Existence of dynamics for a 1D NLS equation perturbed with a generalized point defect

In the present paper we study the well-posedness for the one-dimensional cubic NLS perturbed by a generic point interaction. Point interactions are described as the 4-parameter family of self-adjoint extensions of the symmetric 1D Laplacian defined on the regular functions vanishing at a point, and in the present context can be interpreted as localized defects interacting with the NLS field. A previously treated special case is given by an NLS equation with a δ defect which we generalize and extend, as far as well-posedness is concerned, to the whole family of point interactions. We prove existence and uniqueness of the local Cauchy problem in strong form (initial data and evolution in the operator domain of point interactions), weak form (initial data and evolution in the form domain of point interactions) and L 2(ℝ). Conservation laws of mass and energy are proved for finite energy weak solutions of the problem, which imply global existence of the dynamics. A technical difficulty arises due to the fact that a power nonlinearity does not preserve the form domain for a subclass of point interactions; to overcome it, a technique based on the extension of resolvents of the linear part of the generator to maps between a suitable Hilbert space and the energy space is devised and estimates are given which show the needed regularization properties of the nonlinear flow.