Dynamical Symmetry Approach to Periodic Hamiltonians

We show that dynamical symmetry methods can be applied to Hamiltonians with periodic potentials. We construct dynamical symmetry Hamiltonians for the Scarf potential and its extensions using representations of su(1,1) and so(2,2). Energy bands and gaps are readily understood in terms of representation theory. We compute the transfer matrices and dispersion relations for these systems, and find that the complementary series plays a central role as well as non-unitary representations.Comment: 20 pages, 7 postscript figure

Stress intensity factor in a tapered specimen

The general problem of a tapered specimen containing an edge crack is formulated in terms of a system of singular integral equations. The equations are solved and the stress intensity factor is calculated for a compact and for a slender tapered specimen, the latter simulating the double cantilever beam. The results are obtained primarily for a pair of concentrated forces and for crack surface wedge forces. The stress intensity factors are also obtained for a long strip under uniform tension which contains inclined edge cracks

Interaction between a crack and a soft inclusion

With the application to weld defects in mind, the interaction problem between a planar-crack and a flat inclusion in an elastic solid is considered. The elastic inclusion is assumed to be sufficiently thin so that the thickness distribution of the stresses in the inclusion may be neglected. The problem is reduced to a system of four integral equations having Cauchy-type dominant kernels. The stress intensity factors are calculated and tabulated for various crack-inclusion geometries and the inclusion to matrix modulus ratios, and for general homogeneous loadiong conditions away from the crack-inclusion region

Uncovering the Risk-Return Relation in the Stock Market

There is an ongoing debate in the literature about the apparent weak or negative relation between risk (conditional variance) and return (expected returns) in the aggregate stock market. We develop and estimate an empirical model based on the ICAPM to investigate this relation. Our primary innovation is to model and identify empirically the two components of expected returns--the risk component and the component due to the desire to hedge changes in investment opportunities. We also explicitly model the effect of shocks to expected returns on ex post returns and use implied volatility from traded options to increase estimation efficiency. As a result, the coefficient of relative risk aversion is estimated more precisely, and we find it to be positive and reasonable in magnitude. Although volatility risk is priced, as theory dictates, it contributes only a small amount to the time-variation in expected returns. Expected returns are driven primarily by the desire to hedge changes in investment opportunities. It is the omission of this hedge component that is responsible for the contradictory and counter-intuitive results in the existing literature.

Quantum Antiferromagnetism of Fermions in Optical Lattices with Half-filled p-band

We study Fermi gases in a three-dimensional optical lattice with five fermions per site, i.e. the s-band is completely filled and the p-band with three-fold degeneracy is half filled. We show that, for repulsive interaction between fermions, the system will exhibit spin-3/2 antiferromagnetic order at low temperature. This conclusion is obtained in strong interaction regime by strong coupling expansion which yields an isotropic spin-3/2 Heisenberg model, and also in weak interaction regime by Hatree-Fock mean-field theory and analysis of Fermi surface nesting. We show that the critical temperature for this antiferromagnetism of a p-band Mott insulator is about two orders of magnitudes higher than that of an $s$-band Mott insulator, which is close to the lowest temperature attainable nowadays