The universal eigenvalue bounds of Payne–Pólya–Weinberger, Hile–Protter, and H C Yang

MARK S ASHBAUGH
Department of Mathematics, University of Missouri, Columbia, MO 65211-0001, USA

In this paper we present a unified and simplified approach to the universal eigenvalue inequalities of Payne–Pólya–Weinberger, Hile–Protter, and Yang. We then generalize these results to inhomogeneous membranes and Schrodinger's equation with a nonnegative potential. We also show that Yang's inequality is always better than Hile–Protter's (and hence also better than Payne–Pólya–Weinberger's). In fact, Yang's weaker inequality (which deserves to be better known),

                                  l _k+1 < (1+(4/n))(1/k)S^k_i=1 l _i,

is also strictly better than Hile–Protter's.  Finally, we treat Yang's (and related) inequalities for minimal submaniforlds of a sphere and domains contained in a sphere by our methods.

The Wegner estimate and the integrated density of states for some random operators

J M COMBES^1,2, P D HISLOP^1,3, FRÉDÉRIC KLOPP⁴ and SHU NAKAMURA^5
1Centre de Physique Théorique, CNRS Luminy Case 907, F-13288 Marseille, Cedex 9, France
²Département de Mathématiques, Université de Toulon et du Var, 83130 La Garde, France
³Mathematics Department, University of Kentucky, Lexington KY 40506-0027, USA
⁴L.A.G.A.., Institut Galilée, Université Paris-Nord, F-93430, Villetaneuse, France
⁵Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

The integrated density of states (IDS) for random operators is an important function describing many physical characteristics of a random system.  Properties of the IDS are derived from the Wegner estimate that describes the influence of finite-volume perturbations  on a background system.  In this paper, we present a simple proof of the Wegner estimate applicable to a wide variety of random perturbations of deterministic background operators. The proof yields the correct volume dependence of the upper bound. This implies the local Hölder continuity of the integrated density of states at energies in the unperturbed spectral gap. The proof depends on the  L^p-theory of the spectral shift function (SSF), for p ³ 1, applicable to pairs of self-adjoint operators whose difference is in the trace ideal I_p, for 0 < p £ 1. We present this and other results on the SSF due to other authors. Under an additional condition of the single-site potential, local Holder continuity is proved at all energies. Finally, we present extensions of this work to random potentials with nonsign definite single-site potentials.

Energy transfer in scattering by rotating potentials

VOLKER ENSS¹, VADIM KOSTRYKIN² AND ROBERT SCHRADER^3
1Institut für Reine und Angewandte Mathematik, Rheinisch-Westfälische Technische Hochschule Aachen,Templergraben 55, D-52062 Aachen, Germany
²Fraunhofer-Institut für Lasertechnik, Steinbachstrasse 15, D-52074 Aachen, Germany
³Institut für Theoretische Physik, Freie Universitat Berlin, Arnimallee 14, D-14195 Berlin, Germany
E-mail: enss@rwth-aachen.de; kostrykin@t-online.de; kostrykin@ilt.fraunhofer.de;  schrader@physik.fu-berlin.de

Quantum mechanical scattering theory is studied for time-dependent Schrödinger operators, in particular for particles in a rotating potential. Under various assumptions about the decay rate at infinity we show uniform boundedness in time for the kinetic energy of scattering states, existence and completeness of wave operators, and existence of a conserved quantity under scattering. In a simple model we determine the energy transferred to a particle by collision with a rotating blade.

Magnetic bottles for the Neumann problem: The case of dimension 3

BERNARD HELFFER¹ and ABDEREMANE MORAME^2
1Département de Mathématiques, UMR CNRS 8628, Bât. 425, Université Paris-Sud, F-91405 Orsay Cedex, France
²Département de Mathématiques, UMR CNRS 6629 , Université de Nantes, 2, rue de la Houssinière, B.P. 92208, 44322 Nantes Cedex 3, France
Email: Bernard.Helffer@math.u-psud.fr; morame@math.univ-nantes.fr

The main object of this paper is to analyze the recent results obtained on the Neumann realization of the Schrödinger operator in the case of dimension 3 by Lu and Pan. After presenting a short treatment of their spectral analysis of key-models, we show briefly how to implement the techniques of Helffer–Morame in order to give some localization of the ground state. This leaves open the question of the localization by curvature effect which was solved in the case of dimension 2 in our previous work and will be analysed in the case of dimension 3 in a future paper.

The extradordinary spectral properties of radially periodic Schrödinger operators

ANDREAS M HINZ
Zentrum Mathematik, Technische Universität München, 80290 München, Germany
Email: hinz@appl-math.tu-muenchen.de

Since it became clear that the band structure of the spectrum of periodic Sturm Liouville operators t = – (d²/dr²) + q(r) does not survive a spherically symmetric extension to Schrödinger operators T = – D + V with V(x) = q(|x|) for x Î R^d, d Î N\{1}, a wealth of detailed information about the spectrum of such operators has been acquired. The observation of eigenvalues embedded in the essential spectrum [m₀, ¥] of T with exponentially decaying eigenfunctions provided evidence for the existence of intervals of dense point spectrum, eventually proved by spherical separation into perturbed Sturm–Liouville operators t_c = t + (c/r²). Subsequently, a numerical approach was employed to investigate the distribution of eigenvalues of T more closely. An eigenvalue was discovered below the essential spectrum in the case d = 2, and it turned out that there are in fact infinitely many, accumulating at m₀. Moreover, a method based on oscillation theory made it possible to count eigenvalues of t_c contributing to an interval of dense point spectrum of T. We gained evidence that an asymptotic formula, valid for c ® ¥, does in fact produce correct numbers even for small values of the coupling constant, such that a rather precise picture of the spectrum of radially periodic Schrodinger operators has now been obtained.

On the norm convergence of the self-adjoint Trotter–Kato product formula with error bound

TAKASHI ICHINOSE* and HIDEO TAMURA⁺
*Department of Mathematics, Faculty of Science, Kanazawa University,Kanazawa 920-1192, Japan
⁺Department of Mathematics, Faculty of Science, Okayama University, Okayama 700-8530, Japan

The norm convergence of the Trotter–Kato product formula with error bound is shown for the semigroup generated by that operator sum of two nonnegative self-adjoint operators A and B which is self-adjoint.

On perturbations of eigenvalues embedded at thresholds in a two channel model

ARNE JENSEN
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
On leave from: Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg Ø, Denmark
E-mail: matarne@math.auc.dk

We present some results on the perturbation of eigenvalues embedded at thresholds in a two channel model Hamiltonian with a small off-diagonal perturbation. Examples are given of the various types of behavior of the eigenvalue under perturbation.

On spectral properties of periodic polyharmonic matrix operators

YU E KARPESHINA
Department of Mathematics, University of Alabama at Birmingham,452 Campbell Hall, Birmingham AL 35294-1170, USA

We consider a matrix operator H = (–D )^l + V  in Rⁿ, where n ³ 2, l ³ 1, 4l > n + 1, and V is the operator of multiplication by a periodic in x matrix V(x). We study spectral properties of H in the high energy region. Asymptotic formulae for Bloch eigenvalues and the corresponding spectral projections are constructed. The Bethe–Sommerfeld conjecture, stating that the spectrum of H can have only a finite number of gaps, is proved.

Wegner estimate for sparse and other generalized alloy type potentials

WERNER KIRSCH and IVAN VESELIC
Fakultät für Mathematik, Ruhr-Universität, Bochum, Germany

We prove a Wegner estimate for generalized alloy type models at negative energies (Theorems 8 and 13). The single site potential is assumed to be non-positive. The random potential does not need to be stationary with respect to translations from a lattice. Actually, the set of points to which the individual single site potentials are attached, needs only to satisfy a certain density condition. The distribution of the coupling constants is assumed to have a bounded density only in the energy region where we prove the Wegner estimate.

Lifshitz tails for random perturbations of periodic Schrödinger operators

FRÉDÉRIC KLOPP
Département de Mathématique, Institut Galilée, U.M.R. 7539 C.N.R.S,Université de Paris-Nord, 99 Avenue J.-B. Clément, F-93430 Villetaneuse, France
E-mail: klopp@math.univ-paris13.fr

The present paper is a non-exhaustive review of Lifshitz tails for random perturbations of periodic Schrödinger operators. It is not our goal to review the whole literature on Lifshitz tails; we will concentrate on a single model, the continuous Anderson model.

Smoothness of density of states for random decaying interaction

M KRISHNA
Institute of Mathematical Sciences, Taramani, Chennai 600 113, India
Email: krishna@imsc.ernet.in

In this paper we consider one dimensional random Jacobi operators with decaying independent randomness and show that under some condition on the decay vis-a-vis the distribution of randomness, that the distribution function of the average spectral measures of the associated operators are smooth.

A remark on the Lifshitz tail for Schrödinger operator with random magnetic field

SHU NAKAMURA
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
E-mail: shu@ms.u-tokyo.ac.jp

In this note, we consider the Lifshitz singularity for Schrödinger operator with ergodic random magnetic field. A key estimate is an energy bound for magnetic Schrödinger operators as discussed in Nakamura [8]. Here we remove a technical assumption in [8], namely, the uniform boundedness of the magnetic field.

Truncation method for operators with disconnected essential spectrum

M N N NAMBOODIRI
Department of Mathematics, Cochin University of Science and Technology, Cochin 682 022, India
Email: nambu@cusat.ac.in

In this short paper, the usage of truncation method to get information about essential spectrum of bounded as well as semi-bounded linear operators on separable Hilbert spaces, is investigated. In addition to this, the problem of predicting the gaps in the essential spectrum of self-adjoint operators, linear algebraically is also considered.

Homogenization of a parabolic equation in perforated domain with Neumann boundary condition

A K NANDAKUMARAN and M RAJESH*
Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India
*Corresponding author
E-mail: nands@math.iisc.ernet.in; rajesh@math.iisc.ernet.in

In this article, we study the homogenization of the family of parabolic equations over periodically perforated domains

                      ¶_t b ((x /e) , u_e) – div a ((x /e) , u_e ,Ñu_e) = f(x, t) in W_e ´ (0, T ),
                                                  a((x /e) , u_e ,Ñu_e) .n_e = 0 on ¶S_e ´ (0, T ),
                                                                                u_e = 0 on ¶W ´ (0, T ),
                                                                      u_e (x, 0) = u₀(x) in W_e .                            

Here, W_e = W\S_e is a periodically perforated domain. We obtain the homogenized equation and corrector results. The homogenization of the equations on a fixed domain was studied by the authors [15]. The homogenization for a fixed domain and b((x /e , u_e)) º b(u_e) has been done by Jian [11].

Asymptotic absolute continuity for perturbed time-dependent quadratic Hamiltonians

ERIK SKIBSTED
Institut for Matematiske Fag and MaPhySto, Aarhus Universitet,Ny Munkegade 8000 Aarhus C, Denmark
Email: skibsted@imf.au.dk

We study the notion of asymptotic velocity for a class of perturbed time-dependent quadratic Hamiltonians. In particular we give a sufficient condition for absolute continuity.

Strategies in localization proofs for one-dimensional random Schrödinger operators

GÜNTER STOLZ
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170, USA
E-mail: stolz@math.uab.edu

Recent results on localization, both exponential and dynamical, for various models of one-dimensional, continuum, random Schrödinger operators are reviewed. This includes Anderson models with indefinite single site potentials, the Bernoulli–Anderson model, the Poisson model, and the random displacement model. Among the tools which are used to analyse these models are generalized spectral averaging techniques and results from inverse spectral and scattering theory. A discussion of open problems is included.

High energy asymptotics of the scattering amplitude for the Schrödinger equation

D YAFAEV
Department of Mathematics, University Rennes-1, Campus Beaulieu, 35042 Rennes, France

We find an explicit function approximating at high energies the kernel of the scattering matrix with arbitrary accuracy. Moreover, the same function gives all diagonal singularities of the kernel of the scattering matrix in the angular variables.