Quaternionic differential operators
Stefano De Leo,Gisele Ducati +1 more
TL;DR: In this paper, the authors proposed a method to solve quaternionic and complex linear second order differential equations with constant coefficients, motivated by a quaternion formulation of quantum mechanics.
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Abstract: 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 the presence of quaternionic potentials, represents an interesting application of the mathematical material discussed in this paper.
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Citations
Edge detection of colour image based on quaternion fractional differential
TL;DR: In this paper, the authors extended the real fractional differential (RDF) to quaternion body and put forward a new concept: quaternions fractional differentiation (QFD), and applied it to edge detection of colour image.
78
Solving simple quaternionic differential equations
Stefano De Leo,Gisele Ducati +1 more
TL;DR: In this article, the authors prove existence and uniqueness for quaternionic initial value problems, discuss the reduction of order for homogeneous differential equations and extend to the noncommutative case the method of variation of parameters, and show that the standard Wronskian cannot uniquely be extended to the quaternion case.
Quaternionic quantum mechanics in real Hilbert space
TL;DR: In this article, a formulation of quaternionic quantum mechanics (H QM) is presented in terms of a real Hilbert space, using a physically motivated scalar product, and a novel quaternion Fourier series is obtained.
41
Algebraic tools for the study of quaternionic behavioral systems
TL;DR: In this paper, the authors study behavioral systems whose trajectories are given as solutions of quaternionic difference equations and derive new results concerning their Smith form, based on these results, they obtain a characterization of system theoretic properties such as controllability and stability.
39
Zeros of unilateral quaternionic polynomials
TL;DR: In this paper, the problem of finding the zeros of unilateral n-order polynomials can be solved by determining the eigenvectors of the corresponding companion matrix, which is probably superfluous in the case of quadratic equations for which a closed formula can be given.
References
Quaternions and matrices of quaternions
TL;DR: A brief survey on quaternions and matrices of quaternion is given in this article, where the authors present new proofs for certain known results and discuss the quaternionic analogues of complex matrices.
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Foundations of Quaternion Quantum Mechanics
TL;DR: In this article, a new kind of quantum mechanics using inner products, matrix elements, and coefficients assuming values that are quaternionic (and thus noncommutative) instead of complex is developed.
410
Right eigenvalue equation in quaternionic quantum mechanics
Stefano De Leo,G. Scolarici +1 more
TL;DR: In this paper, the authors studied the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n-dimensional quaternion vector spaces, and they gave a necessary and sufficient condition for the diagonalization of their representations.