Quaternionic Differential Operators Stefano De Leo and Gisele Ducati

Quaternionic differential operators Stefano De Leo and Gisele Ducati

Citation: Journal of Mathematical Physics 42, 2236 (2001); doi: 10.1063/1.1360195 View online: http://dx.doi.org/10.1063/1.1360195 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/42/5?ver=pdfcov Published by the AIP Publishing

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Quaternionic differential operators Stefano De Leoa) Department of Applied Mathematics, University of Campinas, CP 6065, SP 13083-970, Campinas, Brazil Gisele Ducatib) Department of Applied Mathematics, University of Campinas, CP 6065, SP 13083-970, Campinas, Brazil and Department of Mathematics, University of Parana, CP 19081, PR 81531-970, Curitiba, Brazil ͑Received 9 October 2000; accepted for publication 6 February 2001͒ 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 Schro¨- dinger equation, in the presence of quaternionic potentials, represents an interesting application of the mathematical material discussed in this paper. © 2001 Ameri- can Institute of Physics. ͓DOI: 10.1063/1.1360195͔

I. INTRODUCTION There is substantial literature analyzing the possibility to discuss quantum systems by adopting quaternionic wave functions.1–14 This research field has been attacked by a number of people leading to substantial progress. In the last years, many articles,15–31 review papers32–34 and books35–37 provided a detailed investigation of group theory, eigenvalue problem, scattering theory, relativistic wave equations, Lagrangian formalism and variational calculus within a quaternionic formulation of quantum mechanics and field theory. In this context, by observing that the formulation of physical problems in mathematical terms often requires the study of partial differential equations, we develop the necessary theory to solve quaternionic and complex linear differential equations. The main difficulty in carrying out the solution of quaternionic differential equations is obviously represented by the noncommutative nature of the quaternionic field. The standard methods of resolution break down and, consequently, we need to modify the classical approach. It is not our purpose to develop a complete quaternionic theory of differential equations. This exceeds the scope of this paper. The main objective is to include what seemed to be most important for an introduction to this subject. In particular, we restrict ourselves to second order differential equations and give a practical method to solve such equations when quaternionic constant coefficients appear. Some of the results given in this paper can be obtained by translation into a complex formalism.15,16,31 Nevertheless, many subtleties of quaternionic calculus are often lost by using the translation trick. See, for example, the difference between quaternionic and complex geometry in quantum mechanics,32,34 generalization of variational calculus,9,10 the choice of a one-dimensional quaternionic Lorentz group for special relativity,21 the new definitions of transpose and determinant for quaternionic matrices.29 A wholly quaternionic derivation of the general solution of second order differential equations requires a detailed discussion of the fundamental theorem of algebra for quaternions, a revision of the resolution methods and a quaternionic generalization of the complex results.

a͒Electronic mail: [email protected] b͒Electronic mail: [email protected]

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The study of quaternionic linear second order differential equations with constant coefficients is based on the explicit resolution of the characteristic quadratic equation.38–41 We shall show that the loss of fundamental theorem of the algebra for quaternions does not represent a problem in solving quaternionic linear second order differential equations with constant coefficients. From there, we introduce more advanced concepts, like diagonalization and Jordan form for quaternionic and complex linear matrix operators, which are developed in detail in the recent literature22–31 and we apply them to solve quaternionic and complex linear second order differential equations with constant coefficients. As an application of the mathematical material presented in this paper, we discuss the complex linear Schro¨dinger equation in the presence of quaternionic potentials and solve such an equation for stationary states and constant potentials. We also calculate the relation between the reflection and transmition coefficients for the step and square potential and give the quaternionic solution for bound states. This work was intended as an attempt at motivating the study of quaternionic and complex linear differential equations in view of their future applications within a quaternionic formulation of quantum mechanics. In particular, our future objective is to understand the role that such equations could play in developing nonrelativistic quaternionic quantum dynamics4 and the mean- ing that quaternionic potentials15,16 could play in discussing CP violation in the kaon system.36 In order to give a clear exposition and to facilitate access to the individual topics, the sections are rendered as self-contained as possible. In Sec. II, we review some of the standard concepts used in quaternionic quantum mechanics, i.e., state vector, probability interpretation, scalar product and left/right quaternionic operators.29,35,42–45 Section III contains a brief discussion of the momentum operator. In Sec. IV, we summarize without proofs the relevant material on quaternionic eigenvalue equations from Ref. 31. Section V is devoted to the study of the one- dimensional Schro¨dinger equation in quaternionic quantum mechanics. Sections VI and VII pro- vide a detailed exposition of quaternionic and complex linear differential equations. In Sec. VIII, we apply the results of previous sections to the one-dimensional Schro¨dinger equation with quaternionic constant potentials. Our conclusions are drawn in the final section.

II. STATES AND OPERATORS IN QUATERNIONIC QUANTUM MECHANICS In this section, we give a brief survey of the basic mathematical tools used in quaternionic quantum mechanics.32–37 The quantum state of a particle is deﬁned, at a given instant, by a quaternionic wave function interpreted as a probability amplitude given by

⌿͑ ͒ϭ͓ ϩ ͔͑ ͒ ͑ ͒ r f 0 h•f r , 1 ϭ ϭ 3→ ϭ where h (i, j,k), f ( f 1 , f 2 , f 3) and f m :R R, m 0,1,2,3. The probabilistic interpretation of this wave function requires that it belong to the Hilbert vector space of square-integrable functions. We shall denote by F the set of wave functions composed of sufﬁciently regular functions of this vector space. The same function ⌿(r) can be represented by several distinct sets of components, each one corresponding to the choice of a particular basis. With each pair of elements of F, ⌿(r), and ⌽(r), we associate the quaternionic scalar product,

͑⌿,⌽͒ϭ ͵ d3r⌿¯ ͑r͒⌽͑r͒, ͑2͒

where

⌿¯ ͑ ͒ϭ͓ Ϫ ͔͑ ͒ ͑ ͒ r f 0 h•f r 3 represents the quaternionic conjugate of ⌿(r). O ⌿ ෈F A quaternionic linear operator, H , associates with every wave function (r) another O ⌿ ෈F wave function H (r) , the correspondence being linear from the right on H,

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O ͓⌿ ͑ ͒ ϩ⌿ ͑ ͒ ͔ϭ͓O ⌿ ͑ ͔͒ ϩ͓O ⌿ ͑ ͔͒ H 1 r q1 2 r q2 H 1 r q1 H 2 r q2 , ෈ q1,2 H. Due to the noncommutative nature of the quaternionic ﬁeld we need to introduce com- O O plex and real linear quaternionic operators, respectively, denoted by C and R , the correspondence being linear from the right on C and R O ͓⌿ ͑ ͒ ϩ⌿ ͑ ͒ ͔ϭ͓O ⌿ ͑ ͔͒ ϩ͓O ⌿ ͑ ͔͒ C 1 r z1 2 r z2 C 1 r z1 C 2 r z2 , O ͓⌿ ͑ ͒␭ ϩ⌿ ͑ ͒␭ ͔ϭ͓O ⌿ ͑ ͔͒␭ ϩ͓O ⌿ ͑ ͔͒␭ R 1 r 1 2 r 2 H 1 r 1 H 2 r 2 , ෈ ␭ ෈ z1,2 C and 1,2 R. As a concrete illustration of these operators let us consider the case of a ﬁnite, say n-dimensional, quaternionic Hilbert space. The wave function ⌿(r) will then be a column vector,

⌿ 1 ⌿ ⌿ϭ 2 ⌿ ෈F ͩ ͪ , 1,2,...,n . Ӈ ⌿ n

Quaternionic, complex and real linear operators will be represented by nϫn quaternionic matrices A O O ⌿ M n͓ ͔, where represents the space of real operators acting on the components of and Aϭ A A A ( H , C , R) denote the real algebras, A H : ͕1, L, R, L*R͖16 , A ͕ ͖ C : 1, L, Ri , LRi 8 , A ͕ ͖ R : 1, L 4 , generated by the left and right operators,

͑ ͒ ͑ ͒ ͑ ͒ Lª Li ,L j ,Lk , Rª Ri ,R j ,Rk , 4 and by the mixed operators,

͕ ͖ ϭ ͑ ͒ L*Rª L pRq , p,q i, j,k. 5 The action of these operators on the quaternionic wave function ⌿ is given by

L⌿ϵh⌿, R⌿ϵ⌿h.

The operators L and R satisfy the left/right quaternionic algebra,

2ϭ 2ϭ 2ϭ ϭ 2ϭ 2ϭ 2ϭ ϭϪ Li L j Lk LiL jLk Ri R j Rk RkR jRi 1,

and the following commutation relations:

ϭ ͓L p ,Rq͔ 0.

III. SPACE TRANSLATIONS AND QUATERNIONIC MOMENTUM OPERATOR

Space translation operators in quaternionic quantum mechanics are deﬁned in the coordinate representation by the real linear anti-Hermitian operator,36

͒ ͑ ͒ ץ ץ ץ ٢ϵ ͑ x , y , z . 6

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To construct an observable momentum operator we must look for a Hermitian operator that has all the properties of the momentum expected by analogy with the momentum operator in complex quantum mechanics. The choice of the quaternionic linear operator, ͒ ͑ P ϭϪ ប٢ L Li , 7 as a Hermitian momentum operator, would appear completely satisfactory, until we consider the H translation invariance for quaternionic Hamiltonians, q . In fact, due to the presence of the left ͑ ͒ H acting imaginary unit i, the momentum operator 7 does not commute with the j/k-part of q . Thus, although this deﬁnition of the momentum operator gives a Hermitian operator, we must H ϭ ٢ ٢ return to the anti-Hermitian operator to get a translation generator, ͓ , q͔ 0. A second possibility to be considered is represented by the complex linear momentum operator, introduced by Rotelli in Ref. 44, ͒ ͑ P ϭϪ ប٢ R Ri . 8 P O The commutator of R with a quaternionic linear operator H gives ⌿ ٢ P O ⌿ϭប O ͓ R , ͔ ͓ , ͔ i. O H Taking H to be a translation invariant quaternionic Hamiltonian q , we have P H ϭ ͓ R , q͔ 0. However, this second deﬁnition of the momentum operator has the following problem: the com- P plex linear momentum operator R does not represent a quaternionic Hermitian operator. In fact, by computing the difference

⌿ P ⌽͒Ϫ ⌽ P ⌿͒ ͑ , R ͑ , R , P which should vanish for a Hermitian operator R , we find ͒ ͑ ٢⌽͒ ⌿ P ⌽͒Ϫ P ⌿ ⌽͒ϭប ⌿ ͑ , R ͑ R , ͓i,͑ , ͔, 9 which is in general nonvanishing. There is one important case in which the right-hand side of Eq. ͑ ͒ P 9 does vanish. The operator R gives a satisfactory definition of the Hermitian momentum operator when restricted to a complex geometry,45 that is a complex projection of the quaternionic ⌿ P ⌽ scalar product, ( , R )C . Note that the assumption of a complex projection of the quaternionic scalar product does not imply complex wave functions. The state of quaternionic quantum mechanics with complex geometry will be again described by vectors of a quaternionic Hilbert space. In quaternionic quantum mechanics with complex geometry observables can be represented by the quaternionic Hermitian operator, H, obtained taking the spectral decomposition31 of the corre- Ϫ sponding anti-Hermitian operator, A, or simply by the complex linear operator, ARi , obtained by multiplying A by the operator representing the right action of the imaginary unit i. These two possibilities represent equivalent choices in describing quaternionic observables within a quaternionic formulation of quantum mechanics based on complex geometry. In this scenario, the com- P plex linear operator R has all the expected properties of the momentum operator. It satisfies the standard commutation relations with the coordinates. It is a translation generator. Finally, it represents a quaternionic observable. A review of quaternionic and complexified quaternionic quantum mechanics by adopting a complex geometry is found in Ref. 34.

IV. OBSERVABLES IN QUATERNIONIC QUANTUM MECHANICS

In a recent paper,31 we ﬁnd a detailed discussion of eigenvalue equations within a quaternionic formulation of quantum mechanics with quaternionic and complex geometry. Quaternionic eigen-

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value equations for quaternionic and complex linear operators require eigenvalues from the right. In particular, without loss of generality, we can reduce the eigenvalue problem for quaternionic ෈ ͓A O͔ and complex linear anti-Hermitian operators A M n H to ⌿ ϭ⌿ ␭ ϭ ͑ ͒ A m m mi, m 1,2, ...,n, 10 ␭ where m are real eigenvalues. There is an important difference between the structure of Hermitian operators in complex and quaternionic quantum mechanics. In complex quantum mechanics we can always trivially relate an anti-Hermitian operator, A, to a Hermitian operator, H, by removing a factor i, i.e., AϭiH.In general, due to the noncommutative nature of the quaternionic ﬁeld, this does not apply to quaternionic quantum mechanics. ⌿ Let ͕ m͖ be a set of normalized eigenvectors of A with complex imaginary eigenvalues ␭ ͕i m͖. The anti-Hermitian operator A is then represented by

n ϭ ⌿ ␭ ⌿† ͑ ͒ A ͚ r ri r , 11 rϭ1

⌿† ⌿¯ t where ª . It is easy to verify that n n ⌿ ϭ ⌿ ␭ ⌿†⌿ ϭ ⌿ ␭ ␦ ϭ⌿ ␭ A m ͚ r ri r m ͚ r ri rm m mi. rϭ1 rϭ1

In quaternionic quantum mechanics with quaternionic geometry,36 the observable corresponding to the anti-Hermitian operator A is represented by the following Hermitian quaternionic linear operator:

n ϭ ⌿ ␭ ⌿† ͑ ͒ H ͚ r r r . 12 rϭ1

⌿ The action of these operators on the eigenvectors m gives ⌿ ϭ⌿ ␭ H m m m .

The eigenvalues of the operator H are real and eigenvectors corresponding to different eigenvalues are orthogonal. How to relate the Hermitian operator H to the anti-Hermitian operator A? A simple calculation shows that the operators LiH and HLi does not satisfy the same eigenvalue equation of A.In fact,

n n ⌿ ϭͫ ͩ ⌿ ␭ ⌿† ͪͬ⌿ ϭ ⌿ ␭ ⌿†⌿ ϭ ⌿ ␭ LiH m Li ͚ r r r m i ͚ r r r m i m m rϭ1 rϭ1

and

n n ⌿ ϭͫͩ ⌿ ␭ ⌿† ͪ ͬ⌿ ϭ ⌿ ␭ ⌿† ⌿ HLi m ͚ r r r Li m ͚ r r r i m . rϭ1 rϭ1

These problems can be avoided by using the right operator Ri instead of the left operator Li .In fact, the operator HRi satisﬁes the same eigenvalue equation of A,

n n ⌿ ϭͫͩ ⌿ ␭ ⌿† ͪ ͬ⌿ ϭ ⌿ ␭ ⌿†⌿ ϭ⌿ ␭ HRi m ͚ r r r Ri m ͚ r r r mi m mi. rϭ1 rϭ1

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Ϫ The eigenvalues of the operator ARi are real and eigenvectors corresponding to different eigenvalues are orthogonal. The right hermiticity of this operator is recovered within a quaternionic formulation of quantum mechanics based on complex geometry.34 When the space state is finite-dimensional, it is always possible to form a basis with the Ϫ eigenvectors of the operators H and ARi . When the space state is infinite-dimensional, this is no longer necessarily the case. So, it is useful to introduce a new concept, that of an observable. By Ϫ definition, the Hermitian operators H or ARi are observables if the orthonormal system of vectors forms a basis in the state space. In quaternionic quantum mechanics with quaternionic geometry, the Hermitian operator corresponding to the anti-Hermitian operator A of Eq. ͑11͒ is thus given by the operator H of Eq. ͑12͒. By adopting a complex geometry, observables can also be represented by complex linear Hermitian operators obtained by multiplying the corresponding anti-Hermitian operator A by Ϫ Ri . We remark that for complex eigenvectors, the operators LiH, HLi , HRi and A reduce to the same complex operator,

n ϭ ␭ ⌿ ⌿† iH i ͚ r r r . rϭ1

We conclude this section by giving an explicit example of quaternionic Hermitian operators in a ﬁnite two-dimensional space state. Let

Ϫi 3 j Aϭͩ ͪ ͑13͒ 3 ji

be an anti-Hermitian operator. An easy computation shows that the eigenvalues and the eigenvectors of this operator are given by

1 i 1 k ͕2i,4i͖ and ͭ ͩ ͪ , ͩ ͪͮ . & j & 1

It is immediate to verify that iA and Ai are characterized by complex eigenvalues and so cannot represent quaternionic observables. In quaternionic quantum mechanics with quaternionic geometry, the quaternionic observable corresponding to the anti-Hermitian operator of Eq. ͑13͒ is given by the Hermitian operator,

3 k Hϭ⌿ 2⌿†ϩ⌿ 4⌿†ϭͩ ͪ . ͑14͒ 1 1 2 2 Ϫk 3

Within a quaternionic quantum mechanics with complex geometry, a second equivalent deﬁnition of the quaternionic observable corresponding to the anti-Hermitian operator of Eq. ͑13͒ is given by the complex linear Hermitian operator,

Ϫi 3 j H˜ ϭͩ ͪ R . ͑15͒ 3 ji i

V. THE QUATERNIONIC SCHRO¨ DINGER EQUATION

For simplicity, we shall assume a one-dimensional description. In the standard formulation of quantum mechanics, the wave function of a particle whose potential energy is V(x,t) must satisfy the Schro¨dinger equation,

ប2 ϩV͑x,t͒ͪ ⌽͑x,t͒. ͑16͒ ץ ⌽͑x,t͒ϭH ⌽͑x,t͒ϭͩ Ϫ ץ i ប t 2m xx

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Let us modify the previous equation by introducing the quaternionic potential,

͓ ϩ ͔͑ ͒ V h•V x,t . The i-part of this quaternionic potential violates the norm conservation. In fact,

ϩϱ ϩϱ ប ប 1 ⌽¯ ͒ ⌽Ϫ ⌽¯ ͕ ͖ ⌽ͬ ץ͑ ⌽Ϫ ץ ¯⌽¯ ⌽ϭ ͵ ͫ ⌽ ͵ ץ t dx dx i xx xx i i,h V Ϫϱ Ϫϱ 2m 2m ប •

2 ϩϱ ϭ ͵ ⌽¯ ⌽ dx V1 . ប Ϫϱ

4 The j/k-part of h•V is responsible for T-violation. To show that, we brieﬂy discuss the time reversal invariance in quaternionic quantum mechanics. The quaternionic Schro¨dinger equation in the presence of a quaternionic potential which preserves norm conservation, is given by4,15,16,36 ͒ ͑ ͒ ⌽ ͒ϭ HϪ ⌽ ץ ប i t ͑x,t ͓ jW͔ ͑x,t , 17

where W෈C. Evidently, quaternionic conjugation, ͒ ¯⌽¯ ͒ ϭH ⌽¯ ͒ϩ⌽ ץ Ϫប t ͑x,t i ͑x,t ͑x,t jW, does not yield a time-reversed version of the original Schro¨dinger equation ͒ ͑ ͒ ⌽ Ϫ ͒ϭ HϪ ⌽ Ϫ ץ Ϫ ប i t T͑x, t ͓ jW͔ T͑x, t . 18 To understand why the T-violation is proportional to the j/k-part of the quaternionic potential, let us consider a real potential W. Then, the Schro¨dinger equation has a T-invariance. By multiplying Eq. ͑17͒ by j from the left, we have

⌽͑ ͒ϭ͓HϪ ͔ ⌽͑ ͒ ෈ ץ Ϫ ប i t j x,t jW j x,t , W R, which has the same form of Eq. ͑18͒. Thus, ⌽ Ϫ ͒ϭ ⌽ ͒ T͑x, t j ͑x,t .

A similar discussion applies for imaginary complex potential W෈i R. In this case, we ﬁnd ⌽ Ϫ ͒ϭ ⌽ ͒ T͑x, t k ͑x,t . ෈ However, when both V2 and V3 are nonzero, i.e., W C, this construction does not work, and the quaternionic physics is T-violating. The system of neutral kaons is the natural candidate to study ϩ the presence of effective quaternionic potentials, V h•V. In studying such a system, we need of V1 and V2,3 in order to include the decay rates of KS/KL and CP-violation effects. A. Quaternionic stationary states For stationary states,

V͑x,t͒ϭV͑x͒ and W͑x,t͒ϭW͑x͒,

we look for solutions of the Schro¨dinger equation of the form

⌽͑x,t͒ϭ⌿͑x͒␨͑t͒. ͑19͒

Substituting ͑19͒ in the quaternionic Schro¨dinger equation, we obtain

i ប⌿͑x͒␨˙͑t͒ϭ͓HϪ jW͑x͔͒⌿͑x͒␨͑t͒. ͑20͒

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Multiplying by Ϫ⌿¯ (x)i from the left and by ¯␨(t) from the right, we ﬁnd

ប ␨˙͑t͒¯␨͑t͒/͉␨͑t͉͒2ϭ⌿¯ ͑x͓͒Ϫi HϩkW͑x͔͒⌿͑x͒/͉⌿͑x͉͒2. ͑21͒

In this equation we have a function of t on the left-hand side and a function of x on the right-hand side. The previous equality is only possible if

ប ␨˙͑t͒¯␨͑t͒/ ͉␨͑t͉͒2ϭ⌿¯ ͑x͓͒Ϫi HϩkW͑x͔͒⌿͑x͒/͉⌿͑x͉͒2ϭq, ͑22͒

where q is a quaternionic constant. The energy operator Ϫi HϩkW(x) represents an anti- ϭ Hermitian operator. Consequently, its eigenvalues are purely imaginary quaternions, q h•E.By applying the unitary transformation u,

ϭϪ ϭͱ 2ϩ 2ϩ 2 ¯u h•E u iE, E E1 E2 E3, Eq. ͑22͒ becomes

ប ¯u ˙␨͑t͒¯␨͑t͒u/͉␨͑t͉͒2ϭ¯u ⌿¯ ͑x͓͒ϪiHϩkW͑x͔͒⌿͑x͒u/͉⌿͑x͉͒2ϭϪiE. ͑23͒

The solution ⌽(x,t) of the Schro¨dinger equation is not modiﬁed by this similarity transformation. In fact,

⌽͑x,t͒→⌿͑x͒uu¯ ␨͑t͒ϭ⌿͑x͒␨͑t͒.

By observing that ͉⌽(x,t)͉2ϭ͉⌿(x)͉2͉␨(t)͉2, the norm conservation implies ͉␨(t)͉2 constant. Without loss of generality, we can choose ͉␨(t)͉2ϭ1. Consequently, by equating the ﬁrst and the third term in Eq. ͑23͒ and solving the corresponding equation, we ﬁnd

␨͑t͒ϭexp͓ϪiEt/ប͔␨͑0͒, ͑24͒

with ␨(0) unitary quaternion. Note that the position of ␨(0) in Eq. ͑24͒ is very important. In fact, it can be shown that ␨(0)exp͓ϪiEt/ប͔ is not solution of Eq. ͑23͒. Finally, to complete the solution of the quaternionic Schro¨dinger equation, we must determine ⌿(x) by solving the following second order ͑right complex linear͒ differential equation,

ប2 ϪiV͑x͒ϩkW͑x͒ͬ⌿͑x͒ϭϪ⌿͑x͒iE. ͑25͒ ץ iͫ 2m xx B. Real potential

For W(x)ϭ0, Eq. ͑25͒ becomes

ប2 ϪV͑x͕͓͒ͬ⌿͑x͔͒ Ϫ j͓ j⌿͑x͔͒ ͖ϭi ͕͓⌿͑x͔͒ Ϫ j͓ j⌿͑x͔͒ ͖iE. ͑26͒ ץ ͫ 2m xx C C C C

Consequently,

ប2 ,ϪV͑x͓͒ͬ⌿͑x͔͒ ϭϪ͓⌿͑x͔͒ E ץ ͫ 2m xx C C

and

ប2 .ϪV͑x͓͒ͬ j⌿͑x͔͒ ϭ͓ j⌿͑x͔͒ E ץ ͫ 2m xx C C

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By solving these complex equations, we ﬁnd

2m 2m ⌿͑ ͒ϭ ͫͱ ͑ Ϫ ͒ ͬ ϩ ͫ Ϫͱ ͑ Ϫ ͒ ͬ ϩ x exp ប2 V E x k1 exp ប2 V E x k2 j

2m 2m ϫͭ ͫͱ ͑ ϩ ͒ ͬ ϩ ͫ Ϫͱ ͑ ϩ ͒ ͬ ͮ exp ប2 V E x k3 exp ប2 V E x k4 ,

ϭ where kn , n 1,...,4,arecomplex coefﬁcients determined by the initial conditions.

C. Free particles

For free particles, V(x)ϭW(x)ϭ0, the previous solution reduces to

p p p p ⌿͑ ͒ϭ ͫ ͬ ϩ ͫϪ ͬ ϩ ͫ ͬ ϩ ͫϪ ͬ x exp i ប x k1 exp i ប x k2 jͭ exp ប x k3 exp ប x k4ͮ ,

where pϭͱ2mE. For scattering problems with a wave function incident from the left on quaternionic potentials, we have

p p p ⌿͑ ͒ϭ ϩ Ϫ ϩ ͑ ͒ x expͫi ប xͬ r expͫ i ប x ͬ jr˜ expͫ ប xͬ, 27

where ͉r͉2 is the standard coefficient of reflection and ͉˜r exp͓(p/ប) x͔͉2 represents an additional evanescent probability of reflection. In our study of quaternionic potentials, we shall deal with the rectangular potential barrier of width a. In this case, the particle is free for xϽ0, where the solution is given by ͑27͒, and xϾa, where the solution is

p p ⌿͑ ͒ϭ ϩ ˜ Ϫ ͑ ͒ x t expͫi ប x ͬ jtexpͫ ប xͬ. 28

Note that, in Eqs. ͑27͒ and ͑28͒, we have, respectively, omitted the complex exponential solution exp͓Ϫ(p/ប)x͔ and exp͓(p/ប)x͔ which are in conﬂict with the boundary condition that ⌿(x) remain ﬁnite as x→Ϫϱ and x→ϱ. In Eq. ͑28͒, we have also omitted the complex exponential solution exp͓Ϫi(p/ប)x͔ because we are considering a wave incident from the left.

VI. QUATERNIONIC LINEAR DIFFERENTIAL EQUATION

Consider the second order quaternionic linear differential operator,

.ϩb ϩL b෈A O ץϩ͑a ϩL a͒ ץD ϭ H xx 0 • x 0 • H We are interested in ﬁnding the solution of the quaternionic linear differential equation,

D ␸͑ ͒ϭ ͑ ͒ H x 0. 29 In analogy to the complex case, we look for solutions of exponential form

␸͑x͒ϭexp͓qx͔,

where q෈H and x෈R. To satisfy Eq. ͑29͒, the constant q has to be a solution of the quaternionic quadratic equation,38–41

2ϩ͑ ϩ ͒ ϩ ϩ ϭ ͑ ͒ q a0 h•a q b0 h•b 0. 30

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A. Quaternionic quadratic equation

To simplify our discussion, it is convenient to modify Eq. ͑30͒ by removing the real constant ϭ ϩ a0 . To do this, we introduce a new quaternionic constant p deﬁned by p q a0/2. The quadratic equation ͑30͒ then becomes

2ϩ ϩ ϩ ϭ ͑ ͒ p h•a p c0 h•c 0, 31 ϭ Ϫ 2 ϭ Ϫ ͑ ͒ where c0 b0 a0/4 and c b (a0/2)a. We shall give the solution of Eq. 31 in terms of real constant c0 and of the real vectors a and c. Let us analyze the following cases:

Ã ϭ „i… a c 0, Þ Þ ϭ • a 0, c 0: „ii… a•c 0, Ã Þ Þ „iii… a c 0 a•c; • aϭ0, cÞ0; • aÞ0, cϭ0; • aϭcϭ0. Ã ϭ ͑ ͒ • „i… a c 0. In this case a and c are parallel vectors, so Eq. 31 can be easily reduced to a Iϭ ͉ ͉ complex equation. In fact, by introducing the imaginary unit h•a/ a and observing that h•c ϭI ␣, with ␣෈R, we ﬁnd 2ϩI͉ ͉ ϩ ϩI ␣ϭ p a p c0 0,

whose complex solutions are immediately found. ϭ Ã • „ii… a•c 0. By observing that a, c and a c are orthogonal vectors, we can rearrange the Ã imaginary part of p, h•p, in terms of the new basis (a,c,a c), i.e., ϭ ϩ ͑ ϩ ϩ Ã ͒ ͑ ͒ p p0 h• x a y c z a c . 32 Substituting ͑32͒ in Eq. ͑31͒, we obtain the following system of equations for the real variables p0 , x, y and z:

2Ϫ 2ϩ ͉ ͉2Ϫ 2 ͉ ͉2Ϫ 2͉ ͉2͉ ͉2ϩ ϭ R: p0 (x x) a y c z a c c0 0, ϩ ϭ h•a: p0(1 2 x) 0, ϩ Ϫ ͉ ͉2ϭ h•c:12 p0y z a 0, Ã ϩ ϭ h•a c: y 2 p0 z 0. ϩ ϭ ϭ ϭϪ 1 ϭ The second equation, p0(1 2 x) 0, implies p0 0 and/or x 2. For p0 0, it can be shown ͑ ͒ that the solution of Eq. 31 , in terms of p0 , x, y and z, is given by

1 1 p ϭ0, xϭϪ Ϯͱ⌬, yϭ0, zϭ , ͑33͒ 0 2 ͉a͉2

where

1 1 ͉c͉2 ⌬ϭ ϩ ͩ c Ϫ ͪ у0. 4 ͉a͉2 0 ͉a͉2

ϭϪ 1 For x 2, we ﬁnd

2 p 1 ϭϪ 0 ϭ ͑ ͒ y 2ϩ͉ ͉2 , z 2ϩ͉ ͉2 , 34 4 p0 a 4 p0 a

and

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2ϭ 1͓Ϯ ͱ 2ϩ͉ ͉2Ϫ Ϫ͉ ͉2͔ p0 4 2 c0 c 2 c0 a .

It is easily veriﬁed that

͉a͉2 ⌬р0⇒ͱc2ϩ͉c͉2Ϫc у ; 0 0 2

thus

ϭϮ 1 ͱ ͑ͱ 2ϩ͉ ͉2Ϫ ͒Ϫ͉ ͉2 ͑ ͒ p0 2 2 c0 c c0 a . 35 ⌬Þ Þ Summarizing, for 0, we have two quaternionic solutions, p1 p2 , ⌬Ͼ ϭ 0: p0 0,

ϭϪ 1 Ϯͱ⌬ x 2 ,

yϭ0,

1 zϭ ; ͑36͒ ͉a͉2

⌬Ͻ ϭϮ 1 ͱ ͑ͱ 2ϩ͉ ͉2Ϫ ͒Ϫ͉ ͉2 0: p0 2 2 c0 c c0 a ,

ϭϪ 1 x 2 ,

2 p ϭϪ 0 y 2ϩ͉ ͉2 , 4 p0 a

1 ϭ ͑ ͒ z 2ϩ͉ ͉2 . 37 4 p0 a ⌬ϭ ϭ For 0, these solutions tend to the same solution p1 p2 given by

1 1 ⌬ϭ0: p ϭ0, xϭϪ , yϭ0, zϭ . ͑38͒ 0 2 ͉a͉2

Ã Þ Þ ϭ Ϫ ϭ • „iii… a c 0 a•c. In discussing this case, we introduce the vector d c d0 a, d0 a ͉ ͉2 Ã •c/ a and the imaginary part of p in terms of the orthogonal vectors a, d and a d, ϭ ϩ ͑ ϩ ϩ Ã ͒ ͑ ͒ p p0 h• x a y d z a d . 39 By using this decomposition, from Eq. ͑31͒ we obtain the following system of real equations:

2Ϫ 2ϩ ͉ ͉2Ϫ 2 ͉ ͉2Ϫ 2 ͉ ͉2͉ ͉2ϩ ϭ R: p0 (x x) a y d z a d c0 0, ϩ ϩ ϭ h•a: p0(1 2 x) d0 0, ϩ Ϫ ͉ ͉2ϭ h•d;12 p0 y z a 0, Ã ϩ ϭ h•a d: y 2 p0z 0. ϩ ϩ ϭ Þ Þ The second equation of this system, p0(1 2 x) d0 0, implies p0 0 since d0 0. Therefore, we have

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p ϩd 2p 1 ϭϪ 0 0 ϭϪ 0 ϭ ͑ ͒ x , y 2ϩ͉ ͉2 , z 2ϩ͉ ͉2 , 40 2p0 4p0 a 4p0 a

and

͉a͉4 16 w3ϩ8͓͉a͉2ϩ2c ͔w2ϩ4͉ͫa͉2͑c Ϫd2͒ϩ Ϫ͉d͉2ͬwϪd2͉a͉4ϭ0, ͑41͒ 0 0 0 4 0

ϭ 2 ͑ ͒ where w p0. By using the Descartes rule of signs it can be proved that Eq. 41 has only one real 38 ϭ␣2 ␣෈ ϭϮ ␣ positive solution, w , R. This implies p0 . Thus, we also ﬁnd two quaternionic solutions. ϭ Þ Iϭ ͉ ͉ • a 0 and c 0. By introducing the imaginary complex unit h•c/ c , we can reduce Eq. ͑31͒ to the following complex equation:

2ϩ ϩI͉ ͉ϭ p c0 c 0.

• aÞ0 and cϭ0. This case is similar to the previous one. We introduce the imaginary complex Iϭ ͉ ͉ ͑ ͒ unit h•a/ a and reduce Eq. 31 to the complex equation, 2ϩI ͉ ͉ ϩ ϭ p a p c0 0.

• aϭcϭ0. Equation ͑31͒ becomes

2ϩ ϭ p c0 0.

ϭϪ␣2 ␣෈ ϭ␣2 For c0 , R, we find two real solutions. For c0 , we obtain an infinite number of ϭ ͉ ͉ϭ͉␣͉ quaternionic solutions, i.e., p h•p, where p . Let us resume our discussion on a quaternionic linear quadratic equation. For aϭ0 and/or c ϭ0 and for aÃcϭ0 we can reduce quaternionic linear quadratic equations to complex equations. ϭ Ã Þ Þ For non null vectors satisfying a•c 0ora c 0 a•c, we have effective quaternionic equations. ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ϭ In these cases, we always find two quaternionic solutions 36 , 37 and 40 – 41 . For a•c 0 and ⌬ϭ0, these solutions tend to the same solution ͑38͒. Finally, the fundamental theorem of algebra is lost for a restricted class of quaternionic quadratic linear equations, namely

q2ϩ␣2ϭ0, ␣෈R.

B. Second order quaternionic differential equations with constant coefﬁcients

Due to the quaternionic linearity from the right of Eq. ͑29͒, we look for general solutions which are of the form

␸ ͒ϭ␸ ͒ ϩ␸ ͒ ͑x 1͑x c1 2͑x c2 ,

␸ ␸ ͑ ͒ where 1(x) and 2(x) represent two linear independent solutions of Eq. 29 and c1 and c2 are quaternionic constants fixed by the initial conditions. In analogy to the complex case, we can distinguish between quaternionic linear dependent and independent solutions by constructing a Wronskian functional. To do this, we need to define a quaternionic determinant. Due to the noncommutative nature of quaternions, the standard definition of the determinant must be revised. The study of quaternionic, complex and real functionals, extending the complex determinant to quaternionic matrices, has been extensively developed in quaternionic linear algebra.46–49 In a recent paper,50 we find an interesting discussion on the impossibility to obtain a quaternionic functional with the main properties of the complex determinant. For quaternionic matrices, M,a real positive functional, ͉detM͉ϭͱdet͓MM†͔, which reduces to the absolute value of the standard

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determinant for complex matrices, was introduced by Study51 and its properties axiomatized by Dieudonne´.52 The details can be found in the excellent survey paper of Aslaksen.53 This functional allows us to construct a real positive Wronskian,31

␸ ͑x͒ ␸ ͑x͒ W͑ ͒ϭͯ ͩ 1 2 ͪͯ x det ␸ ͒ ␸ ͒ ˙ 1͑x ˙ 2͑x ϭ͉␸ ͑ ͒ʈ␸ ͑ ͒Ϫ␸ ͑ ͒␸Ϫ1͑ ͒␸ ͑ ͉͒ 1 x ˙ 2 x ˙ 1 x 1 x 2 x ϭ͉␸ ͑ ͒ʈ␸ ͑ ͒Ϫ␸ ͑ ͒␸Ϫ1͑ ͒␸ ͑ ͉͒ 2 x ˙ 1 x ˙ 2 x 2 x 1 x ϭ͉␸ ͑ ͒ʈ␸ ͑ ͒Ϫ␸ ͑ ͒␸Ϫ1͑ ͒␸ ͑ ͉͒ ˙ 1 x 2 x 1 x ˙ 1 x ˙ 2 x ϭ͉␸ ͑ ͒ʈ␸ ͑ ͒Ϫ␸ ͑ ͒␸Ϫ1͑ ͒␸ ͑ ͉͒ ˙ 2 x 1 x 2 x ˙ 2 x ˙ 1 x .

Solutions of Eq. ͑29͒,

a0 ␸ ͑x͒ϭexp͓q x͔ϭexpͫͩ p Ϫ ͪ xͬ, 1,2 1,2 1,2 2

͑ ͒ are given in terms of the solutions of the quadratic equation 31 , p1,2 , and of the real variable x. In this case, the Wronskian becomes

W ͒ϭ͉ Ϫ ʈ ʈ ͉ ͑x p1 p2 exp͓q1x͔ exp͓q2x͔ .

This functional allows us to distinguish between quaternionic linear dependent (Wϭ0) and independent (WÞ0) solutions. A generalization for quaternionic second order differential equations with nonconstant coefﬁcients should be investigated. Þ ͑ ͒ For p1 p2 , the solution of Eq. 29 is then given by

a0 ␸͑x͒ϭexpͫϪ x͕ͬexp͓p x͔c ϩexp͓p x͔c ͖. ͑42͒ 2 1 1 2 2

As observed at the end of the previous subsection, the fundamental theorem of algebra is lost for a restricted class of quaternionic quadratic equation, i.e., p2ϩ␣2ϭ0 where ␣෈R. For these ϭ ␣ ͉␣͉2ϭ␣2 equations we ﬁnd an inﬁnite number of solutions, p h• with . Nevertheless, the general solution of the second order differential equation,

␸¨ ͑x͒ϩ␣2 ␸͑x͒ϭ0, ͑43͒

is also expressed in terms of two linearly independent exponential solutions,

␸ ͒ϭ ␣ ϩ Ϫ ␣ ͑ ͒ ͑x exp͓i x͔c1 exp͓ i x͔c2 . 44

͓ ␣ ͔ Note that any other exponential solution, exp h• x , can be written as a linear combination of exp͓i ␣ x͔ and exp͓Ϫi ␣ x͔,

1 exp͓h ␣ x͔ϭ ͕exp͓i ␣ x͔͑␣Ϫi h ␣͒ϩexp͓Ϫi ␣ x͔͑␣ϩi h ␣͖͒. • 2␣ • •

As a consequence, the loss of the fundamental theorem of algebra for quaternions does not

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represent an obstacle in solving second order quaternionic linear differential equations with con- ϭ stant coefﬁcients. To complete our discussion, we have to examine the case p1 p2 . From Eq. ͑38͒ we ﬁnd

Ã h a 1 a0 p ϭp ϭϪ ϩ h aÃͩ bϪ aͪ , 1 2 2 ͉a͉2 • 2

Thus, a ﬁrst solution of the differential equation ͑29͒ is

Ã a b a a0 ␰͑x͒ϭexpͭͫh ͩ Ϫ ͪ Ϫ ͬxͮ . • ͉a͉2 2 2

For aÃbϭ0, we can immediately obtain a second linearly independent solution by multiplying exp͓Ϫ (a/2)x͔ by x, ␩(x)ϭx ␰(x). For aÃbÞ0, the second linearly independent solution takes a more complicated form, i.e.,

h a ␩͑x͒ϭͩ xϩ • ͪ ␰͑x͒. ͑45͒ ͉a͉2

It can easily be shown that ␩(x) is a solution of the differential equation ͑29͒,

␩¨ ͑x͒ϩa ␩˙ ͑x͒ϩb ␩͑x͒ h a h a ϭͫx͑q2ϩaqϩb͒ϩ2 qϩaϩ • ͑q2ϩaq͒ϩb • ͬ ␰͑x͒ ͉a͉2 ͉a͉2 h a ϭͩ 2 qϩaϩͫb, • ͬͪ ␰͑x͒ ͉a͉2 aÃb h a ϭͩ 2 h ϩͫh b, • ͬͪ ␰͑x͒ϭ0. • ͉a͉2 • ͉a͉2

ϭ ϭ ϭ Ã ͉ ͉2 Ϫ ͑ ͒ Thus, for p1 p2 p h•((a b)/ a a/2, the general solution of the differential equation 29 is given by

a0 h a ␸͑x͒ϭexpͫϪ xͬͭ exp͓px͔c ϩͩ xϩ • ͪ exp͓px͔c ͮ . ͑46͒ 2 1 ͉a͉2 2

C. Diagonalization and Jordan form

To ﬁnd the general solution of linear differential equations, we can also use quaternionic formulations of eigenvalue equations, matrix diagonalization and Jordan form. The quaternionic linear differential equation ͑29͒ can be written in matrix form as follows:

⌽˙ ͑x͒ϭM ⌽͑x͒, ͑47͒

where

01 ␸͑x͒ Mϭͩ ͪ and ⌽͑x͒ϭͫ ͬ. Ϫb Ϫa ␸˙ ͑x͒

By observing that x is real, the formal solution of the matrix equation ͑47͒ is given by

⌽͑x͒ϭexp͓Mx͔⌽͑0͒, ͑48͒

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where ⌽(0) represents a constant quaternionic column vector determined by the initial conditions ␸ ␸ ͓ ͔ϭ͚ϱ ͓ n ͔ (0), ˙ (0) and exp Mx nϭ0 (Mx) /n! . In the sequel, we shall use right eigenvalue equations for quaternionic linear matrix operators equations,

M ⌽ϭ⌽ q. ͑49͒

Without loss of generality, we can work with complex eigenvalue equations. By setting ⌿ ϭ⌽u, from the previous equation, we have

M ⌿ϭM ⌽uϭ⌽ quϭ⌽uu¯ quϭ⌿ z, ͑50͒

31 where z෈C and u is a unitary quaternion. In a recent paper, we ﬁnd a complete discussion of the eigenvalue equation for quaternionic matrix operators. In such a paper was shown that the complex counterpart of the matrix M has an eigenvalue spectrum characterized by eigenvalues which ⌿ ⌿ appear in conjugate pairs ͕z1 ,¯z1 ,z2 ,¯z2͖. Let 1 and 2 be the quaternionic eigenvectors corresponding to the complex eigenvalues z1 and z2 , ⌿ ϭ⌿ ⌿ ϭ⌿ M 1 1 z1 and M 2 2 z2 . ͉ ͉Þ͉ ͉ ⌿ ⌿ It can be shown that for z1 z2 , the eigenvectors 1 and 2 are linearly independent on H and ϫ ϭ ⌿ ⌿ consequently there exists a 2 2 quaternionic matrix S ͓ 1 2͔ which diagonalizes M,

z1 0 exp͓z1x͔ 0 exp͓Mx͔ϭS expͫͩ ͪ xͬ SϪ1ϭS ͩ ͪ SϪ1. 0 z2 0 exp͓z2x͔

In this case, the general solution of the quaternionic differential equation can be written in terms Ϫ1 of the elements of the matrices S and S and of the complex eigenvalues z1 and z2 ,

Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͒ ␸͑x͒ S11 exp͓z1x͔ S12 exp͓z2x͔ S11 0 S12 ˙ 0 ͫ ͬϭͩ ͪ ͫ ͬ. ␸˙ ͑x͒ ͓ ͔ ͓ ͔ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͒ S21 exp z1x S22 exp z2x S21 0 S22 ˙ 0

Hence,

␸͑ ͒ϭ ͓ ͔͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ x S11 exp z1 x S11 0 S12 ˙ 0 ϩ ͓ ͔͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ S12 exp z2 x S21 0 S22 ˙ 0 ϭ ͓ ͑ ͒Ϫ1 ͔ ͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ exp S11 z1 S11 x S11 S11 0 S12 ˙ 0 ϩ ͓ ͑ ͒Ϫ1 ͔ ͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ exp S12 z2 S12 x S12 S21 0 S22 ˙ 0 ϭ ͓ ͑ ͒Ϫ1 ͔ ͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ exp S21 S11 x S11 S11 0 S12 ˙ 0 ϩ ͓ ͑ ͒Ϫ1 ͔ ͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ ͑ ͒ exp S22 S12 x S12 S21 0 S22 ˙ 0 . 51

We remark that a different choice of the eigenvalue spectrum does not modify the solution ͑51͒. In fact, by taking the following quaternionic eigenvalue spectrum:

ϭ ͉ ͉Þ͉ ͉ ͑ ͒ ͕q1 ,q2͖ ͕¯u1z1u1 ,¯u2z2u2͖, q1 q2 , 52

and observing that the corresponding linearly independent eigenvectors are given by

⌽ ϭ⌿ ⌽ ϭ⌿ ͑ ͒ ͕ 1 1u1 , 2 2u2͖, 53

we obtain

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ϭ ⌽ ⌽ ⌽ ⌽ Ϫ1 M ͓ 1 2͔diag͕q1 ,q2͖͓ 1 2͔ ϭ ⌿ ⌿ ⌿ ⌿ Ϫ1 ͓ 1u1 2u2͔diag͕¯u1z1u1 ,¯u2z2u2͖͓ 1u1 2u2͔ ϭ ⌿ ⌿ ⌿ ⌿ Ϫ1 ͓ 1 2͔diag͕z1 ,z2͖͓ 1 2͔ . ͉ ͉ϭ͉ ͉ ⌿ ⌿ Let us now discuss the case z1 z2 . If the eigenvectors ͕ a , b͖, corresponding to the eigenvalue spectrum ͕z,z͖, are linearly independent on H, we can obviously repeat the previous dis- ϫ ϭ ⌿ ⌿ cussion and diagonalize the matrix operator M by the 2 2 quaternionic matrix U ͓ 1 2͔. Then, we ﬁnd

␸͑ ͒ϭ ͓ ͑ ͒Ϫ1 ͔ ͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ x exp U11 z U11 x U11 U11 0 U12 ˙ 0 ϩ ͓ ͑ ͒Ϫ1 ͔ ͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ exp U12 z U12 x U12 U21 0 U22 ˙ 0 ϭ ͓ ͑ ͒Ϫ1 ͔ ͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ exp U21 U11 x U11 11 0 U12 ˙ 0 ϩ ͓ ͑ ͒Ϫ1 ͔ ͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ ͑ ͒ exp U22 U12 x U12 U21 0 U22 ˙ 0 . 54

For linearly dependent eigenvectors, we cannot construct a matrix which diagonalizes the matrix operator M. Nevertheless, we can transform the matrix operator M to Jordan form,

z 1 MϭJ ͩ ͪ JϪ1. ͑55͒ 0 z

It follows that the solution of our quaternionic differential equation can be written as

ϩ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͒ z 1 J11 xJ11 J12 J11 0 J12 ˙ 0 ⌽͑x͒ϭJ expͫͩ ͪ xͬ JϪ1 ⌽͑0͒ϭͩ ͪ exp͓zx͔ͫ ͬ. ϩ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͒ 0 z J21 xJ21 J22 J21 0 J22 ˙ 0

Thus,

␸͑ ͒ϭ ͓ ͔͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ x J11 exp zx J11 0 J12 ˙ 0 ϩ͑ ϩ ͒ ͓ ͔͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ xJ11 J12 exp zx J21 0 J22 ˙ 0 ϭ ͓ ͑ ͒Ϫ1 ͔ ͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ exp J11 z J11 x J11 J11 0 J12 ˙ 0 ϩ ϩ ͒Ϫ1 ͒Ϫ1 ͓x J12͑J11 ͔exp͓J11 z͑J11 x͔ ϫ ͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ J11 J21 0 J22 ˙ 0 ϭ ͓ ͑ ͒Ϫ1 ͔ ͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ exp J21 J11 x J11 J11 0 J12 ˙ 0 ϩ ϩ ͒Ϫ1 ͒Ϫ1 ͓x J12͑J11 ͔exp͓J21͑J11 x͔ ϫ ͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ ͑ ͒ J11 J21 0 J22 ˙ 0 . 56

The last equality in the previous equation follows from the use of Eq. ͑55͒ and the deﬁnition of M. Finally, the general solution of the quaternionic differential equation ͑29͒ can be given by solving the corresponding eigenvalue problem. We conclude this section by observing that the quaternionic exponential solution, exp͓qx͔, can also be written in terms of complex exponential solutions, u exp͓zx͔uϪ1, where qϭuzuϪ1. The elements of the similarity transformations S, U or J and the complex eigenvalue spectrum of M determine the quaternion u and the complex number z. This form for exponential solutions will be very useful in solving complex linear differential equations with constant coefﬁcients. In fact, due to the presence of the right acting operator Ri , we cannot use quaternionic exponential solutions for complex linear differential equations.

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VII. COMPLEX LINEAR QUATERNIONIC DIFFERENTIAL EQUATIONS

Consider now the second order complex linear quaternionic differential operator,

ץ͔ D ϭ͓a ϩL a ϩ͑b ϩL b ͒R C 02 • 2 02 • 2 i xx ץ͔ ͒ ϩ͓ ϩ ϩ͑ ϩ a01 L•a1 b01 L•b1 Ri x ϩ ϩ ϩ͑ ϩ ͒ a00 L•a0 b00 L•b0 Ri ෈A O C ,

and look for solutions of the complex linear quaternionic differential equation,

D ␸͑ ͒ϭ ͑ ͒ C x 0. 57 ͑ ͒ Due to the presence of Ri in 57 , the general solution of the complex linear quaternionic differential equation cannot be given in terms of quaternionic exponentials. In matrix form, Eq. ͑57͒ reads as

⌽˙ ͑ ͒ϭ ⌽͑ ͒ ͑ ͒ x M C x , 58

where

01 ␸͑x͒ M ϭͩ ͪ and ⌽͑x͒ϭͫ ͬ. C Ϫ Ϫ ␸͑ ͒ bC aC ˙ x

The complex counterpart of complex linear quaternionic matrix operator M C has an eigenvalue ͕ ͖ spectrum characterized by four complex eigenvalues z1 ,z2 ,z3 ,z4 . It can be shown that M C is diagonalizable if and only if its complex counterpart is diagonalizable. For diagonalizable matrix

operator M C , we can ﬁnd a complex linear quaternionic linear similarity transformation SC which 31 reduces the matrix operator M C to diagonal form,

z ϩ¯z z Ϫ¯z 1 2 ϩ 1 2 Ri 0 2 2i Ϫ M ϭS S 1 . C C ͩ ͪ C ϩ Ϫ z3 ¯z4 z3 ¯z4 0 ϩ R 2 2i i

It is immediate to verify that

1 j 0 0 ͭͩ ͪ , ͩ ͪ , ͩ ͪ , ͩ ͪͮ 0 0 1 j

are eigenvectors of the diagonal matrix operator,

ϩ Ϫ z1 ¯z2 z1 ¯z2 ϩ R 0 2 2i i ͩ ͪ , ϩ Ϫ z3 ¯z4 z3 ¯z4 0 ϩ R 2 2i i

with right complex eigenvalues z1 , z2 , z3 and z4 . The general solution of the differential equation ͑57͒ can be given in terms of these complex eigenvalues,

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ϩ Ϫ z1 ¯z2 z1 ¯z2 Ϫ Ϫ ␸͑x͒ϭS expͫͩ ϩ R ͪ x͓ͬS 1 ␸͑0͒ϩS 1 ␸˙ ͑0͔͒ C 11 2 2i i C 11 C 12 ϩ Ϫ z3 ¯z4 z3 ¯z4 Ϫ Ϫ ϩS expͫͩ ϩ R ͪ x͓ͬS 1 ␸͑0͒ϩS 1 ␸˙ ͑0͔͒ C 12 2 2i i C 21 C 22 ϭ ϩ u1 exp͓z1 x͔k1 u2 exp͓z2 x͔k2 ϩ ϩ ͑ ͒ u3 exp͓z3 x͔k3 u4 exp͓z4 x͔k4 , 59

where kn are complex coefﬁcients determined by the initial conditions. This solution holds for diagonalizable matrix operator M C . For nondiagonalizable matrix operators we need to ﬁnd the similarity transformation JC which reduces M C to the Jordan form. For instance, it can be shown ϭ ͑ ͒ that for equal eigenvalues, z1 z2 , the general solution of the differential equation 57 is ␸ ͒ϭ ϩ ϩ ͒ ϩ ϩ ͑ ͒ ͑x u exp͓zx͔k1 ͑ux ˜u exp͓zx͔k2 u3 exp͓z3 x͔k3 u4 exp͓z4 x͔k4 . 60

A. Schro¨ dinger equation

Let us now examine the complex linear Schro¨dinger equation in the presence of a constant quaternionic potential,

ប2 ϪVϩ jWͬ⌿͑x͒ϭi ⌿͑x͒iE. ͑61͒ ץ ͫ 2m xx

In this case, the complex linear matrix operator,

01 M ϭͩ ͪ , b ϭVϪ jWϩiER , C Ϫ C i bC 0 represents a diagonalizable operator. Consequently, the general solution of the Schro¨dinger equation is given by

␸ ͒ϭ ϩ ϩ ϩ ͑ ͒ ͑x u1 exp͓z1 x͔k1 u2 exp͓z2 x͔k2 u3 exp͓z3 x͔k3 u4 exp͓z4 x͔k4 . 62

The quaternions un and the complex eigenvalues zn are obtained by solving the eigenvalue equa- ͑ ͒ tion for the complex linear operator M C . We can also obtain the general solution of Eq. 61 by substituting u exp͓ͱ2m/ប2 zx͔ in the Schro¨dinger equation. We ﬁnd the following quaternionic equation:

uz2Ϫ͑VϪ jW͒uϪiEuiϭ0,

ϭ ϩ where u zu jz˜u . This equation can be written as two complex equations:

2Ϫ Ϫ ͒ Ϫ ¯ ϭ 2Ϫ ϩ ͒ ϩ ϭ ͓z ͑V E ͔zu W˜zu ͓z ͑V E ͔˜zu Wzu 0.

An easy calculation shows that z satisﬁes the complex equation,

z4Ϫ2 Vz2ϩV2ϩ͉W͉2ϪE2ϭ0, ͑63͒

whose roots are

ϭϮ ͱ Ϫͱ 2Ϫ͉ ͉2ϭϮ ϭϮ ͱ ϩͱ 2Ϫ͉ ͉2ϭϮ ͑ ͒ z1,2 V E W zϪ and z3,4 V E W zϩ . 64 ϭ Ϫ ϭ By setting (u1,2)C ( ju3,4)C 1, we ﬁnd

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W W¯ uϪϭͩ 1ϩ j ͪ and uϩϭͩ ϩ j ͪ . ͑65͒ EϩͱE2Ϫ͉W͉2 EϩͱE2Ϫ͉W͉2

The solution of the complex linear quaternionic Schro¨dinger equation is then given by

2m 2m ⌿͑ ͒ϭ ͭ ͫͱ ͬ ϩ ͫ Ϫ ͱ ͬ ͮ x uϪ exp ប2 zϪ x k1 exp ប2 zϪ x k2

2m 2m ϩ ͭ ͫͱ ͬ ϩ ͫ Ϫ ͱ ͬ ͮ ͑ ͒ uϩ exp ប2 zϩ x k3 exp ប2 zϩ x k4 . 66

Equation ͑63͒ can also be obtained by multiplying the complex linear Schro¨dinger equation ͑61͒ from the left by the operator,

ប2 .ϪVϪ jW ץ 2m xx

This gives

ប2 2 ប2 ប2 ϪVϩ jWͬ⌿͑x͒iE ץ ϩV2ϩ͉W͉2ͬ⌿͑x͒ϭiͫ ץ Ϫ2 V ץ ͪ ͩͫ 2m xxxx 2m xx 2m xx ϭE2 ⌿͑x͒.

By substituting the exponential solution u exp͓ͱ2m/ប2 zx͔ in the previous equation, we immediately re-obtain Eq. ͑63͒.

VIII. QUATERNIONIC CONSTANT POTENTIALS Of all Schro¨dinger equations the one for a constant potential is mathematically the simplest. The reason for resuming the study of the Schro¨dinger equation with such a potential is that the qualitative features of a physical potential can often be approximated reasonably well by a potential which is pieced together from a number of constant portions. A. The potential step Let us consider the quaternionic potential step,

0, xϽ0, V͑x͒Ϫ jW͑x͒ϭͭ VϪ jW, xϾ0,

where V and W represent constant potentials. For scattering problems with a wave function incident from the left on the quaternionic potential step, the complex linear quaternionic Schro¨- dinger equation has the solution

p p p Ͻ ϩ Ϫ ϩ x 0: expͫ i ប x ͬ r expͫ i ប x ͬ jr˜ expͫ ប x ͬ;

2m 2m ⌿͑x͒ϭ Ͼ ͫͱ ͬϩ ˜ ͫ Ϫ ͱ ͬ ͓EϾͱV2ϩ͉W͉2͔, x 0: uϪ t exp ប2 zϪ x uϩ t exp ប2 zϩ x Ά ͑67͒ 2m 2m ͫ Ϫͱ ͬϩ ˜ ͫ Ϫ ͱ ͬ ͓EϽͱV2ϩ͉W͉2͔, uϪ t exp ប2 zϪ x uϩ t exp ប2 zϩ x

where r, ˜r, t and ˜t are complex coefﬁcients to be determined by matching the wave function ⌿(x) and its slope at the discontinuity of the potential xϭ0.

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For EϾͱV2ϩ͉W͉2, the complex exponential solutions of the quaternionic Schro¨dinger equation are characterized by

ͱͱ 2 2 ͱͱ 2 2 zϪϭi E Ϫ͉W͉ ϪV෈i R and zϩϭ E Ϫ͉W͉ ϩV෈R.

The complex linearly independent solutions,

2m 2m ͫ Ϫͱ ͬ ͫͱ ͬ uϪ exp ប2 zϪ x and uϩ exp ប2 zϩ x ,

ϭ ϭ ͑ ͒ have been omitted, k2 k3 0in 66 , because we are considering a wave incident from the left 2 and because the second complex exponential solution, exp͓ͱ2m/ប zϩ x͔, is in conﬂict with the boundary condition that ⌿(x) remain ﬁnite as x→ϱ. The standard result of complex quantum mechanics are immediately recovered by considering Wϭ0 and taking the complex part of the quaternionic solution. For EϽͱV2ϩ͉W͉2, the complex exponential solutions of the quaternionic Schro¨dinger equation are characterized by

ͱ ͱ 2 2 ͱ ͱ 2 2 zϪϭ VϪ E Ϫ͉W͉ , zϩϭ Vϩ E Ϫ͉W͉ ෈R ͓EϾ͉W͉͔,

␪ ͱ͉W͉2ϪE2 ϭ͑ 2ϩ͉ ͉2Ϫ 2͒1/4 ͫϮ ͬ ␪ϭ ෈ ͓ Ͻ͉ ͉͔ zϮ V W E exp i , tan E W . 2 V C

The complex linearly independent solutions, ͫͱ2m ͬ ͫͱ2m ͬ uϪ exp ប2 zϪ x and uϩ exp ប2 zϩ x ,

ϭ ϭ ͑ ͒ have been omitted, k1 k3 0in 66 , because they are in conflict with the boundary condition that ⌿(x) remain finite as x→ϱ. A relation between the complex coefficients of reflection and transmission can immediately be obtained by the continuity equation,

͒ ͑ ϭ͒ ץ␳ ͒ϩ ץ t ͑x,t xJ͑x,t 0, 68

where

␳͑x,t͒ϭ⌽¯ ͑x,t͒⌽͑x,t͒,

and

ប .⌽͑x,t͖͒ ץ ⌽¯ ͑x,t͔͒i ⌽͑x,t͒Ϫ⌽¯ ͑x,t͒i ץ͓͕ J͑x,t͒ϭ 2m x x

Note that, due to the noncommutative nature of the quaternionic wave functions, the position of the imaginary unit i in the probability current density J(x,t) is important to recover a continuity equation in quaternionic quantum mechanics. For stationary states, ⌽(x,t) ϭ⌿(x)exp͓Ϫi (E/ប)t͔␨(0), it can easily be shown that the probability current density,

ប E E ,⌿͑x͖͒ expͫϪi t ͬ␨͑0͒ ץ ⌿¯ ͑x͔͒i ⌿͑x͒Ϫ⌿¯ ͑x͒i ץ͓͕ͬ J͑x,t͒ϭ ¯␨͑0͒expͫ i t 2m ប x x ប

must be independent of x, J(x,t)ϭ f (t). Hence,

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ប E E ,␣⌿͑x͖͒ϭexpͫϪi t ͬ ␨͑0͒f ͑t͒¯␨͑0͒expͫ i t ͬϭ ץ ⌿¯ ͑x͔͒i ⌿͑x͒Ϫ⌿¯ ͑x͒i ץ͓͕ 2m x x ប ប

where ␣ is a real constant. This implies that the quantity

p ,⌿͑x͖͒ ץ ⌿¯ ͑x͔͒i ⌿͑x͒Ϫ⌿¯ ͑x͒i ץ͓͕ Jϭ 2m x x

has the same value at all points x. In the free potential region, xϽ0, we ﬁnd

p 2 JϪϭ ͑1Ϫ͉r͉ ͒. m

In the potential region, xϾ0, we obtain

2 2 ͉W͉ ͱ ͑ͱE2Ϫ͉W͉2ϪV͒ ͫ 1Ϫͩ ͪ ͬ ͉t͉2 ͓EϾͱV2ϩ͉W͉2͔, J ϭ m ϩͱ 2Ϫ͉ ͉2 ϩ ͭ E E W 0 ͓EϽͱV2ϩ͉W͉2͔.

Finally, for stationary states, the continuity equation leads to

2 ͱE2Ϫ͉W͉2ϪV ͉W͉ ͉r͉2ϩ ͫ 1Ϫͩ ͪ ͬ ͉t͉2ϭ1 ͓EϾͱV2ϩ͉W͉2͔, E EϩͱE2Ϫ͉W͉2 ͑69͒ ͉r͉2ϭ1 ͓EϽͱV2ϩ͉W͉2͔.

Thus, by using the concept of a probability current, we can define the following coefficients of transmission and reflection:

2 ͱE2Ϫ͉W͉2ϪV ͉W͉ Rϭ͉r͉2, Tϭ ͫ 1Ϫͩ ͪ ͬ ͉t͉2 ͓EϾͱV2ϩ͉W͉2͔, E EϩͱE2Ϫ͉W͉2

Rϭ͉r͉2, Tϭ0 ͓EϽͱV2ϩ͉W͉2͔.

These coefﬁcients give the probability for the particle, arriving from xϭϪϱ, to pass the potential step at xϭ0 or to turn back. The coefﬁcients R and T depend only on the ratios E/V and ͉W͉/V. The predictions of complex quantum mechanics are recovered by setting Wϭ0.

B. The rectangular potential barrier

In our study of quaternionic potentials, we now reach the rectangular potential barrier,

0, xϽ0, V͑x͒Ϫ jW͑x͒ϭͭ VϪ jW,0ϽxϽa, 0, xϾa.

For scattering problems with a wave function incident from the left on the quaternionic potential barrier, the complex linear quaternionic Schro¨dinger equation has the solution

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p p p Ͻ ϩ Ϫ ϩ x 0: expͫ i ប x ͬ r expͫ i ប x ͬ jr˜ expͫ ប x ͬ; 2m 2m Ͻ Ͻ ͭ ͫͱ ͬ ϩ ͫ Ϫ ͱ ͬ ͮ 0 x a: uϪ exp ប2 zϪ x k1 exp ប2 zϪ x k2 ⌿͑x͒ϭ ͑70͒ 2m 2m ϩ ͭ ͫͱ ͬ ϩ ͫ Ϫ ͱ ͬ ͮ uϩ exp ប2 zϩ x k3 exp ប2 zϩ x k4 ; p p Ͼ ͫ ͬϩ ˜ ͫϪ ͬ ¦ x a: t exp i ប x jtexp ប x . The complex coefficients r, ˜r, t and ˜t are determined by matching the wave function ⌿(x) and its slope at the discontinuity of the potential xϭ0 and will depend on ͉W͉. By using the continuity equation, we immediately find the following relation between the transmission, Tϭ͉t͉2, and reflection, Rϭ͉r͉2, coefficients

RϩTϭ1. ͑71͒

C. The rectangular potential well

Finally, we brieﬂy discuss the quaternionic rectangular potential well,

0, xϽ0, V͑x͒Ϫ jW͑x͒ϭͭ ϪVϩ jW,0ϽxϽa, 0, xϾa.

In the potential region, the solution of the complex linear quaternionic Schro¨dinger equation is then given by

2m 2m ⌿͑ ͒ϭ ͭ ͫͱ ͬ ϩ ͫ Ϫ ͱ ͬ ͮ x uϪ exp ប2 zϪ x k1 exp ប2 zϪ x k2

2m 2m ϩ ͭ ͫͱ ͬ ϩ ͫ Ϫ ͱ ͬ ͮ ͑ ͒ uϩ exp ប2 zϩ x k3 exp ប2 zϩ x k4 , 72

where

W W¯ uϪϭͩ 1Ϫ j ͪ , uϩϭͩ jϪ ͪ , EϩͱE2Ϫ͉W͉2 EϩͱE2Ϫ͉W͉2

and

ͱͱ 2 2 ͱͱ 2 2 zϪϭi E Ϫ͉W͉ ϩV, zϩϭ E Ϫ͉W͉ ϪV.

Depending on whether the energy is positive or negative, we distinguish two separate cases. If EϾ0, the particle is unconﬁned and is scattered by the potential; if EϽ0, it is conﬁned and in a bound state. We limit ourselves to discussing the case EϽ0. For ͉W͉Ͻ͉E͉ϽͱV2ϩ͉W͉2, solution ͑72͒ becomes

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2m 2m ͭ ͫ ͱ ͱͱ 2Ϫ͉ ͉2ϩ ͬ ϩ ͫ Ϫ ͱ ͱͱ 2Ϫ͉ ͉2ϩ ͬ ͮ uϪ exp i E W Vx k exp i E W Vx k ប2 1 ប2 2

2m 2m ϩ ͭ ͫ ͱ ͱ Ϫͱ 2Ϫ͉ ͉2 ͬ ϩ ͫ Ϫ ͱ ͱ Ϫͱ 2Ϫ͉ ͉2 ͬ ͮ uϩ exp i V E W x k exp i V E W x k . ប2 3 ប2 4 ͑73͒

For ͉E͉Ͻ͉W͉, the solution is given by

2m ␪ϩ␲ 2m ␪Ϫ␲ ͭ ͫͱ ␳ ͫ ͬͬ ϩ ͫ Ϫͱ ␳ ͫ ͬͬ ͮ uϪ exp exp i x k exp exp i x k ប2 2 1 ប2 2 2

2m ␲Ϫ␪ 2m ␪ϩ␲ ϩ ͭ ͫͱ ␳ ͫ ͬ ͬ ϩ ͫ Ϫ ͱ ␳ ͫϪ ͬͬ ͮ ͑ ͒ uϩ exp exp i x k exp exp i x k , 74 ប2 2 3 ប2 2 4

where ␳ϭͱV2ϩ͉W͉2ϪE2 and tan ␪ϭͱ͉W͉2ϪE2/V. In the region of zero potential, by using the boundary conditions at large distances, we ﬁnd

2m 2m Ͻ ͫ ͱ ͉ ͉ ͬ ϩ ͫ Ϫ ͱ ͉ ͉ ͬ x 0: exp ប2 E x c1 j exp i ប2 E x c4 ; ⌿͑x͒ϭ ͑75͒ Ά 2m 2m Ͼ ͫ Ϫ ͱ ͉ ͉ ͬ ϩ ͫ ͱ ͉ ͉ ͬ x a: exp ប2 E x d2 j exp i ប2 E x d3 .

The matching conditions at the discontinuities of the potential yield the energy eigenvalues.

IX. CONCLUSIONS D ␸ ϭ In this paper, we have discussed the resolution of quaternionic, H (x) 0, and complex, D ␸ ϭ C (x) 0, linear differential equations with constant coefficients within a quaternionic formulation of quantum mechanics. We emphasize that the only quaternionic quadratic equation in- volved in the study of second order linear differential equations with constant coefficients is given ͑ ͒ D ␸ ϭ by Eq. 30 following from H (x) 0. Due to the right action of the factor i in complex linear differential equations, we cannot factorize a quaternionic exponential and consequently we are not D ␸ ϭ able to obtain a quaternionic quadratic equation from C (x) 0. Complex linear differential equations can be solved by searching for quaternionic solutions of the form q exp͓zx͔, where q ෈H and z෈C. The complex exponential factorization gives a complex quartic equation. A similar D ␸ ϭ discussion can be extended to real linear differential equations, R (x) 0. In this case, the D presence of left/right operators L and R in R requires quaternionic solutions of the form q exp͓␭ x͔, where q෈H and ␭෈R. A detailed discussion of real linear differential equations deserves a further investigation. The use of quaternionic mathematical structures in solving the complex linear Schro¨dinger equation could represent an important direction for the search of new physics. The open question of whether quaternions could play a significant role in quantum mechanics is strictly related to the whole understanding of resolutions of quaternionic differential equations and eigenvalue problems. The investigation presented in this work is only a first step towards a whole theory of quaternionic differential, integral and functional equations. Obviously, due to the great variety of problems in using a noncommutative field, it is very difficult to define the precise limit of the subject.

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ACKNOWLEDGMENTS The authors acknowledge the Department of Physics and INFN, University of Lecce, for the hospitality and ﬁnancial support. In particular, they thank P. Rotelli, G. Scolarici, and L. Solom- brino for helpful comments and suggestions. Stefano De Leo was partially supported by the FAPESP Grant No. 99/09008-5. Gisele Ducati was supported by a CAPES Ph.D. fellowship.

APPENDIX A: QUATERNIONIC LINEAR QUADRATIC EQUATIONS In this appendix, we give some examples of quaternionic linear quadratic equations; see cases ͑i͒–͑iii͒ and ﬁnd their solutions. 2ϩ& ϩ Ϫ Ϫ & ϩ ϭ • „i…: p (i j)p 1 2 (i j) 0. In solving such an equation we observe that aϭ(&,&,0) and cϭϪ(2 &,2 &,0) are parallel vectors, cϭϪ2 a. Consequently, by introducing the complex imaginary unit Iϭ(iϩ j)/&, we can reduce the quadratic quaternionic equation to the following complex equation:

p2ϩ2 I pϪ1Ϫ4 Iϭ0, ϭϪIϮ ͱI whose solutions are p1,2 2 . It follows that the quaternionic solutions are

iϩ j p ϭϮ&Ϫ͑1ϯ&͒ . 1,2 &

2ϩ ϩ 1 ϭ ⌬ϭ • „ii…: p ip 2 k 0, 0. ϭ ϭ 1 ⌬ϭ We note that a (1,0,0) and c (0,0, 2) are orthogonal vectors and 0. So, we ﬁnd two coincident quaternionic solutions given by

1 iϩ j pϭϪ h aϩh aÃcϭϪ . 2 • • 2

2ϩ ϩ Ϫ ϭ ⌬Ͼ • „ii…: p jp 1 k 0, 0. ϭ ϭ Ϫ ϭ ⌬ϭ In this case, a (0,1,0) and c (0,0, 1) are orthogonal vectors, c0 1 and 1/4. So,

ϭ ϭϪ 1 Ϯ 1 ϭ ϭ p0 0, x 2 2 , y 0, z 1.

By observing that ϭ ϭϪ Ã ϭϪ h•a j, h•c k, h•a c i, we ﬁnd the following quaternionic solutions: ϭϪ ϭϪ ϩ ͒ p1 i and p2 ͑i j . 2ϩ ϩ ϭ ⌬Ͻ • „ii…: p kp j 0, 0. ϭ ϭ ϭ ϭ ⌬ϭϪ We have a (0,0,1), c (0,1,0) and c0 0. Then a•c 0 and 3/4. So, ϭϮ 1 ϭϪ 1 ϭϯ 1 ϭ 1 p0 2 , x 2 , y 2 , z 2 .

In this case, ϭ ϭ Ã ϭϪ h•a k, h•c j, h•a c i; thus, the solutions are given by

ϭ 1 ͑Ϯ Ϫ ϯ Ϫ ͒ p1,2 2 1 i j k .

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2ϩ ϩ ϩ ϩ ϭ • „iii…: p ip 1 i k 0. ϭ ϭ ϭ Þ We have a (1,0,0), c (1,0,1) and c0 1. In this case a•c 0, so we introduce the quater- ϩ ϭ ϩ ϭ Ϫ ϭ nion d0 h•d 1 k, whose vectorial part d c d0a (0,0,1) is orthogonal to a. The imaginary part of our solution will be given in terms of the imaginary quaternions, ϭ ϭ Ã ϭϪ h•a i, h•d k, h•a d j. The real part of p is determined by solving the equation

6ϩ 4Ϫ 2Ϫ ϭ 16 p0 24 p0 3 p0 1 0.

2ϭ 1 The real positive solution is given by p0 4. Consequently, ϭϮ 1 ϭϪ 1 ϯ ϭϯ 1 ϭ 1 p0 2 , x 2 1, y 2 , z 2 .

The quaternionic solutions are

ϭ 1 ͑ Ϫ Ϫ Ϫ ͒ ϭϪ 1 ͑ Ϫ ϩ Ϫ ͒ p1 2 1 3i j k and p2 2 1 i j k .

APPENDIX B: QUATERNIONIC LINEAR DIFFERENTIAL EQUATIONS

We solve quaternionic linear differential equations whose characteristic equations are given by examples „i…–„iii… in Appendix A. 1ϩk ␸͑ ͒ϩ&͑ ϩ ͒␸͑ ͒Ϫ͓ ϩ &͑ ϩ ͔͒␸͑ ͒ϭ ␸͑ ͒ϭ ␸͑ ͒ϭ • „i…: ¨ x i j ˙ x 1 2 i j x 0, 0 i, ˙ 0 & .

The exponential exp͓px͔ is solution of the previous differential equation if and only if the quaternion p satisﬁes the following quadratic equation:

p2ϩ&͑iϩ j͒pϪ1Ϫ2&͑iϩ j͒ϭ0,

whose solutions are given by

iϩ j p ϭϮ&Ϫ͑1ϯ &͒ . 1,2 &

Consequently,

iϩ j iϩ j ␸͑x͒ϭexpͭͫ &Ϫ͑1Ϫ&͒ ͬ x ͮ c ϩexpͭͫ Ϫ&Ϫ͑1ϩ&͒ ͬ x ͮ c . & 1 & 2

By using the initial conditions, we ﬁnd

iϩ j iϩ j ␸͑x͒ϭexpͫ Ϫ x ͬ cosh ͫͩ &ϩ ͪ x ͬi. & &

2ϩiϩk 1ϩiϩ j • ii : ␸¨ ͑x͒ϩ͑1ϩi͒␸˙ ͑x͒ϩ ␸͑x͒ϭ0, ␸͑0͒ϭ0, ␸˙ ͑0͒ϭϪ . „ … 4 2

␸ ϭ ͓ ͔ϭ ͓ Ϫ 1 ͔ We look for exponential solutions of the form (x) exp qx exp (p 2)x . The quaternion p must satisfy the quadratic equation,

2ϩ ϩ 1 ϭ p ip 2 k 0.

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This equation implies

iϩ j p ϭp ϭϪ . 1 2 2

Thus,

1ϩiϩ j ␸ ͑x͒ϭexpͫϪ x ͬ. 1 2

The second linearly independent solution is given by

1ϩiϩ j ␸ ͑x͒ϭ͑xϩi͒expͫϪ x ͬ. 2 2

By using the initial conditions, we ﬁnd

␸͑x͒ϭ͕ exp͓qx͔ϩ͑xϩi͒exp͓ qx͔i ͖͓1ϩqϪ1͑1ϩiq͒i͔Ϫ1,

where qϭϪ(1ϩiϩ j)/2.

1Ϫi • ii : ␸¨ (x)ϩ(2ϩ j)␸˙ (x)ϩ(2ϩ jϪk)␸(x)ϭ0, ␸(0)ϭ , ␸˙ (0)ϭ j. „ … 2 The exponential solution ␸(x)ϭexp͓qx͔ϭexp͓(pϪ1)x͔ leads to

p2ϩ jpϩ1Ϫkϭ0,

whose solutions are

ϭϪ ϭϪ ϩ ͒ p1 i and p2 ͑i j .

Consequently,

␸ ͒ϭ Ϫ Ϫ ϩ Ϫ ϩ ͒ ͑x exp͓ x͔͕ exp͓ ix͔c1 exp͓ ͑i j x͔c2͖.

The initial conditions yield

3ϪiϪ2 j ␸͑x͒ϭexp͓Ϫ x͔ͭ exp͓Ϫ ix͔ ϩexp͓Ϫ ͑iϩ j͒x͔͑jϪ1͒ͮ . 2 ␸ ϩ ␸ ϩ ␸ ϭ ␸ ϭ ϩ ␸ ϭ • „ii…: ¨ (x) k ˙ (x) j (x) 0, (0) i k, ˙ (0) 1. The characteristic equation is

p2ϩkpϩ jϭ0,

whose solutions are

ϭ 1 ͑Ϯ Ϫ ϯ Ϫ ͒ p1,2 2 1 i j k .

Thus, the general solution of our differential equation reads as

1ϪiϪ jϪk 1ϩiϪ jϩk ␸͑x͒ϭexpͫ x ͬ c ϩexpͫϪ x ͬ c . 2 1 2 2

By using the initial conditions, we obtain

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1ϪiϪ jϪk 1ϩiϪ jϩk iϩk ␸͑x͒ϭͭ expͫ x ͬϩexpͫϪ x ͬ ͮ . 2 2 2 ␸ ϩ Ϫ ␸ ϩ ϩ ␸ ϭ ␸ ϭ ␸ ϭ • „iii…: ¨ (x) (i 2) ˙ (x) (2 k) (x) 0, (0) 0, ˙ (0) j. By substituting ␸(x)ϭexp͓qx͔ϭexp͓(pϩ1)x͔ in the previous differential equation, we ﬁnd

p2ϩipϩ1ϩiϩkϭ0.

The solutions of this quadratic quaternionic equation are

ϭ 1 ͑ Ϫ Ϫ Ϫ ͒ ϭϪ 1 ͑ Ϫ ϩ Ϫ ͒ p1 2 1 3i j k and p2 2 1 i j k .

So, the general solution of the differential equation is

1Ϫ3iϪ jϪk 1Ϫiϩ jϪk ␸͑x͒ϭexpͫ x ͬc ϩexpͫϪ x ͬc . 2 1 2 2

By using the initial conditions, we obtain

1Ϫ3iϪ jϪk 1Ϫiϩ jϪk jϪiϩ2k ␸͑x͒ϭͭ expͫ x ͬϪexpͫϪ x ͬ ͮ . 2 2 6

APPENDIX C: DIAGONALIZATION AND JORDAN FORM

In this appendix, we ﬁnd the solution of quaternionic and complex linear differential equations by using diagonalization and Jordan form.

1. Quaternionic linear differential equation

By using the discussion about quaternionic quadratic equation, it can immediately be shown that the solution of the following second order equation:

␸¨ ͑x͒ϩ͑kϪi͒␸˙ ͑x͒Ϫ j ␸͑x͒ϭ0,

with initial conditions

k j ␸͑0͒ϭ , ␸˙ ͑0͒ϭ1ϩ , 2 2

is given by

k ␸͑x͒ϭͩ xϩ ͪ exp͓ix͔. 2

Let us solve this differential equation by using its matrix form ͑47͒, with

01 Mϭͩ ͪ . jiϪk

This quaternionic matrix can be reduced to its Jordan form,

i 1 MϭJ ͩ ͪ JϪ1, 0 i

by the matrix transformation

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k 3ϩ j iϩk 1 Ϫ 2 4 4 Jϭͩ ͪ , JϪ1ϭͩ ͪ . j iϩk 1Ϫ j i 1ϩ Ϫ 2 2 2

The solution of the quaternionic linear quaternionic differential equation is then given by

␸͑ ͒ϭ ͓ ͔͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ x J11 exp ix J11 0 J12 ˙ 0 ϩ͑ ϩ ͒ ͓ ͔͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ xJ11 J12 exp ix J21 0 J22 ˙ 0 ϭ ϩ ͒ ͑xJ11 J12 exp͓ix͔ k ϭͩ xϩ ͪ exp͓ix͔. 2

2. Complex linear differential equations

Let us now consider the complex linear quaternionic differential equation,

␸¨ ͑x͒Ϫ j ␸͑x͒iϭ0,

with initial conditions

␸͑0͒ϭ j, ␸˙ ͑0͒ϭk.

To ﬁnd particular solutions, we set ␸(x)ϭq exp͓zx͔. Consequently,

qz2Ϫ jqiϭ0.

The solution of the complex linear second order differential equation is

␸͑ ͒ϭ 1͓͑ ϩ ͒ ͓Ϫ ͔ϩ͑ Ϫ ͒ ϩ͑ Ϫ ͒ ͔ x 2 i j exp ix j i cosh x k 1 sinh x .

This solution can also be obtained by using the matrix

01 ϭͩ ͪ M C , jRi 0

and its diagonal form

Ϫ iRi 0 Ϫ M ϭS ͩ ͪ S 1 , C C 0 i C

where

1ϪiϪ jϪk 1Ϫiϩ jϩk 1ϩiϪ jϩk 1ϩiϩ jϪk ϩ R Ϫ R 2 2 i 2 2 i ϭ SC ͩ ͪ 1ϩiϩ jϪk 1ϩiϪ jϩk 1ϪiϪ jϪk 1Ϫiϩ jϩk Ϫ R Ϫ ϩ R 2 2 i 2 2 i

and

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1ϩiϩ jϩk 1ϩiϪ jϪk 1ϪiϪ jϩk 1Ϫiϩ jϪk Ϫ ϩ Ri Ri Ϫ 1 2 2 2 2 S 1ϭ . C ͩ ͪ 4 1Ϫiϩ jϪk 1ϪiϪ jϩk 1ϩiϩ jϩk 1ϩiϪ jϪk ϩ R Ϫ Ϫ R 2 2 i 2 2 i

The solution of the complex linear quaternionic differential equation is then given by

␸͑ ͒ϭ ͓Ϫ ͔͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ x SC 11 exp iRi x SC 11 0 SC 12 ˙ 0 ϩ ͓ ͔͓ Ϫ1 ␸͑ ͒ϩ Ϫ1 ␸͑ ͔͒ SC 12 exp ix SC 21 0 SC 22 ˙ 0 1 ϭ ͕͑1Ϫiϩ jϪk͒exp͓Ϫx͔Ϫ͑1ϩiϪ jϪk͒exp͓x͔͖ 4 iϩ j ϩ exp͓Ϫix͔. 2

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