We investigate spectral properties of Gesztesy-Šeba realizations DX,α and D X,β of the 1-D Dirac differential expression D with point interactions on a discrete set X = {xn} ∞ n=1 ⊂ R. Here α := {αn} ∞ n=1 and β := {βn} ∞ n=1 ⊂ R. The Gesztesy-Šeba realizations DX,α and D X,β are the relativistic counterparts of the corresponding Schrödinger operators HX,α and H X,β with δ-and δ ′ -interactions, respectively. We define the minimal operator DX as the direct sum of the minimal Dirac operators on the intervals (xn−1, xn). Then using the regularization procedure for direct sum of boundary triplets we construct an appropriate boundary triplet for the maximal operator D * X in the case d * (X) := inf{|xi − xj| , i = j} = 0. It turns out that the boundary operators BX,α and B X,β parameterizing the realizations DX,α and D X,β are Jacobi matrices. These matrices substantially differ from the ones appearing in spectral theory of Schrödinger operators with point interactions. We show that certain spectral properties of the operators DX,α and D X,β correlate with the corresponding spectral properties of the Jacobi matrices BX,α and B X,β , respectively. Using this connection we investigate spectral properties (self-adjointness, discreteness, absolutely continuous and singular spectra) of Gesztesy-Šeba realizations. Moreover, we investigate the non-relativistic limit as the velocity of light c → ∞. Most of our results are new even in the case d * (X) > 0. are naturally associated with (1.1). Namely, assuming that I = (a, +∞) and I = N one defines the operators H 0

We consider a two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity. In both the focusing and defocusing case we prove local well-posedness, i.e., existence and uniqueness of the solution for short times, as well as energy and mass conservation. In addition, we prove that this implies global existence in the defocusing case, irrespective of the power of the nonlinearity, while in the focusing case blowing-up solutions may arise.Date: May 31, 2018.

In this paper we study the nonlinear Dirac (NLD) equation on noncompact metric graphs with localized Kerr nonlinearities, in the case of Kirchhoff-type conditions at the vertices. Precisely, we discuss existence and multiplicity of the bound states (arising as critical points of the NLD action functional) and we prove that, in the L 2 -subcritical case, they converge to the bound states of the NLS equation in the nonrelativistic limit. 1The attention recently attracted by the linear and the nonlinear Dirac equations is due to their applications, as effective equations, in many physical models, as in solid state physics and nonlinear optics [33,34].While originally the NLDE appeared as a field equation for relativistic interacting fermions [38], then it was used in particle physics to simulate features of quark confinement, acoustic physics, and in the context of Bose-Einstein condensates [34].Recently, it also appeared that some properties of physical models, as thin carbon structures, are well described using as an effective equation for non-relativistic electronic properties , the Dirac equation. We mention, thereupon, the seminal papers by Fefferman and Weinstein [28,29], the work of Arbunich and Sparber [10] (where a rigorous justification of linear and nonlinear equations in two-dimensional honeycomb structures is given) and the referenced therein. In addition, we

We consider the Schrödinger equation in dimension two with a fixed, pointwise, focusing nonlinearity and show the occurrence of a blow-up phenomenon with two peculiar features: first, the energy threshold under which all solutions blow up is strictly negative and coincides with the infimum of the energy of the standing waves. Second, there is no critical power nonlinearity, i.e. for every power there exist blow-up solutions. This last property is uncommon among the conservative Schrödinger equations with local nonlinearity.