Some basic properties of infinite dimensional Hamiltonian systems

We consider some fundamental properties of infinite dimensional Hamiltonian systems, both linear and nonlinear. For exemple, in the case of linear systems, we prove a symplectic version of the teorem of M. Stone. In the general case we establish conservation of energy and the moment function for system with symmetry. (The moment function was introduced by B. Kostant and J .M. Souriau). For infinite dimensional systems these conservation laws are more delicate than those for finite dimensional systems because we are dealing with partial as opposed to ordinary differential equations

Quantum Canonical Transformations revisited

A preferred form for the path integral discretization is suggested that allows the implementation of canonical transformations in quantum theory.Comment: 8 pages, LaTe

Recent revisions to corporate profits: what we know and when we knew it

Initial estimates in the National Income and Product Accounts significantly overstated U.S. corporate profits for the 1998-2000 period. Subsequent revisions reveal that the profitability of the nation's corporate sector in the late 1990s was substantially weaker than "real-time" data indicated. An unexpected surge in employee stock options exercised-and perhaps, in some sectors, firms' inflated statements of profit-may help explain the large downward revisions.Corporate profits ; Stock options ; Statistics ; Economic indicators

Sufficient conditions for the anti-Zeno effect

The ideal anti-Zeno effect means that a perpetual observation leads to an immediate disappearance of the unstable system. We present a straightforward way to derive sufficient conditions under which such a situation occurs expressed in terms of the decaying states and spectral properties of the Hamiltonian. They show, in particular, that the gap between Zeno and anti-Zeno effects is in fact very narrow.Comment: LatEx2e, 9 pages; a revised text, to appear in J. Phys. A: Math. Ge

Geometrical Description of Quantum Mechanics - Transformations and Dynamics

In this paper we review a proposed geometrical formulation of quantum mechanics. We argue that this geometrization makes available mathematical methods from classical mechanics to the quantum frame work. We apply this formulation to the study of separability and entanglement for states of composite quantum systems.Comment: 22 pages, to be published in Physica Script

Pseudospectral Calculation of the Wavefunction of Helium and the Negative Hydrogen Ion

We study the numerical solution of the non-relativistic Schr\"{o}dinger equation for two-electron atoms in ground and excited S-states using pseudospectral (PS) methods of calculation. The calculation achieves convergence rates for the energy, Cauchy error in the wavefunction, and variance in local energy that are exponentially fast for all practical purposes. The method requires three separate subdomains to handle the wavefunction's cusp-like behavior near the two-particle coalescences. The use of three subdomains is essential to maintaining exponential convergence. A comparison of several different treatments of the cusps and the semi-infinite domain suggest that the simplest prescription is sufficient. For many purposes it proves unnecessary to handle the logarithmic behavior near the three-particle coalescence in a special way. The PS method has many virtues: no explicit assumptions need be made about the asymptotic behavior of the wavefunction near cusps or at large distances, the local energy is exactly equal to the calculated global energy at all collocation points, local errors go down everywhere with increasing resolution, the effective basis using Chebyshev polynomials is complete and simple, and the method is easily extensible to other bound states. This study serves as a proof-of-principle of the method for more general two- and possibly three-electron applications.Comment: 23 pages, 20 figures, 2 tables, Final refereed version - Some references added, some stylistic changes, added paragraph to matrix methods section, added last sentence to abstract

Dynamics and Lax-Phillips scattering for generalized Lamb models

This paper treats the dynamics and scattering of a model of coupled oscillating systems, a finite dimensional one and a wave field on the half line. The coupling is realized producing the family of selfadjoint extensions of the suitably restricted self-adjoint operator describing the uncoupled dynamics. The spectral theory of the family is studied and the associated quadratic forms constructed. The dynamics turns out to be Hamiltonian and the Hamiltonian is described, including the case in which the finite dimensional systems comprises nonlinear oscillators; in this case the dynamics is shown to exist as well. In the linear case the system is equivalent, on a dense subspace, to a wave equation on the half line with higher order boundary conditions, described by a differential polynomial $p(\partial_x)$ explicitely related to the model parameters. In terms of such structure the Lax-Phillips scattering of the system is studied. In particular we determine the incoming and outgoing translation representations, the scattering operator, which turns out to be unitarily equivalent to the multiplication operator given by the rational function $-p(i\kappa)^*/p(i\kappa)$, and the Lax-Phillips semigroup, which describes the evolution of the states which are neither incoming in the past nor outgoing in the future

Spectral properties on a circle with a singularity

We investigate the spectral and symmetry properties of a quantum particle moving on a circle with a pointlike singularity (or point interaction). We find that, within the U(2) family of the quantum mechanically allowed distinct singularities, a U(1) equivalence (of duality-type) exists, and accordingly the space of distinct spectra is U(1) x [SU(2)/U(1)], topologically a filled torus. We explore the relationship of special subfamilies of the U(2) family to corresponding symmetries, and identify the singularities that admit an N = 2 supersymmetry. Subfamilies that are distinguished in the spectral properties or the WKB exactness are also pointed out. The spectral and symmetry properties are also studied in the context of the circle with two singularities, which provides a useful scheme to discuss the symmetry properties on a general basis.Comment: TeX, 26 pages. v2: one reference added and two update

Behavioral implications of shortlisting procedures

We consider two-stage “shortlisting procedures” in which the menu of alternatives is first pruned by some process or criterion and then a binary relation is maximized. Given a particular first-stage process, our main result supplies a necessary and sufficient condition for choice data to be consistent with a procedure in the designated class. This result applies to any class of procedures with a certain lattice structure, including the cases of “consideration filters,” “satisficing with salience effects,” and “rational shortlist methods.” The theory avoids background assumptions made for mathematical convenience; in this and other respects following Richter’s classical analysis of preference-maximizing choice in the absence of shortlisting