Exact response of the non-relativistic harmonic oscillator

Using Green$'$s function and operator techniques we give a closed expression for the response of a non-relativistic system interacting through confining, harmonic forces. The expression for the incoherent part permits rapid evaluation of coefficients in a 1/q expansion. A comparison is made with standard approximation methods.Comment: 9p.; INFN-ISS 93/2; WIS-93/50/Jun-P

Elliptic theory for sets with higher co-dimensional boundaries

Many geometric and analytic properties of sets hinge on the properties of harmonic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1. To this end, we turn to degenerate elliptic equations. Let $\Gamma \subset \mathbb R^n$ be an Ahlfors regular set of dimension $d<n-1$ (not necessarily integer) and $\Omega = \mathbb R^n \setminus \Gamma$. Let $L = - {\rm div} A\nabla$ be a degenerate elliptic operator with measurable coefficients such that the ellipticity constants of the matrix $A$ are bounded from above and below by a multiple of ${\rm dist}(\cdot, \Gamma)^{d+1-n}$. We define weak solutions; prove trace and extension theorems in suitable weighted Sobolev spaces; establish the maximum principle, De Giorgi-Nash-Moser estimates, the Harnack inequality, the H\"older continuity of solutions (inside and at the boundary). We define the Green function and provide the basic set of pointwise and/or $L^p$ estimates for the Green function and for its gradient. With this at hand, we define harmonic measure associated to $L$, establish its doubling property, non-degeneracy, change-of-the-pole formulas, and, finally, the comparison principle for local solutions. In another article to appear, we will prove that when $\Gamma$ is the graph of a Lipschitz function with small Lipschitz constant, we can find an elliptic operator $L$ for which the harmonic measure given here is absolutely continuous with respect to the $d$-Hausdorff measure on $\Gamma$ and vice versa. It thus extends Dahlberg's theorem to some sets of codimension higher than 1.Comment: 122 page

A harmonic/anharmonic energy partition method for lattice statics computations

A method of lattice statics analysis is developed. Consideration of anharmonic effects is restricted to finite regions surrounding lattice defects. All displacements of the crystal are expressed as the effect of unknown forces applied to a perfect harmonic lattice of infinite extent. Displacements are related to the unknown applied forces by means of the Green function of the perfect harmonic lattice, so that equilibrating forces need only be applied to the anharmonic region. The unknown forces are determined so as to maximize the complementary energy of the crystal, which yields a lower bound to the potential energy. The method does not require the explicit enforcement of equilibrium or compatibility conditions across the boundary between the harmonic and anharmonic regions. The performance of the method is assessed on the basis of selected numerical examples. The rate of convergence of the method with increasing domain size is found to be cubic. This is one or two orders of magnitude faster than rigid boundary methods based on the harmonic and continuum solutions, respectively