Direct integrals and spectral averaging

A one parameter family of selfadjoint operators gives rise to a corresponding direct integral. We show how to use the Putnam Kato theorem to obtain a new method for the proof of a spectral averaging result

Wegner bounds for a two-particle tight binding model

We consider a quantum two-particle system on a d-dimensional lattice with interaction and in presence of an IID external potential. We establish Wegner-typer estimates for such a model. The main tool used is Stollmann's lemma

Absence of continuous spectral types for certain nonstationary random models

We consider continuum random Schr\"odinger operators of the type $H_{\omega} = -\Delta + V_0 + V_{\omega}$ with a deterministic background potential $V_0$. We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of $-\Delta +V_0$. The models we treat include random surface potentials as well as sparse or slowly decaying random potentials. In particular, we establish absence of absolutely continuous surface spectrum for random potentials supported near a one-dimensional surface (``random tube'') in arbitrary dimension.Comment: 14 pages, 2 figure

Multi-Particle Anderson Localisation: Induction on the Number of Particles

This paper is a follow-up of our recent papers \cite{CS08} and \cite{CS09} covering the two-particle Anderson model. Here we establish the phenomenon of Anderson localisation for a quantum $N$-particle system on a lattice $\Z^d$ with short-range interaction and in presence of an IID external potential with sufficiently regular marginal cumulative distribution function (CDF). Our main method is an adaptation of the multi-scale analysis (MSA; cf. \cite{FS}, \cite{FMSS}, \cite{DK}) to multi-particle systems, in combination with an induction on the number of particles, as was proposed in our earlier manuscript \cite{CS07}. Similar results have been recently obtained in an independent work by Aizenman and Warzel \cite{AW08}: they proposed an extension of the Fractional-Moment Method (FMM) developed earlier for single-particle models in \cite{AM93} and \cite{ASFH01} (see also references therein) which is also combined with an induction on the number of particles. An important role in our proof is played by a variant of Stollmann's eigenvalue concentration bound (cf. \cite{St00}). This result, as was proved earlier in \cite{C08}, admits a straightforward extension covering the case of multi-particle systems with correlated external random potentials: a subject of our future work. We also stress that the scheme of our proof is \textit{not} specific to lattice systems, since our main method, the MSA, admits a continuous version. A proof of multi-particle Anderson localization in continuous interacting systems with various types of external random potentials will be published in a separate papers

Leaky quantum graphs: approximations by point interaction Hamiltonians

We prove an approximation result showing how operators of the type $-\Delta -\gamma \delta (x-\Gamma)$ in $L^2(\mathbb{R}^2)$, where $\Gamma$ is a graph, can be modeled in the strong resolvent sense by point-interaction Hamiltonians with an appropriate arrangement of the $\delta$ potentials. The result is illustrated on finding the spectral properties in cases when $\Gamma$ is a ring or a star. Furthermore, we use this method to indicate that scattering on an infinite curve $\Gamma$ which is locally close to a loop shape or has multiple bends may exhibit resonances due to quantum tunneling or repeated reflections.Comment: LaTeX 2e, 31 pages with 18 postscript figure

Nonrelativistic hydrogen type stability problems on nonparabolic 3-manifolds

We extend classical Euclidean stability theorems corresponding to the nonrelativistic Hamiltonians of ions with one electron to the setting of non parabolic Riemannian 3-manifolds.Comment: 20 pages; to appear in Annales Henri Poincar

A matrix-valued point interactions model

We study a matrix-valued Schr\"odinger operator with random point interactions. We prove the absence of absolutely continuous spectrum for this operator by proving that away from a discrete set its Lyapunov exponents do not vanish. For this we use a criterion by Gol'dsheid and Margulis and we prove the Zariski denseness, in the symplectic group, of the group generated by the transfer matrices. Then we prove estimates on the transfer matrices which lead to the H\"older continuity of the Lyapunov exponents. After proving the existence of the integrated density of states of the operator, we also prove its H\"older continuity by proving a Thouless formula which links the integrated density of states to the sum of the positive Lyapunov exponents