Properties of the Zeros of the Sum of two Polynomials

Some properties—including relations having a Diophantine character—of the zeros of the sum of two polynomials are reported.     

Polynomials Satisfying Functional and Differential Equations and Diophantine Properties of Their Zeros

By investigating the behavior of two solvable isochronous N-body problems in the immediate vicinity of their equilibria, functional equations satisfied by the para-Jacobi polynomial ${p_{N} \left(0, 1; \gamma; x \right)}$ and by the Jacobi polynomial ${P_{N}^{\left(-N-1,-N-1 \right)} \left(x \right)}$ (or, equivalently, by the Gegenbauer polynomial ${C_{N}^{-N-1/2}\left( x \right) }$ ) are identified, as well as Diophantine properties of the zeros and coefficients of these polynomials.     

Many-Body Problem with Quadratic and/or Inversely-Quadratic Potentials in One- and More-Dimensional Spaces: Some Retrospective Remarks

In the context of an ambient space with an arbitrary number $d$ of dimensions, the many-body problem consisting of an arbitrary number $N$ of particles confined by a common, external harmonic potential (realizing a container with soft walls) and interacting among themselves and with the environment with arbitrary conservative repulsive forces scaling as the inverse cube of distances, displays a peculiar behaviour: its effective volume oscillates isochronously without damping. We recently discovered this remarkable phenomenon (valid in the context of both classical and quantum mechanics) and discussed its implications in the context of statistical mechanics and thermodynamics; but after publishing these findings we were informed that essentially analogous results had been previously obtained by Lyndell-Bell and Lyndell-Bell. In the present paper, motivated by the need we felt to acknowledge this fact, we also offer some retrospective remarks on the $N$ -body problem with quadratic and/or inversely-quadratic potentials in one- and more-dimensional space.     

Properties of the Zeros of the Polynomials Belonging to the Askey Scheme

In this paper, we provide properties—which are, to the best of our knowledge, new—of the zeros of the polynomials belonging to the Askey scheme. These findings include Diophantine relations satisfied by these zeros when the parameters characterizing these polynomials are appropriately restricted.     

Two Peculiar Classes of Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations

In this paper we identify certain peculiar systems of 2 discrete-time evolution equations,

which are algebraically solvable. Here l is the “discrete-time” independent variable taking integer values (l = 0, 1, 2, . . .), x_n ≡ x_n(l) are 2 dependent variables, and are the corresponding 2 updated variables. In a previous paper the 2 functions F⁽ⁿ⁾(x₁, x₂), n = 1, 2, were defined as follows: F⁽ⁿ⁾(x₁, x₂) = P₂ (x_n, x_n+1), n = 1, 2 mod[2], with P₂(x₁, x₂) a specific second-degree homogeneous polynomial in the 2 (indistinguishable!) dependent variables x₁(l) and x₂(l). In the present paper we further clarify some aspects of that model and we present its extension to the case when a specific homogeneous function of arbitrary (integer) degree k (hence a polynomial of degree k when k > 0) in the 2 dependent variables x₁(l) and x₂(l).     

New evolution PDEs with many isochronous solutions

Certain nonlinear evolution PDEs in 1+1 variables (time and space) are identified, featuring a positive parameter ω and evolving, for a large class of initial data, periodically with the fixed period T=2π/ω (or perhaps with p a small integer). They are autonomous (i.e., they do not feature any explicit dependence on the time variable), but they generally (although not quite all of them) depend explicitly on the space variable hence are not translation-invariant. They are integrable, having been obtained by applying an appropriate change of dependent and independent variables to certain nonlinear evolution PDEs whose integrable character has been recently ascertained. Solutions of some of these PDEs are exhibited.     

The peculiar (monic) polynomials,the zeros of which equal their coefficients

We evaluate the number of complex monic polynomials, of arbitrary degree N, the zeros of which are equal to their coefficients. In the following, we call polynomials with this property peculiar polynomials. We further show that the problem of determining the peculiar polynomials of degree N simplifies when any of the coefficients is either 0 or 1. We proceed to estimate the numbers of peculiar polynomials of degree N having one coefficient zero, or one coefficient equal to one, or neither.     

A Hamiltonian yielding damped motion in an homogeneous magnetic field: quantum treatment

In earlier work, a Hamiltonian describing the classical motion of a particle moving in two dimensions under the combined influence of a perpendicular magnetic field and of a damping force proportional to the particle velocity, was indicated. Here we derive the quantum propagator for the Hamiltonian in different representations, one corresponding to momentum space, the other to position, and the third to a natural choice of “velocity” variables. We call attention to the following noteworthy fact: the Hamiltonian contains three parameters which do not in any way influence the motion of the position of the particle. However, at the quantum level, the propagator, even in the position representation, depends in an intricate way on these classically irrelevant parameters. This creates considerable doubt as to the validity of such a quantization procedure, as the physical results predicted differ for various Hamiltonians, all of which describe the dissipative dynamics equally well.