) algebra of Colombeau. © World Scientific Publishing Company.