A numerical investigation of the solution of a class of fourth-order eigenvalue problems

This paper is concerned with the accurate numerical approximation of the spectral properties of the biharmonic operator on various domains in two dimensions. A number of analytic results concerning the eigenfunctions of this operator are summarized and their implications for numerical approximation are discussed. In particular, the asymptotic behaviour of the first eigenfunction is studied since it is known that this has an unbounded number of oscillations when approaching certain types of corners on domain boundaries. Recent computational results of Bjorstad & Tjostheim, using a highly accurate spectral Legendre-Galerkin method, have demonstrated that a number of these sign changes may be accurately computed on a square domain provided sufficient care is taken with the numerical method. We demonstrate that similar accuracy is also achieved using an unstructured finite-element solver which may be applied to problems on domains with arbitrary geometries. A number of results obtained from this mixed finite-element approach are then presented for a variety of domains. These include a family of circular sector regions, for which the oscillatory behaviour is studied as a function of the internal angle, and another family of (symmetric and non-convex) domains, for which the parity of the least eigenfunction is investigated. The paper not only verifies existing asymptotic theory, but also allows us to make a new conjecture concerning the eigenfunctions of the biharmonic operator

Semi-classical analysis of non self-adjoint transfer matrices in statistical mechanics. I

We propose a way to study one-dimensional statistical mechanics models with complex-valued action using transfer operators. The argument consists of two steps. First, the contour of integration is deformed so that the associated transfer operator is a perturbation of a normal one. Then the transfer operator is studied using methods of semi-classical analysis. In this paper we concentrate on the second step, the main technical result being a semi-classical estimate for powers of an integral operator which is approximately normal.Comment: 28 pp, improved the presentatio

Constrained Curve Fitting

We survey techniques for constrained curve fitting, based upon Bayesian statistics, that offer significant advantages over conventional techniques used by lattice field theorists.Comment: Lattice2001(plenary); plenary talk given by G.P. Lepage at Lattice 2001 (Berlin); 9 pages, 5 figures (postscript specials

Microwave-induced resistance oscillations and zero-resistance states in 2D electron systems with two occupied subbands

We report on theoretical studies of recently discovered microwave-induced resistance oscillations and zero resistance states in Hall bars with two occupied subbands. In the same results, resistance presents a peculiar shape which appears to have a built-in interference effect not observed before. We apply the microwave-driven electron orbit model, which implies a radiation-driven oscillation of the two-dimensional electron system. Thus, we calculate different intra and inter-subband electron scattering rates and times that are revealing as different microwave-driven oscillations frequencies for the two electronic subbands. Through scattering, these subband-dependent oscillation motions interfere giving rise to a striking resistance profile. We also study the dependence of irradiated magnetoresistance with power and temperature. Calculated results are in good agreement with experiments.Comment: 7 pages, 6 figure

The heavy quark's self energy from moving NRQCD on the lattice

We present a calculation of the heavy quark's self energy in moving NRQCD to one-loop in perturbation theory. Results for the energy shift and external momentum renormalisation are discussed and compared with non-perturbative results. We show that the momentum renormalisation is small, which is the result of a remnant of re-parameterisation invariance on the lattice.Comment: Talk given at Lattice2004(heavy), Fermilab, June 21-26, 200