Riccardo Adami, Diego Noja
Existence of dynamics for a 1-d NLS equation perturbed with a
generalized point defect.
(101K, LATeX 2e)
ABSTRACT. 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 1-d 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 a NLS equation with a delta 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 and weak form. 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.