Conic degeneration and the determinant of the Laplacian

We investigate the behavior of various spectral invariants, particularly the determinant of the Laplacian, on a family of smooth Riemannian manifolds which undergo conic degeneration; that is, which converge in a particular way to a manifold with a conical singularity. Our main result is an asymptotic formula for the determinant up to terms which vanish as the degeneration parameter goes to zero. The proof proceeds in two parts; we study the fine structure of the heat trace on the degenerating manifolds via a parametrix construction, and then use that fine structure to analyze the zeta function and determinant of the Laplacian.Comment: 41 pages, 7 figures. Version 2: bug fixed in Theorem 2 statement, other minor change

Experimentally Probing the Shape of Extra Dimensions

In brane world scenarios in which only gravity can propagate in the extra dimensions, effects on the gravitational force may be experimentally testable if there are two or three large extra dimensions. The strength of the force at distances smaller than the compactification radius will be sensitive to the volume of the extra dimensions, but the determination of the shape requires knowing the gravitational potential at intermediate scales. We determine the dependence of the potential vs. distance as a function of both the relative size of the extra dimensions and the possible angle between the extra dimensional unit vectors, and show that high precision measurements of the gravitational force will allow the determination of the shape of the extra dimensions.Comment: Much more pedagogical version. Version to be published in the American Journal of Physic

SET based experiments for HTSC materials: II

The cuprates seem to exhibit statistics, dimensionality and phase transitions in novel ways. The nature of excitations [i.e. quasiparticle or collective], spin-charge separation, stripes [static and dynamics], inhomogeneities, psuedogap, effect of impurity dopings [e.g. Zn, Ni] and any other phenomenon in these materials must be consistently understood. In this note we further discuss our original suggestion of using Single Electron Tunneling Transistor [SET] based experiments to understand the role of charge dynamics in these systems. Assuming that SET operates as an efficient charge detection system we can expect to understand the underlying physics of charge transport and charge fluctuations in these materials for a range of doping. Experiments such as these can be classed in a general sense as mesoscopic and nano characterization of cuprates and related materials. In principle such experiments can show if electron is fractionalized in cuprates as indicated by ARPES data. In contrast to flux trapping experiments SET based experiments are more direct in providing evidence about spin-charge separation. In addition a detailed picture of nano charge dynamics in cuprates may be obtained.Comment: 10 pages revtex plus four figures; ICMAT 2001 Conference Symposium P: P10-0

The heat kernel on curvilinear polygonal domains in surfaces

We construct the heat kernel on curvilinear polygonal domains in arbitrary surfaces for Dirichlet, Neumann, and Robin boundary conditions as well as mixed problems, including those of Zaremba type. We compute the short time asymptotic expansion of the heat trace and apply this expansion to demonstrate a collection of results showing that corners are spectral invariants