Geometry of Riccati equations over normed division algebras

Abstract

This work presents and studies Riccati equations over finite-dimensional normed division algebras. We prove that a Riccati equation over a finite-dimensional normed division algebra A is a particular case of conformal Riccati equation on an Euclidean space and it can be considered as a curve in a Lie algebra of vector fields V so(dim A + 1, 1). Previous results on known types of Riccati equations are recovered from a new viewpoint. A new type of Riccati equations, the octonionic Riccati equations, are extended to the octonionic projective line OP 1 . As a new physical application, quaternionic Riccati equations are applied to study quaternionic Schrödinger equations on 1+1 dimensions.

Coherent and Squeezed States: Introductory Review of Basic Notions, Properties, and Generalizations

“…Besides, they have close formal similarities with general superpotentials leading to isospectral potentials in supersymmetric quantum mechanics [63][64][65][66][67][68][69]. Recent applications include propagation of optical beams in parabolic media [171][172][173] and Kerr media [174][175][176][177] as well, studies of the geometry of the Riccati equation [178] and the fourth-order Schrödinger equation with the energy spectrum of the Pöschl-Teller system [179]. Further discussion on the subject can be found in the Schuch's book on a nonlinear perspective to quantum theory [180].…”

Quasi-Lie schemes for PDEs

“…As a consequence, superposition rules simplify the application of numerical methods [67], and they enable us to analyse the properties of Lie systems, e.g. their periodic orbits [35], without knowing their general solutions.…”