Semiclassical Approach to Finite-N Matrix Models

We reformulate the zero-dimensional hermitean one-matrix model as a (nonlocal) collective field theory, for finite~$N$. The Jacobian arising by changing variables from matrix eigenvalues to their density distribution is treated {\it exactly\/}. The semiclassical loop expansion turns out {\it not\/} to coincide with the (topological) ${1\over N}$~expansion, because the classical background has a non-trivial $N$-dependence. We derive a simple integral equation for the classical eigenvalue density, which displays strong non-perturbative behavior around $N\!=\!\infty$. This leads to IR singularities in the large-$N$ expansion, but UV divergencies appear as well, despite remarkable cancellations among the Feynman diagrams. We evaluate the free energy at the two-loop level and discuss its regularization. A simple example serves to illustrate the problems and admits explicit comparison with orthogonal polynomial results.Comment: 27 pages / 3 figures (ps file fixed

Asymptotic flexibility of globally hyperbolic manifolds

In this short note, a question of patching together globally hyperbolic manifolds is adressed which appeared in the context of the construction of Hadamard states.Comment: 2 pages, submitted to 'Mathematische Zeitschrift

Degeneracies and scaling relations in general power-law models for gravitational lenses

The time delay in gravitational lenses can be used to derive the Hubble constant in a relatively simple way. The results of this method are less dependent on astrophysical assumptions than in many other methods. The most important uncertainty is related to the mass model used. We discuss a family of models with a separable radial power-law and an arbitrary angular dependence for the potential psi = r^beta * F(theta). Isothermal potentials are a special case of these models with beta=1. An additional external shear is used to take into account perturbations from other galaxies. Using a simple linear formalism for quadruple lenses, we can derive H0 as a function of the observables and the shear. If the latter is fixed, the result depends on the assumed power-law exponent according to H0 proportional to (2-beta)/beta. The effect of external shear is quantified by introducing a `critical shear' gamma_c as a measure for the amount of shear that changes the result significantly. The analysis shows, that in the general case H0 and gamma_c do not depend on the position of the lens galaxy. We discuss these results and compare with numerical models for a number of real lens systems.Comment: accepted for publication in MNRAS, 10 pages, 4 figures (eps included), uses mn2e.cls, amsmath.sty, times.st

Eigenvalues in Spectral Gaps of a Perturbed Periodic Manifold

We consider a non-compact Riemannian periodic manifold such that the corresponding Laplacian has a spectral gap. By continuously perturbing the periodic metric locally we can prove the existence of eigenvalues in a gap. A lower bound on the number of eigenvalue branches crossing a fixed level is established in terms of a discrete eigenvalue problem. Furthermore, we discuss examples of perturbations leading to infinitely many eigenvalue branches coming from above resp. finitely many branches coming from below.Comment: 30 pages, 3 eps-figures, LaTe

4. Wochenbericht M77/1

Im Pazifischen Ozean erstreckt sich westlich von Peru und Ecuador ein riesiges Gebiet, in dem lebenswichtiger Sauerstoff Mangelware ist. Dieses Gebiet ist Ziel der Expedition „M77“, zu der das deutsche Forschungsschiff METEOR am 22. Oktober 2008 ausläuft. Sie steht unter der Leitung von Kieler Meereswissenschaftlern des Sonderforschungsbereichs (SFB) 754, an dem das Leibniz-Institut für Meereswissenschaften (IFM-GEOMAR) und die Christian-Albrechts-Universität zu Kiel (CAU) beteiligt sind. In einem Weblog berichten die Forscher direkt von Bord der METEOR über ihrer Arbeit. METEOR Cruise 77/1 Talcahuano (Chile) – Callao (Peru) Weekly Report No. 4: 10. - 16. 11. 0