MPEJ Volume 5, No.1, 16pp
Received: June 19, 1998, Accepted: Jan 6, 1999

I. Guarneri, H. Schulz-Baldes
Lower bounds on wave packet propagation by packing dimensions
of spectral measures

ABSTRACT:  We prove that, for any quantum evolution in $\ell^2(\ZZ^D)$,
there exist arbitrarily long time scales on which the $q$th moment
of the position operator increases at least as fast as a power of time
given by $q/D$ times the packing dimension of the spectral measure.  
Packing dimensions of measures and their connections to scaling exponents
and box-counting dimensions are also discussed.

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MPEJ Volume 5, No.2, 22pp
Received: Oct 19, 1998, Accepted: Apr 8, 1999

G.D. Raikov
Eigenvalue Asymptotics for the Dirac Operator
in Strong Constant Magnetic Fields

ABSTRACT:  We consider the three-dimensional Dirac operator $H$ with
constant magnetic field and electric potential which decays at infinity.
We study the asymptotic behaviour of the discrete spectrum of $H$ as the
norm of the magnetic field grows unboundedly.

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MPEJ Volume 5, No.3, 17pp
Received: Mar 24, 1999, Revised: Jun 23, 1999, Accepted: Jun 24, 1999

Stephan De Bievre, Joseph V. Pule
Propagating Edge States for a Magnetic Hamiltonian

ABSTRACT:  We study the quantum motion of a charged particle in a half plane,
subject to a perpendicular constant magnetic field $B$ and to an arbitrary
weak impurity potential $W_B$ (i.e. $||W_B||_\infty < \delta B$, for some
$\delta$ small enough). We show that there exist states propagating with a
speed of size $B^{1/2}$ along the edge, no matter how fast $W_B$ fluctuates.
As a consequence, the spectrum of the Hamiltonian is purely absolutely
continuous in a spectral interval of size $\gamma B$ ($0 < \gamma < 1$)
between the Landau levels of the system without edge or potential, so that the
corresponding eigenstates are extended. This then provides a rigorous proof
of a phenomenon pointed out by Halperin in his work on the quantum Hall effect.

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MPEJ Volume 5, No.4, 8pp
Received: Jul 7, 1999, Accepted: Sep 29, 1999

Federico Bonetto, Guido Gentile
On a conjecture for the critical behaviour of KAM tori

ABSTRACT:  At the light of recent results in literature we review a
conjecture formulated in Math. Phys. Electron. J. 1 (1995), paper 5, 1-13,
about the mechanism of breakdown of invariant sets in KAM problems
and the identification of the dominant terms in the perturbative expansion
of the conjugating function. We show that some arguments developed therein
can be carried out further only in some particular directions, so limiting
a possible future research program, and that the mechanism of break down
of invariant tori has to be more complicated than as conjectured
in the quoted paper.

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MPEJ Volume 5, No.5, 11pp
Received: Jan 14, 1999, Revised: May 12, 1999, Accepted: Nov 18, 1999

Y. Lacroix
Local perturbations of energy and Kac's return time theorem

ABSTRACT:  We introduce the notion of local perturbations for normalized
energies and study their effect on the level of equilibrium measures.
Using coupling technics and Kac's return time theorem, we obtain some
$\bar{d}$-estimates for the equilibrium measures.
These reveal stability of certain energies under local perturbations.
They also show how some weak-$\star$ convergence of equilibrium may be
obtained in absence of $\norm_\infty$-accuracy of the energies.

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MPEJ Volume 5, No.6, 11pp
Received: Sep 21, 1999, Accepted: Dec 7, 1999

Georg Hoever, Heinz Siedentop
Stability of the Brown-Ravenhall Operator

ABSTRACT:  The Brown-Ravenhall Hamiltonian is a model for the behavior of
$N$ electrons in a field of $K$ fixed nuclei having the atomic numbers
${\bf Z}=(Z_1,\ldots,Z_K)$, which is written, in appropriate units, as
$$B=\Lambda_{+,N}\left(\sum_{n=1}^N D_0^{(n)} +\alpha V_c\right)\Lambda_{+,N}$$
acting on the $N$-fold antisymmetric tensor product $\mathfrak H_N$ of
$\Lambda_+(L^2({\mathbb R}^3)\otimes {\mathbb C}^4)$, where $D_0^{(n)}$
denotes the free Dirac operator $D_0$ acting on the $n$-th particle,
$\Lambda_+$ denotes the projection onto the positive spectral subspace of
$D_0$, $\Lambda_{+,N}$ the projection onto $\mathfrak H_N$ and the potential
$V_c$ is the usual Coulomb interaction of the particles, coupled by
the constant $\alpha$. It is proved in the massless case that for any
$\gamma <2/(2/\pi+\pi/2)$ there exists an $\alpha_0$ such that for all
$\alpha<\alpha_0$ and $\alpha Z_k\leq\gamma$ $(k=1,\ldots K)$ we have stability,
i.e., $B\geq 0$. Using numerical calculations we get stability for the
physical value $\alpha\approx 1/137$ up to $Z_k\leq 88$ $(k=1,\ldots K)$.