22,179 research outputs found
Quaternionic differential operators
Motivated by a quaternionic formulation of quantum mechanics, we discuss
quaternionic and complex linear differential equations. We touch only a few
aspects of the mathematical theory, namely the resolution of the second order
differential equations with constant coefficients. We overcome the problems
coming out from the loss of the fundamental theorem of the algebra for
quaternions and propose a practical method to solve quaternionic and complex
linear second order differential equations with constant coefficients. The
resolution of the complex linear Schrodinger equation, in presence of
quaternionic potentials, represents an interesting application of the
mathematical material discussed in this paper.Comment: 25 pages, AMS-Te
Solving simple quaternionic differential equations
The renewed interest in investigating quaternionic quantum mechanics, in
particular tunneling effects, and the recent results on quaternionic
differential operators motivate the study of resolution methods for
quaternionic differential equations. In this paper, by using the real matrix
representation of left/right acting quaternionic operators, we prove existence
and uniqueness for quaternionic initial value problems, discuss the reduction
of order for quaternionic homogeneous differential equations and extend to the
non-commutative case the method of variation of parameters. We also show that
the standard Wronskian cannot uniquely be extended to the quaternionic case.
Nevertheless, the absolute value of the complex Wronskian admits a
non-commutative extension for quaternionic functions of one real variable.
Linear dependence and independence of solutions of homogeneous (right) H-linear
differential equations is then related to this new functional. Our discussion
is, for simplicity, presented for quaternionic second order differential
equations. This involves no loss of generality. Definitions and results can be
readily extended to the n-order case.Comment: 9 pages, AMS-Te
Right eigenvalue equation in quaternionic quantum mechanics
We study the right eigenvalue equation for quaternionic and complex linear
matrix operators defined in n-dimensional quaternionic vector spaces. For
quaternionic linear operators the eigenvalue spectrum consists of n complex
values. For these operators we give a necessary and sufficient condition for
the diagonalization of their quaternionic matrix representations. Our
discussion is also extended to complex linear operators, whose spectrum is
characterized by 2n complex eigenvalues. We show that a consistent analysis of
the eigenvalue problem for complex linear operators requires the choice of a
complex geometry in defining inner products. Finally, we introduce some
examples of the left eigenvalue equations and highlight the main difficulties
in their solution.Comment: 24 pages, AMS-Te
Quaternions and Special Relativity
We reformulate Special Relativity by a quaternionic algebra on reals. Using
{\em real linear quaternions}, we show that previous difficulties, concerning
the appropriate transformations on the space-time, may be overcome. This
implies that a complexified quaternionic version of Special Relativity is a
choice and not a necessity.Comment: 17 pages, latex, no figure
Partial immersions and partially free maps
In a recent paper~\cite{DDL10} we studied basic properties of partial
immersions and partially free maps, a generalization of free maps introduced
first by Gromov in~\cite{Gro70}. In this short note we show how to build
partially free maps out of partial immersions and use this fact to prove that
the partially free maps in critical dimension introduced in Theorems 1.1-1.3
of~\cite{DDL10} for three important types of distributions can actually be
built out of partial immersions. Finally, we show that the canonical contact
structure on \bR^{2n+1} admits partial immersions in critical dimension for
every .Comment: 8 pages, submitted to the proceedings of the conference DGA201
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