Invariant manifolds and orbit control in the solar sail three-body problem

In this paper we consider issues regarding the control and orbit transfer of solar sails in the circular restricted Earth-Sun system. Fixed points for solar sails in this system have the linear dynamical properties of saddles crossed with centers; thus the fixed points are dynamically unstable and control is required. A natural mechanism of control presents itself: variations in the sail's orientation. We describe an optimal controller to control the sail onto fixed points and periodic orbits about fixed points. We find this controller to be very robust, and define sets of initial data using spherical coordinates to get a sense of the domain of controllability; we also perform a series of tests for control onto periodic orbits. We then present some mission strategies involving transfer form the Earth to fixed points and onto periodic orbits, and controlled heteroclinic transfers between fixed points on opposite sides of the Earth. Finally we present some novel methods to finding periodic orbits in circumstances where traditional methods break down, based on considerations of the Center Manifold theorem

Quasi-exact solvability beyond the SL(2) algebraization

We present evidence to suggest that the study of one dimensional quasi-exactly solvable (QES) models in quantum mechanics should be extended beyond the usual \sla(2) approach. The motivation is twofold: We first show that certain quasi-exactly solvable potentials constructed with the \sla(2) Lie algebraic method allow for a new larger portion of the spectrum to be obtained algebraically. This is done via another algebraization in which the algebraic hamiltonian cannot be expressed as a polynomial in the generators of \sla(2). We then show an example of a new quasi-exactly solvable potential which cannot be obtained within the Lie-algebraic approach.Comment: Submitted to the proceedings of the 2005 Dubna workshop on superintegrabilit

Spin-phonon induced magnetic order in Kagome ice

We study the effects of lattice deformations on the Kagome spin ice, with Ising spins coupled by nearest neighbor exchange and long range dipolar interactions, in the presence of in-plane magnetic fields. We describe the lattice energy according to the Einstein model, where each site distortion is treated independently. Upon integration of lattice degrees of freedom, effective quadratic spin interactions arise. Classical MonteCarlo simulations are performed on the resulting model, retaining up to third neighbor interactions, under different directions of the magnetic field. We find that, as the effect of the deformation is increased, a rich plateau structure appears in the magnetization curves.Comment: 7 pages, 8 figure

Exceptional orthogonal polynomials and the Darboux transformation

We adapt the notion of the Darboux transformation to the context of polynomial Sturm-Liouville problems. As an application, we characterize the recently described $X_m$ Laguerre polynomials in terms of an isospectral Darboux transformation. We also show that the shape-invariance of these new polynomial families is a direct consequence of the permutability property of the Darboux-Crum transformation.Comment: corrected abstract, added references, minor correction

The Inverse Amplitude Method and Adler Zeros

The Inverse Amplitude Method is a powerful unitarization technique to enlarge the energy applicability region of Effective Lagrangians. It has been widely used to describe resonances from Chiral Perturbation Theory as well as for the Strongly Interacting Symmetry Breaking Sector. In this work we show how it can be slightly modified to account also for the sub-threshold region, incorporating correctly the Adler zeros required by chiral symmetry and eliminating spurious poles. These improvements produce negligible effects on the physical region.Comment: 17 pages, 4 figure

A conjecture on Exceptional Orthogonal Polynomials

Exceptional orthogonal polynomial systems (X-OPS) arise as eigenfunctions of Sturm-Liouville problems and generalize in this sense the classical families of Hermite, Laguerre and Jacobi. They also generalize the family of CPRS orthogonal polynomials. We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux-Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X2-OPS. The classification includes all cases known to date plus some new examples of X2-Laguerre and X2-Jacobi polynomials