MPEJ Volume 4, No.1, 16pp Received: Dec 3, 1997, Revised: Dec 30, 1997, Accepted: Jan 9, 1998 M. Guzzo, F. Fasso`, G. Benettin On the Stability of Elliptic Equilibria ABSTRACT: We consider stability of elliptic equilibria in Hamiltonian systems in the frame of Nekhoroshev's theory, recovering the steepness assumption, in the form of convexity, from an appropriate treatment of the higher orders. The singularity of the action-angle coordinates is overcome by using Cartesian coordinates. We introduce an essential refinement of the perturbative technique used in a previous work on the subject, and obtain significant improvements of results, namely better values of the exponents controlling the stability time and the confinement around equilibrium, in case the equilibrium frequency satisfy stronger nonresonance conditions. Within the same nonresonance assumptions the new method provides instead independent informations, namely one gets a better confinement on a reduced time scale. ------------------------------ MPEJ Volume 4, No.2, 19pp Received: Dec 17, 1997, Revised: Mar 18, 1998, Accepted: May 14, 1998 N. Ripamonti Classical Limit of the Matrix Elements on Quantized Lobachevskii Plane ABSTRACT: It is proved that the matrix elements $\wh F_{n,n+k}$ between harmonic oscillator eigenvectors of any smooth observable in the quantized Lobachevskii plane converge to the Fourier coefficients $F_{k}$ of the corresponding classical observable $F(A,\phi)$ at the classical limit $n\to\infty, \hbar\to 0, n\hbar\to A$, $k$ fixed, where $A$, $\phi$ are the oscillator action-angle variables. The Wigner functions are then defined and, as a consequence of the above result, their convergence to $\delta(A-A_{0})e^{-ik\phi}$ at the classical limit is proved when computed on the harmonic oscillator eigenstates $n$ and $n+k$. ------------------------------ MPEJ Volume 4, No.3, 20pp Received: May 18, 1998, Revised: Jul 10, 1998, Accepted: Aug 5, 1998 J.-P. Eckmann, C.E. Wayne Non-Linear Stability Analysis of Higher Order Dissipative Partial Differential Equations ABSTRACT: We extend the invariant manifold method for analyzing the asymptotics of dissipative partial differential equations on unbounded spatial domains to treat equations in which the linear part has order greater than two. One important example of this type of equation which we analyze in some detail is the Cahn-Hilliard equation. We analyze the marginally stable solutions of this equation in some detail. A second context in which such equations arise is in the Ginzburg-Landau equation, or other pattern forming equations, near a codimension-two bifurcation. ------------------------------ MPEJ Volume 4, No.4, 16pp Received: Apr 3, 1998, Revised: Aug 21, 1998, Accepted: Sep 14, 1998 Bernard Helffer, Heinz Siedentop Form Perturbations of the Second Quantized Dirac Field ABSTRACT. We give a criterion on when a form perturbation of the free Dirac operator defines a form perturbation of the second quantized free Dirac field. Moreover, we show that the potentials allowing this quantization are regular in the sense of Klaus and Scharf. Furthermore we prove that only non-local potentials allow for this construction in three dimensions. In two dimensions, though, all local potentials with finite Dirichlet norm are allowed. ------------------------------ MPEJ Volume 4, No.5, 67pp Received: Apr 1, 1998, Revised: Oct 3, 1998, Accepted: Oct 5, 1998 Gregory F. Lawler Strict Concavity of the Intersection Exponent for Brownian Motion in Two and Three Dimensions ABSTRACT: The intersection exponent for Brownian motion is a measure of how likely Brownian motion paths in two and three dimensions do not intersect. We consider the intersection exponent $\xi(\lambda) = \xi_d(k,\lambda)$ as a function of $\lambda$ and show that $\xi$ has a continuous, negative second derivative. As a consequence, we improve some estimates for the intersection exponent; in particular, we give the first proof that the intersection exponent $\xi_3(1,1)$ is strictly greater than the mean field prediction. The results here are used in a later paper to analyze the multifractal spectrum of the harmonic measure of Brownian motion paths. ------------------------------ MPEJ Volume 4, No.6, 8pp Received: Sep 11, 1998, Accepted: Oct 23, 1998 Cheng-Hung Chang, Dieter Mayer The period function of the nonholomorphic Eisenstein series for PSL(2,Z) ABSTRACT: We calculate the period function of Lewis of the automorphic Eisenstein series $E(s,w)=\frac{1}{2}v^s\,\sum_{n,m\neq (0,0)}(mw+n)^{-2s}$ for the modular group $PSL(2,Z)$. This function turns out to be the function $B(\frac{1}{2},s+\frac{1}{2})\psi_s(z)$, where $B(x,y)$ denotes the beta function and $\psi_s$ a function introduced some time ago by Zagier and given for $\Re s>1$ by the series $\psi_s(z)=\sum_{n,m\geq 1}(mz+n)^{-2s}+\frac{1}{2}\zeta(2s)\,(1+z^{-2s})$. The analytic extension of $\psi_s$ to negative integers $s$ gives just the odd part of the period functions in the Eichler, Shimura, Manin theory for the holomorphic Eisenstein forms of weight $-2s+2$. We find this way an interesting connection between holomorphic and nonholomorphic Eisenstein series on the level of their respective period functions.