Related papers: Spherical geometry and integrable systems
Variables adapted to the quantum dynamics of spherically symmetric models are introduced, which further simplify the spherically symmetric volume operator and allow an explicit computation of all matrix elements of the Euclidean and…
Integrable models are often constructed with real systems in mind. The exact solvability of the models leads to results which are unambiguous and provide the correct physical picture. In this review, we discuss the physical basis of some…
Integrable discretizations are introduced for the rational and hyperbolic spin Ruijsenaars--Schneider models. These discrete dynamical systems are demonstrated to belong to the same integrable hierarchies as their continuous--time…
Schroedinger equations with position dependent mass which are scale invariant and admit second order integrals of motion are classified.
We suggest the notion of perfect integrability for quantum spin chains and conjecture that quantum spin chains are perfectly integrable. We show the perfect integrability for Gaudin models associated to simple Lie algebras of all finite…
The theory of Lie systems has recently been applied to Quantum Mechanics and additionally some integrability conditions for Lie systems of differential equations have also recently been analysed from a geometric perspective. In this paper…
An object in the bounded derived category D^b(Coh(X)) of coherent sheaves on a complex projective K3 surface X is spherical if it is rigid and simple. Although spherical objects form only a discrete set in the moduli stack of complexes,…
The integrability in quadratures of normality equation for spatially homogeneous dynamical systems in two-dimensional space is shown. The classical symmetries of this equation are calculated and the corresponding self-similar solutions are…
Given a spherical spacelike three-geometry, there exists a very simple algebraic condition which tells us whether, and in which, Schwarzschild solution this geometry can be smoothly embedded. One can use this result to show that any given…
A class of two-dimensional superintegrable systems on a constant curvature surface is considered as the natural generalization of some well known one-dimensional factorized systems. By using standard methods to find the shape-invariant…
Recently, Theophilou (J. Chem.Phys {\bf 149} 074104 (2018)) showed that a set of spherically symmetric densities determines uniquely the external potential in molecules and solids. Here, spherically symmetric Kohn-Sham-like equations are…
Non-orientable nanostructures are becoming feasable today. This lead us to the study of spin in these geometries. Hence a physically sound definition of spin is suggested. Using our definition, we study the question of the number of…
In this paper, we give definitions and characterizations of normal and spherical curves in the dual space. We show that normal curves are also spherical curves in D^3.
How do symmetries induce natural and useful quantum structures? This question is investigated in the context of models of three interacting particles in one-dimension. Such models display a wide spectrum of possibilities for dynamical…
It has been known for almost 200 years that some angles cannot be trisected by straightedge and compass alone. This paper studies the set of such angles as well as its complement $\mathcal{T}$, both regarded as subsets of the unit circle…
We describe the notion of stability of coherent systems as a framework to deal with redundancy. We define stable coherent systems and show how this notion can help the design of reliable systems. We demonstrate that the reliability of…
Quantum dynamics of integrable systems is discussed. Localized wave packets generalizing the conventional coherent states of minimal uncertainty are constructed. The wave packet moves along a certain trajectory and does not change its shape…
A few years ago Selivanova gave an existence proof for some integrable models, in fact geodesic flows on two dimensional manifolds, with a cubic first integral. However the explicit form of these models hinged on the solution of a nonlinear…
Integrable two-dimensional models which possess an integral of motion cubic or quartic in velocities are governed by a single prepotential, which obeys a nonlinear partial differential equation. Taking into account the latter's invariance…
Integer geometry on a plane deals with objects whose vertices are points in $\mathbb Z^2$. The congruence relation is provided by all affine transformations preserving the lattice $\mathbb Z^2$. In this paper we study circumscribed circles…