Volume 112, Issue 1
February 2002, pages 1-255
pp 1-1
pp 3-30
The Universal Eigenvalue Bounds of Payne–Pólya–Weinberger, Hile–Protter, and H C Yang
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 Schrödinger'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),
$$𝜆_{k+1} < \left(1+\frac{4}{n}\right)\frac{1}{k}\sum^k_{i=1} 𝜆_i,$$
is also strictly better than Hile–Protter's. Finally, we treat Yang's (and related) inequalities for minimal submanifolds of a sphere and domains contained in a sphere by our methods.
pp 31-53
The Wegner Estimate and the Integrated Density of States for some Random Operators
J M Combes P D Hislop Frédéric Klopp Shu Nakamura
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 ≥ 1, applicable to pairs of self-adjoint operators whose difference is in the trace ideal $\mathcal{I}_p$, 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 Hölder continuity is proved at all energies. Finally, we present extensions of this work to random potentials with nonsign definite single-site potentials.
pp 55-70
Energy Transfer in Scattering by Rotating Potentials
Volker Enss Vadim Kostrykin Robert Schrader
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.
pp 71-84
Magnetic Bottles for the Neumann Problem: The Case of Dimension 3
Bernard Helffer Abderemane Morame
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.
pp 85-98
The Extraordinary Spectral Properties of Radially Periodic Schrödinger Operators
Since it became clear that the band structure of the spectrum of periodic Sturm Liouville operators 𝑡 = - (d^{2}/d𝑟^{2}) + 𝑞(𝑟) does not survive a spherically symmetric extension to Schrödinger operators 𝑇 = - 𝛥 + 𝑉 with 𝑉(𝑥) = 𝑞(|𝑥|) for $x \in \mathbb{R}^d, d \in \mathbb{N}\backslash\{1\}$, a wealth of detailed information about the spectrum of such operators has been acquired. The observation of eigenvalues embedded in the essential spectrum [𝜇_{0}, ∞] 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/𝑟^{2}). 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 𝜇_{0}. 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 Schrödinger operators has now been obtained.
pp 99-106
On the Norm Convergence of the Self-Adjoint Trotter–Kato Product Formula with Error Bound
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.
pp 107-116
On Perturbation of Eigenvalues Embedded at Thresholds in a Two Channel Model
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.
pp 117-130
On Spectral Properties of Periodic Polyharmonic Matrix Operators
We consider a matrix operator 𝐻 = (-𝛥)^{𝑙} + 𝑉 in 𝑅^{𝑛}, where 𝑛 ≥ 2, 𝑙 ≥ 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.
pp 131-146
Wegner Estimate for Sparse and other Generalized Alloy Type Potentials
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.
pp 147-162
Lifshitz Tails for Random Perturbations of Periodic Schrödinger Operators
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.
pp 163-181
Smoothness of Density of States for Random Decaying Interaction
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.
pp 183-187
A Remark on the Lifshitz Tail for Schrödinger Operator with Random Magnetic Field
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.
pp 189-193
Truncation Method for Operators with Disconnected Essential Spectrum
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.
pp 195-207
Homogenization of a Parabolic Equation in Perforated Domain with Neumann Boundary Condition
In this article, we study the homogenization of the family of parabolic equations over periodically perforated domains
\begin{align*}𝜕_tb\left(\frac{x}{𝜀}, u_{𝜀}\right)-\mathrm{div} a\left(\frac{x}{𝜀}, u_{𝜀},\nabla u_{𝜀}\right) & = f(x,t)\quad\text{in}\quad 𝛺_{𝜀}×(0, T),\\ a\left(\frac{x}{𝜀},u_{𝜀},\nabla u_{𝜀}\right).𝜐_{𝜀} & = 0 \quad\text{on}\quad 𝜕 S_{𝜀}×(0, T),\\ u_{𝜀} & = 0 \quad\text{on}\quad 𝜕𝛺×(0, T),\\ u_{𝜀}(x,0) & = u_0(x) \quad\text(in)\quad 𝛺_{𝜀}.\end{align*}
Here, $𝛺_{𝜀} = 𝛺 \ S_{𝜀}$ 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 $b\left(\frac{x}{𝜀},u_{𝜀}\right)≡ b(u_{𝜀})$ has been done by Jian [11].
pp 209-228
Asymptotic Absolute Continuity for Perturbed Time-Dependent Quadratic Hamiltonians
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.
pp 229-243
Strategies in Localization Proofs for One-Dimensional Random Schrödinger Operators
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.
pp 245-255
High Energy Asymptotics of the Scattering Amplitude for the Schrödinger Equation
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.
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