Authors:Kenichi Ito, Arne Jensen Abstract: Reviews in Mathematical Physics, Ahead of Print. We consider the one-dimensional discrete Schrödinger operator on ℤ, and study the relation between the generalized eigenstates and the asymptotic expansion of the resolvent for the threshold 0. We decompose the generalized zero eigenspace into subspaces, some of which correspond to the bound states or the resonance states, only by their growth properties at infinity, and precisely describe the first few leading coefficients in the expansion using these subspaces. The generalized zero eigenspace we consider is the largest possible one, consisting of all solutions to the eigenequation. For the resolvent expansion, we implement the expansion scheme of Jensen–Nenciu [Rev. Math. Phys.13 (2001) 717–754] and [Rev. Math. Phys.16 (2004) 675–677] in its full generality. Citation: Reviews in Mathematical Physics PubDate: Wed, 11 Feb 2015 07:17:41 GMT DOI: 10.1142/S0129055X15500026

Authors:Elena Cordero, Fabio Nicola, Luigi Rodino Abstract: Reviews in Mathematical Physics, Ahead of Print. We consider Schrödinger equations with real-valued smooth Hamiltonians, and non-smooth bounded pseudo-differential potentials, whose symbols may not even be differentiable. The well-posedness of the Cauchy problem is proved in the frame of the modulation spaces, and results of micro-local propagation of singularities are given in terms of Gabor wave front sets. Citation: Reviews in Mathematical Physics PubDate: Wed, 21 Jan 2015 02:59:12 GMT DOI: 10.1142/S0129055X15500014

Authors:Philippe Kerdelhué, Jimena Royo-Letelier Abstract: Reviews in Mathematical Physics, Volume 26, Issue 10, November 2014. In a semi-classical regime, we study a periodic magnetic Schrödinger operator in ℝ2. This is inspired by recent experiments on artificial magnetism with ultra cold atoms in optical lattices, and by the new interest for the operator on the hexagonal lattice describing the behavior of an electron in a graphene sheet. We first review some results for the square (Harper), triangular and hexagonal lattices. Then, we study the case when the periodicity is given by the kagome lattice considered by Hou. Following the techniques introduced by Helffer–Sjöstrand and Carlsson, we reduce this problem to the study of a discrete operator on ℓ2(ℤ2;ℂ3) and a pseudo-differential operator on L2(ℝ;ℂ3), which keep the symmetries of the kagome lattice. We estimate the coefficients of these operators in the case of a weak constant magnetic field. Plotting the spectrum for rational values of the magnetic flux divided by 2πh where h is the semi-classical parameter, we obtain a picture similar to Hofstadter's butterfly. We study the properties of this picture and prove the symmetries of the spectrum and the existence of flat bands, which do not occur in the case of the three previous models. Citation: Reviews in Mathematical Physics PubDate: Thu, 11 Dec 2014 07:08:42 GMT DOI: 10.1142/S0129055X14500202

Authors:Pedro Freitas, Petr Siegl Abstract: Reviews in Mathematical Physics, Ahead of Print. We study the spectral properties of the two-dimensional Dirac operator on bounded domains together with the appropriate boundary conditions which provide a (continuous) model for graphene nanoribbons. These are of two types, namely, the so-called armchair and zigzag boundary conditions, depending on the line along which the material was cut. In the former case, we show that the spectrum behaves in what might be called a classical way; while in the latter, we prove the existence of a sequence of finite multiplicity eigenvalues converging to zero and which correspond to edge states. Citation: Reviews in Mathematical Physics PubDate: Wed, 19 Nov 2014 01:59:58 GMT DOI: 10.1142/S0129055X14500184

Authors:Dafeng Zuo et al Abstract: Reviews in Mathematical Physics, Ahead of Print.
In this paper, we are interested in a series of sub-hierarchies of the KP hierarchy introduced by Date et al. in [4], which we call the ℬð’žr-KP hierarchy for r ∈ ℤ≥0. In a unified way, we construct additional symmetries of the ℬð’žr-KP hierarchy and show that all of them form a -algebra. Then, we introduce the constrained ℬð’žr-KP hierarchy, and construct its additional symmetries and show that all of them form a Witt algebra. PubDate: Wed, 19 Nov 2014 01:59:58 GMT DOI: 10.1142/S0129055X14500196