Related papers: A remark on rational isochronous potentials
A class of periodic differential $n$-dimensional systems with patch structure with (possibly infinite) delay and nonlinear impulses is considered. These systems incorporate very general nonlinearities and impulses whose signs may vary.…
The purpose of this paper is to study various monotonicity conditions of the period function $T(c)$ (energy-dependent) for potential systems $\ddot x + g(x)=0$ with a center at the origin 0. We had before identified a family of new criteria…
New families of time-dependent potentials related to the parametric oscillator are introduced. This is achieved by introducing some general time-dependent operators that factorize the appropriate constant of motion (quantum invariant) of…
Using the formalism of supersymmetric quantum mechanics, we obtain a large number of new analytically solvable one-dimensional periodic potentials and study their properties. More specifically, the supersymmetric partners of the Lame…
We reconsider the conjecture by Gepner that the fusion ring of a rational conformal field theory is isomorphic to a ring of polynomials in $n$ variables quotiented by an ideal of constraints that derive from a potential. We show that in a…
Time analysis of oscillations of a particle between wells in the one-dimensional double-well potential with infinite high outside walls, based on wave packet use and energy spectrum analysis, is presented. For the double-well potential of…
We examine the dynamics of a particle in a general rotating quadratic potential, not necessarily stable or isotropic, using a general complex mode formalism. The problem is equivalent to that of a charged particle in a quadratic potential…
The gauge equivalent counterparts of the some (1+1)-, or (2+0)-dimensional sigma models with potentials are found. The gauge equivalence between the some soliton equations of spin-phonon systems and the Yajima-Oikawa and Ma equations are…
This paper is concerned with the monotonicity of the period function for closed orbits of systems of the Li\'enard II type equation given by $\ddot{x} + f(x)\dot{x}^{2} + g(x) = 0$. We generalize Chicone's result regarding the monotonicity…
Periods are defined as integrals of semialgebraic functions defined over the rationals. Periods form a countable ring not much is known about. Examples are given by taking the antiderivative of a power series which is algebraic over the…
We show that all bounded trajectories in the two dimensional classical system with the potential $V(r,\phi)=\omega^2 r^2+ \frac{\al k^2}{r^2 \cos^2 {k \phi}}+ \frac{\beta k^2}{r^2 \sin^2 {k \phi}}$ are closed for all integer and rational…
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…
We consider motion in a periodic potential in a classical, quantum, and semiclassical context. Various results on the distribution of asymptotic velocities are proven.
A set of exactly solvable one-dimensional quantum mechanical potentials is described. It is defined by a finite-difference-differential equation generating in the limiting cases the Rosen-Morse, harmonic, and P\"oschl-Teller potentials.…
Motivated by the concept of shape invariance in supersymmetric quantum mechanics, we obtain potentials whose spectrum consists of two shifted sets of equally spaced energy levels. These potentials are similar to the Calogero-Sutherland…
We consider an open connected set $\Omega$ and a smooth potential $U$ which is positive in $\Omega$ and vanishes on $\partial\Omega$. We study the existence of orbits of the mechanical system \[ \ddot{u}=U_x(u), \] that connect different…
We prove a Lagrangian analogue of the Conley conjecture: given a 1-periodic Tonelli Lagrangian with global flow on a closed configuration space, the associated Euler-Lagrange system has infinitely many periodic solutions. More precisely, we…
We review basic ideas and basic examples of the theory of the inverse spectral problems.
The quantum mechanical bound states of the $-{\alpha}/x^2$ potential are truly anomalous. We revisit this problem by adopting a slightly modified version of this potential, one that adopts a cutoff in the potential arbitrarily close to the…
In this paper, we propose an analytical non-polynomial potential system which has infinitely many critical periodic orbits in phase plane. By showing the existence of infinitely many $2\pi-$ periodic solutions, the proof bases on…