ar X iv : 0 70 7 . 17 85 v 1 [ m at h . A P ] 1 2 Ju l 2 00 7 INSTABILITIES FOR SUPERCRITICAL SCHRÖDINGER EQUATIONS IN ANALYTIC MANIFOLDS by

ar X iv : 0 70 7 . 03 46 v 1 [ m at h - ph ] 3 J ul 2 00 7 THE ONE - DIMENSIONAL SCHRÖDINGER - NEWTON EQUATIONS

We prove an existence and uniqueness result for ground states of one-dimensional Schrödinger-Newton equations.

ar X iv : 0 70 7 . 39 09 v 1 [ m at h - ph ] 2 6 Ju l 2 00 7 Schrödinger operators on armchair nanotubes . I

We consider the Schrödinger operator with a periodic potential on quasi-1D models of armchair single-wall nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe all eigenfunctions with the same eigenvalue. We define a Lyapunov function, which is analytic on so...

ar X iv : 0 70 7 . 33 03 v 1 [ m at h . O A ] 2 3 Ju l 2 00 7 OPERATOR - VALUED FRAMES

ar X iv : 0 70 7 . 45 25 v 1 [ m at h . A G ] 3 1 Ju l 2 00 7 Analytic Classification of Plane Branches up to Multiplicity 4

We perform the analytic classification of plane branches of multiplicity less or equal than four. This is achieved by computing a Standard basis for the modules of Kähler differentials of such branches by means of the algorithm we developed in [9] and then applying the classification method for plane branches presented in [10]. Subject classification: Primary 14H20; Secondary 14Q05, 14Q20, 32S10

ar X iv : 0 70 7 . 40 46 v 1 [ m at h . A G ] 2 7 Ju l 2 00 7 CLIFFORD ’ S THEOREM FOR COHERENT SYSTEMS

Let C be an algebraic curve of genus g ≥ 2. We prove an analogue of Clifford’s theorem for coherent systems on C and some refinements using results of Re and Mercat.