Solving simple quaternionic differential equations - Unicamp

JOURNAL OF MATHEMATICAL PHYSICS

VOLUME 44, NUMBER 5

MAY 2003

Solving simple quaternionic differential equations Stefano De Leoa) Department of Applied Mathematics, State University of Campinas, SP 13083-970, Campinas, Brazil

Gisele C. Ducatib) Department of Mathematics, University of Parana, PR 81531-970, Curitiba, Brazil

共Received 10 December 2002; accepted 27 January 2003兲 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 noncommutative 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 noncommutative extension for quaternionic functions of one real variable. Linear dependence and independence of solutions of homogeneous 共right兲 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. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1563735兴

I. INTRODUCTION

Let R, C⬅span 兵 1,i 其 , and H⬅span 兵 1,i, j,k 其 be the real, complex, and quaternionic field,3 i 2 ⫽ j 2 ⫽k 2 ⫽i jk⫽⫺1 and F : R→R be the set of real functions of real variable. Through the paper, quaternionic functions of real variable, ⌿(x)苸H 丢 F, will be denoted by greek letters and constant quaternionic coefficients by Roman letters. To shorten notation the prime and double prime in the quaternionic functions shall, respectively, indicate the first and second derivative of quaternionic functions with respect to the real variable x, ⌿ ⬘ª

d⌿ dx

and ⌿ ⬙ ª

d2 ⌿ . dx 2

Due to the noncommutative nature of quaternions, it is convenient to distinguish between the left and right action of the quaternionic imaginary units i, j, and k by introducing the operators L q and R p whose action on quaternionic functions ⌿ is given by a兲

Electronic mail: [email protected] Electronic mail: [email protected]

b兲

0022-2488/2003/44(5)/2224/10/$20.00

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© 2003 American Institute of Physics

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J. Math. Phys., Vol. 44, No. 5, May 2003

Solving quaternionic differential equations

L q ⌿⫽q ⌿

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and R p ⌿⫽⌿ p,

共1兲

These 共left/right acting兲 quaternionic operators satisfy L q L p ⫽L qp ,

R q R p ⫽R pq ,

and 关 L q , R p 兴 ⫽0,

共2兲

and admit for q⫽q 0 ⫹i q 1 ⫹ j q 2 ⫹k q 3 ,

p⫽p 0 ⫹i p 1 ⫹ j p 2 ⫹k p 3 ,

⌿⫽⌿ 0 ⫹i ⌿ 1 ⫹ j ⌿ 2 ⫹k ⌿ 3 ,

the following real matrix representation4 – 6

L q↔

冉

q0

⫺q 1

⫺q 2

⫺q 3

q1

q0

⫺q 3

q2

q2

q3

q0

⫺q 1

q3

⫺q 2

q1

q0

冊冉 ,

R p↔

p0

⫺p1

⫺p2

⫺p3

p1

p0

p3

⫺p2

p2

⫺p3

p0

p1

p3

p2

⫺p1

p0

冊冋册

⌿0 ⌿1 ⌿↔ 苸R4 丢 F. ⌿2 ⌿3

,

共3兲

II. EXISTENCE AND UNIQUENESS

In this section we discuss existence and uniqueness for the quaternionic initial value problem ⌿ ⬙ ⫽ ␣ ⌿ ⬘ ⫹ ␤ ⌿⫹ ␳ ,

⌿共 x0兲⫽ f ,

⌿ ⬘ 共 x 0 兲 ⫽g,

共4兲

with ␣ (x), ␤ (x), ␳ (x)苸H 丢 F, x 0 苸I:(x ⫺ ,x ⫹ ) and f ,g苸H. Theorem 1: Let ␣, ␤, and ␳ in Eq. 共4兲 be continuous functions of x on an open interval I containing the point x⫽x 0 . Then, the initial value problem 共4兲 has a solution ⌿ on this interval and this solution is unique. Proof: By using the real matrix representation 共3兲, we can immediately rewrite the quaternionic initial value problem 共4兲 in the following vector form:

冋册冉

␣0 ⌿0 ⬙ ␣1 ⌿1 ⫽ ⌿2 ␣2 ⌿3 ␣3

with

⫺␣1

⫺␣2

⫺␣3

␣0

⫺␣3

␣2

␣3

␣0

⫺␣1

⫺␣2

␣1

␣0

冊冋册冉

␤0 ⌿0 ⬘ ␤1 ⌿1 ⫹ ⌿2 ␤2 ⌿3 ␤3

⫺␤1

⫺␤2

⫺␤3

␤0

⫺␤3

␤2

␤3

␤0

⫺␤1

⫺␤2

␤1

␤0

冋册冋册冋册冋册 ⌿ 0共 x 0 兲 f0 ⌿ 1共 x 0 兲 f1 ⫽ ⌿ 2共 x 0 兲 f2 ⌿ 3共 x 0 兲 f3

and

⌿ 0⬘ 共 x 0 兲 g0 ⌿ ⬘1 共 x 0 兲 g1 ⫽ . g2 ⌿ 2共 x 0 兲 g3 ⌿ 3⬘ 共 x 0 兲

冊冋册冋册 ⌿0 ␳0 ⌿1 ␳1 ⫹ ⌿2 ␳2 ⌿3 ␳3

共5兲

共6兲

Equation 共5兲 represents a 共nonhomogeneous兲 linear system with ␣ m , ␤ m , ␳ m 苸R 丢 F, where m ⫽0,1,2,3. These functions are 共see hypothesis of Theorem 1兲 continuous 共real兲 functions of x on an open interval I containing the point x⫽x 0 . Then, by a well-known theorem of analysis, see, for example, Ref. 7, the linear system 共5兲 has a solution

冋册

⌿0 ⌿1 苸R4 丢 F ⌿2 ⌿3


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S. De Leo and G. C. Ducati

on this interval satisfying 共6兲, and this solution is unique.

䊏

III. LINEAR INDEPENDENCE AND DEPENDENCE OF SOLUTIONS

Let us now analyze the linear independence and dependence of the solutions of second order homogeneous differential equations ⌿ ⬙ ⫽ ␣ ⌿ ⬘ ⫹ ␤ ⌿,

共7兲

where ␣ and ␤ are 共quaternionic兲 continuous functions of x on an open interval I. Equation 共7兲 is linear over H from the right. Consequently, if ␸ is a solution of Eq. 共7兲 only the function obtained by right multiplication by constant quaternionic coefficients, ␸ u, still represent a solution of such an equation. The general solution of Eq. 共7兲 is given in terms of a pair of linearly independent solutions ␸ and ␰ by ⌿⫽ ␸ u⫹ ␰ v ,

共8兲

where ␸ ⫽ ␸ 0 ⫹i ␸ 1 ⫹ j ␸ 2 ⫹k ␸ 3 , ␰ ⫽ ␰ 0 ⫹i ␰ 1 ⫹ j ␰ 2 ⫹k ␰ 3 苸H 丢 F, and u, v 苸H. In the standard complex theory ( ␸ ⫽ ␸ 0 ⫹i ␸ 1 and ␰ ⫽ ␰ 0 ⫹i ␰ 1 苸C 丢 F) a useful criterion to establish linear independence and dependence of two solutions of homogeneous second order differential equation, uses the concept of Wronskian of these solutions defined by W⫽ ␸ ␰ ⬘ ⫺ ␸ ⬘ ␰ ,

W苸C 丢 F.

共9兲

This definition cannot be extended to quaternionic functions. Let us consider two linearly dependent solutions of Eq. 共7兲,

␰ ⫽ ␸ q,

␸ , ␰ 苸H 丢 F,

q苸H.

共10兲

By substituting ␰ ⫽ ␸ q and ␰ ⬘ ⫽ ␸ ⬘ q in the Wronskian 共9兲, we find

␸ ␰ ⬘ ⫺ ␸ ⬘ ␰ ⫽ ␸ ␸ ⬘ q⫺ ␸ ⬘ ␸ q⫽0. Observe that a quaternionic function and its first derivative do not, in general, commute. Thus, the definition 共9兲, and all its possible factor combinations cannot be extended to the quaternionic case. Let us now use the linear dependence condition 共10兲 to investigate the possibility to define a quaternionic functional which extends 共in a nontrivial way兲 the standard 共complex兲 Wronskian to the noncommutative case. From Eq. 共10兲 and its derivative, we get q⫽ ␸ ⫺1 ␰ ⫽ 共 ␸ ⬘ 兲 ⫺1 ␰ ⬘ , where ␸ ⫺1 ⬅1/␸ and ( ␸ ⬘ ) ⫺1 ⬅1/␸ ⬘ . Consequently, for linearly dependent quaternionic solutions, we have

␰ ⬘ ⫺ ␸ ⬘ ␸ ⫺1 ␰ ⫽0.

共11兲

To recover, in the complex limit, the standard definition 共9兲 we multiply ␰ ⬘ ⫺ ␸ ⬘ ␸ ⫺1 ␰ by ␸. Due to the noncommutative nature of quaternions, we have to consider the following possibilities: W L ⫽ ␸ 共 ␰ ⬘ ⫺ ␸ ⬘ ␸ ⫺1 ␰ 兲

and W R ⫽ 共 ␰ ⬘ ⫺ ␸ ⬘ ␸ ⫺1 ␰ 兲 ␸ .

共12兲

Obviously two other similar definitions can be obtained by ␸ ↔ ␰ , ˜ L ⫽⫺ ␰ 共 ␸ ⬘ ⫺ ␰ ⬘ ␰ ⫺1 ␸ 兲 ⫽⫺W L 关 ␸ ↔ ␰ 兴 W

˜ R ⫽⫺ 共 ␸ ⬘ ⫺ ␰ ⬘ ␰ ⫺1 ␸ 兲 ␰ ⫽⫺W R 关 ␸ ↔ ␰ 兴 . and W 共13兲




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The quaternionic functionals 共12兲 and 共13兲, which give in the complex limit the standard definition, extend a first important property of Wronskian. Two solutions of Eq. 共7兲 are linearly depen˜ L(R) 兴 is zero on I. To avoid ambiguity in defining the Wronskian, we shall dent on I if W L(R) 关 W introduce a 共real兲 functional, ˜ L兩 2⫽ 兩 W ˜ R兩 2, 兩 W 兩 2⫽ 兩 W L兩 2⫽ 兩 W R兩 2⫽ 兩 W which extends the squared absolute value of the Wronskian. This unique functional is 兩 W 兩 2 ⫽ 兩 ␸ 兩 2 兩 ␰ ⬘ 兩 2 ⫹ 兩 ␰ 兩 2 兩 ␸ ⬘ 兩 2 ⫺ ␸ ⬘ ␸ c ␰ ␰ ⬘c ⫺ ␰ ⬘ ␰ c ␸ ␸ ⬘c

苸R 丢 F,

共14兲

where ␸ c ⫽ ␸ 0 ⫺i ␸ 1 ⫺ j ␸ 2 ⫺k ␸ 3 and ␰ c ⫽ ␰ 0 ⫺i ␰ 1 ⫺ j ␰ 2 ⫺k ␰ 3 are, respectively, the quaternionic conjugate functions of ␸ and ␰. Observe that Eq. 共14兲 can also be obtained as an application of the Dieudonne´ theory of quaternionic determinants.8 –12 In fact, 兩 W 兩 2 ⫽ 关 Det共 M 兲兴 2 ªdet共 M M ⫹ 兲 ,

共15兲

where M⫽

冉

␸

␰

␸⬘

␰⬘

冊

.

Theorem 2: Let ␣ and ␤ in Eq. 共7兲 be continuous functions of x on an open interval I : (a,b). Then, two solutions ␸ and ␰ of Eq. 共7兲 on I are linearly dependent on I if and only if the absolute value of the Wronskian, 兩 W 兩 , is zero at some x 0 in I. The proof will be divided into three steps: 共a兲共b兲共c兲

If ␸ and ␰ are linearly dependent on I then 兩 W 兩 ⫽0. If 兩 W 兩 ⫽0 at some x 0 in I then 兩 W 兩 ⫽0 on I. If 兩 W 兩 ⫽0 at some x 0 in I then ␸ and ␰ are linearly dependent on I. Proof (a): If ␸ and ␰ are linearly dependent on I, then Eq. 共10兲 holds on I. From Eq. 共10兲, we

get 兩 W 兩 2 ⫽ 兩 ␸ 兩 2 兩 ␸ ⬘ 兩 2 兩 q 兩 2 ⫹ 兩 ␸ 兩 2 兩 ␸ ⬘ 兩 2 兩 q 兩 2 ⫺ ␸ ⬘ ␸ c ␸ 兩 q 兩 2 ␸ ⬘c ⫺ ␸ ⬘ 兩 q 兩 2 ␸ c ␸␸ ⬘c ⫽0,

then 兩 W 兩 ⫽0. Proof (b): Let us consider Eq. 共14兲. By calculating the first derivative of the left-hand- and right-hand-side terms, we obtain 2 兩 W 兩兩 W 兩 ⬘ ⫽ ␸ ⬘ ␸ c ␰ ⬘ ␰ c⬘ ⫹ ␸␸ c⬘ ␰ ⬘ ␰ c⬘ ⫹ ␸␸ c ⌿ 2⬙ ␰ c⬘ ⫹ ␸␸ c ␰ ⬘ ␰ c⬙ ⫹ ␰ ⬘ ␰ c ␸ ⬘ ␸ c⬘ ⫹ ␰␰ c⬘ ␸ ⬘ ␸ c⬘ ⫹ ␰␰ c ⌿ 1⬙ ␸ c⬘ ⫹ ␰␰ c ␸ ⬘ ␸ c⬙ ⫺⌿ 1⬙ ␸ c ␰␰ c⬘ ⫺ ␸ ⬘ ␸ c⬘ ␰␰ c⬘ ⫺ ␸ ⬘ ␸ c ␰ ⬘ ␰ c⬘ ⫺ ␸ ⬘ ␸ c ␰␰ c⬙ ⫺⌿ 2⬙ ␰ c ␸␸ c⬘ ⫺ ␰ ⬘ ␰ c⬘ ␸␸ c⬘ ⫺ ␰ ⬘ ␰ c ␸ ⬘ ␸ c⬘ ⫺ ␰ ⬘ ␰ c ␸␸ c⬙ ⫽ 兩 ␸ 兩 2 共 ⌿ 2⬙ ␰ c⬘ ⫹ ␰ ⬘ ␰ c⬙ 兲 ⫹ 兩 ␰ 兩 2 共 ⌿ 1⬙ ␸ c⬘ ⫹ ␸ ⬘ ␸ c⬙ 兲 ⫺⌿ 1⬙ ␸ c ␰␰ c⬘ ⫺ ␸ ⬘ ␸ c ␰␰ c⬙ ⫺⌿ 2⬙ ␰ c ␸␸ c⬘ ⫺ ␰ ⬘ ␰ c ␸␸ c⬙ ⫽ 兩 ␸ 兩 2 共 ␣ 兩 ␰ ⬘ 兩 2 ⫹ ␤ ␰␰ c⬘ ⫹h.c.兲 ⫹ 兩 ␰ 兩 2 共 ␣ 兩 ␸ ⬘ 兩 2 ⫹ ␤ ␸␸ c⬘ ⫹h.c.兲 ⫺ 关共 ␣ ␸ ⬘ ␸ c ⫹ ␤ 兩 ␸ 兩 2 兲 ␰␰ c⬘ ⫹h.c.兴 ⫺ 关共 ␣ ␰ ⬘ ␰ c ⫹ ␤ 兩 ␰ 兩 2 兲 ␸␸ c⬘ ⫹h.c.兴 ⫽2 Re关 ␣ 兴共兩 ␸ 兩 2 兩 ␰ ⬘ 兩 2 ⫹ 兩 ␰ 兩 2 兩 ␸ ⬘ 兩 2 ⫺ ␸ ⬘ ␸ c ␰␰ c⬘ ⫺ ␰ ⬘ ␰ c ␸␸ c⬘ 兲 ⫽2 Re关 ␣ 兴兩 W 兩 2 .


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By a simple integration, we find

冋冕

兩 W 共 x 兲兩 ⫽exp

x

x0

册

Re关 ␣ 共 y 兲兴 dy 兩 W 共 x 0 兲兩 .

共16兲

This proves the statement 共b兲. Proof (c): From the statement 共b兲, we have 兩 W 共 x 0 兲兩 ⫽0 ⇒ 兩 W 共 x 兲兩 ⫽0,

x苸I.

This implies that the quaternionic matrix

冉

␸

␰

␸⬘

␰⬘

冊

is not invertible on I. 12 Hence the linear system

␸ q 1 ⫹ ␰ q 2 ⫽0,

␸ ⬘ q 1 ⫹ ␰ ⬘ q 2 ⫽0,

in the unknowns q 1,2苸H, has a solution (q 1 ,q 2 ) where q 1 and q 2 are not both zero. Recalling that ␸ and ␰ are linearly independent on an interval I if

␸ 共 x 兲 q 1 ⫹ ␰ 共 x 兲 q 2 ⫽0 ⇒ q 1 ⫽q 2 ⫽0, the fact that q 1 and q 2 are not both zero guarantees the linear dependence of ␸ and ␰ on I. Example 1: Show that ␸ ⫽exp关⫺ix兴 and ␰ ⫽exp关(i⫺j)x兴 form a basis of solutions of ⌿ ⬙ ⫹ j ⌿ ⬘ ⫹ 共 1⫺k 兲 ⌿⫽0,

䊏共17兲

on any interval. Solution: Substitution shows that they are solutions, 关 ⫺1⫹ j 共 ⫺i 兲 ⫹1⫺k 兴 exp关 ⫺ix 兴 ⫽0, 关 ⫺2⫹ j 共 i⫺ j 兲 ⫹1⫺k 兴 exp关共 i⫺ j 兲 x 兴 ⫽0,

and linear independence follows from Theorem 2, since 兩 W 兩 ⫽ 冑兩 i⫺ j 兩 2 ⫹ 兩 i 兩 2 ⫹i 共 j⫺i 兲 ⫺ 共 i⫺ j 兲 i⫽ 冑5. IV. HOMOGENEOUS EQUATIONS: REDUCTION OF ORDER

Let ␸ be solution of Eq. 共7兲 on some interval I. Looking for a solution in the form

␰⫽␸ ␶ and substituting ␰ and its derivatives

␰ ⬘⫽ ␸ ⬘ ␶ ⫹ ␸ ␶ ⬘

and ␰ ⬙ ⫽ ␸ ⬙ ␶ ⫹2 ␸ ⬘ ␶ ⬘ ⫹ ␸ ␶ ⬙

into Eq. 共7兲, we obtain

␶ ⬙ ⫽ 共 ␸ ⫺1 ␣ ␸ ⫺2 ␸ ⫺1 ␸ ⬘ 兲 ␶ ⬘ .

共18兲




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It is important to observe that, for quaternionic functions, we cannot give a formal solution of the previous equation. Only in particular cases, Eq. 共18兲 can be immediately integrated. For example, for homogeneous second order equations with constant coefficients,

␣ 共 x 兲 →a苸H and ␤ 共 x 兲 →b苸H, at least one solution is in the form of a quaternionic exponential, ␸ ⫽exp关qx兴, and consequently Eq. 共18兲 reduces to

␸ ␶ ⬙ ⫽ 共 a⫺2 q 兲 ␸ ␶ ⬘ .

共19兲

Let us introduce the quaternionic function

␴ ⫽ ␸ ␶ ⬘. Observing that

␴ ⬘ ⫽ ␸ ⬘ ␶ ⬘ ⫹ ␸ ␶ ⬙ ⫽q ␸ ␶ ⬘ ⫹ ␸ ␶ ⬙ , Eq. 共19兲 can be rewritten as follows:

␴ ⬘ ⫽ 共 a⫺q 兲 ␴ .

共20兲

This equation can be immediately integrated, its solution reads

␴ ⫽exp关共 a⫺q 兲 x 兴 . Thus, the second solution of the homogeneous second order differential equation with constant coefficients is given by

␰ ⫽exp关 qx 兴

冕

exp关 ⫺qx 兴 exp关共 a⫺q 兲 x 兴 dx.

共21兲

In the complex limit (a,q苸C) we find the well-known results ␰ ⬀exp关(a⫺q)x兴 if 2q⫽a and ␰ ⬀x exp关qx兴 if 2q⫽a. In the quaternionic case (a,q苸C), the integral which appears in 共21兲 must be treated with care. The solution of this integral will give interesting information about the second solution of quaternionic differential equations with constant coefficients when the associated characteristic quadratic equation has a unique solution. To solve the integral in Eq. 共21兲, we start by observing that 关 e ux e v x 兴 ⬘ ⫽u e ux e v x ⫹e ux e v x v ⫽ 共 L u ⫹R v 兲 e ux e v x .

If the operator L u ⫹R v is invertible the previous equality implies

冕

e ux e v x dx⫽ 共 L u ⫹R v 兲 ⫺1 e ux e v x .

This result guarantees that, if the operator L ⫺q ⫺R a⫺q is invertible the second solution can be written in the form

␰ ⫽exp关 qx 兴共 L ⫺q ⫹R a⫺q 兲 ⫺1 exp关 ⫺qx 兴 exp关共 a⫺q 兲 x 兴 ⫽exp关 qx 兴共 R a⫺q ⫺L q 兲 ⫺1 exp关 ⫺qx 兴 exp关共 a⫺q 兲 x 兴 .

共22兲

If the operator L ⫺q ⫹R a⫺q is not invertible, we need to solve the integral which appears in 共21兲 by using the polar decomposition of quaternions 共see example 3兲 and a term linearly dependent on x


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will appear. In the complex case (a,q苸C), the operator L ⫺q ⫹R a⫺q is not invertible if and only if 2q⫽a. In the quaternionic (a,q苸H), the condition 2q⫽a does not guarantee that the operator is invertible. Example 2: Knowing that ␸ ⫽exp关⫺ix兴 is solution of the homogeneous second order equation 共17兲, find 共by using the method of reduction of order兲 a second independent solution, ␰. Solution: We have q⫽⫺i and a⫽⫺ j. To use Eq. 共22兲 we have to prove that the operator L ⫺q ⫹R a⫺q ⫽L i ⫹R i⫺ j is invertible. A simple algebraic calculation shows that 共 L i ⫺R i⫺ j 兲共 L i ⫹R i⫺ j 兲 ⫽1.

Thus, 共 L ⫺q ⫹R a⫺q 兲 ⫺1 ⫽L i ⫺R i⫺ j .

We are now ready to calculate ␰ from Eq. 共22兲,

␰ ⫽exp关 ⫺ix 兴共 L i ⫺R i⫺ j 兲 exp关 ix 兴 exp关共 i⫺ j 兲 x 兴 ⫽ 共 L i ⫺R i⫺ j 兲 exp关共 i⫺ j 兲 x 兴 ⫽exp关共 i⫺ j 兲 x 兴 j. Due to the H linearity 共from the right兲 of Eq. 共17兲 the right factor j can be ignored recovering the solution of example 1. Example 3: Inspection shows that k ⌿ ⬙ ⫹i ⌿ ⬘ ⫹ ⫽0 2

共23兲

has ␸ ⫽exp兵⫺ 关(i⫹j)/2兴 x 其 as a first solution. Find the second linear independent solution. Solution: We have q⫽⫺ (i⫹ j)/2 and a⫽⫺i. In this case, the operator L ⫺q ⫹R a⫺q ⫽L 共 i⫹ j 兲 /2⫹R 共 j⫺i 兲 /2 is not invertible. This is easily seen by using, for example, the real matrix representation 共3兲. Thus, the integral in Eq. 共21兲 cannot be expressed in terms of an exponential product. Let us explicitly calculate ␰ from Eq. 共21兲. We find

冋冋冋

␰ ⫽exp ⫺

i⫹ j x 2

⫽exp ⫺

i⫹ j x 2

⫽exp ⫺

i⫹ j x 2

册冕冋册冋册册冕冉冊冉册冕 exp

i⫹ j j⫺i x exp x dx 2 2

cos

x

&

⫹

i⫹ j &

sin

x

&

兵 1⫺k exp关 ⫺ 共 i⫹ j 兲 x 兴其

cos

x &

⫹

j⫺i &

sin

x &

冊

dx

1⫹k dx. 2

Due to the H linearity 共from the right兲 of Eq. 共23兲 the right factor (1⫹k)/2 can be removed. After integration, we find

冋

␰ ⫽exp ⫺

i⫹ j x 2

册再

x⫺k

冎冉

冊冋

册

i⫹ j i⫺ j i⫹ j exp关 ⫺ 共 i⫹ j 兲 x 兴 ⫽ x⫹ x . exp ⫺ 2 2 2

Observe that the quaternionic factor (i⫺ j)/2 appears on the left of the quaternionic exponential and consequently cannot be removed. It is a fundamental part of the solution. Inspection shows that




冋

␰ ⫽x exp ⫺

i⫹ j x 2

2231

册

is not the solution of Eq. 共23兲.

V. NONHOMOGENEOUS EQUATIONS: VARIATION OF PARAMETERS

A general solution of the nonhomogeneous equation 共4兲 is a solution of the form ⌿⫽⌿ h ⫹⌿ p ,

共24兲

where ⌿ h⫽ ␸ q 1⫹ ␰ q 2 is a general solution of the homogeneous equation 共7兲 and ⌿ p is any particular solution of 共4兲 containing no arbitrary constants. In this section we discuss the so-called method of variation of parameters to find a particular solution for quaternionic nonhomogeneous differential equations. A method to solve a homogeneous second order quaternionic differential equations with constant coefficients has been recently developed.2 Quaternionic differential equations with nonconstant coefficients are under investigation. We suppose to know two independent solutions of the homogeneous equation associated with Eq. 共7兲. We wish to investigate if the method of variation of parameters still works in the quaternionic case. The method of variation of parameters involves replacing the constant q 1 and q 2 by quaternionic functions ␯ 1 (x) and ␯ 2 (x) to be determined so that the resulting function ⌿ p⫽ ␸ ␯ 1⫹ ␰ ␯ 2 is a particular solution of Eq. 共4兲. By differentiating ⌿ p we obtain ⌿ ⬘p ⫽ ␸ ⬘ ␯ 1 ⫹ ␰ ⬘ ␯ 2 ⫹ ␸ ␯ ⬘1 ⫹ ␰ ␯ ⬘2 . The requirement that ⌿ p satisfies Eq. 共4兲 imposes only one condition on ␯ 1 and ␯ 2 . Hence, we can impose a second arbitrary condition, that is

␸ ␯ 1⬘ ⫹ ␰ ␯ ⬘2 ⫽0.

共25兲

This reduces ⌿ ⬘p to the form ⌿ ⬘p ⫽ ␸ ⬘ ␯ 1 ⫹ ␰ ⬘ ␯ 2 . By differentiating this function we have ⌿ ⬙p ⫽ ␸ ⬙ ␯ 1 ⫹ ␸ ⬘ ␯ 1⬘ ⫹ ␰ ⬙ ␯ 2 ⫹ ␰ ⬘ ␯ ⬘2 . Substituting ⌿ p , ⌿ ⬘p , and ⌿ ⬙p in Eq. 共4兲 we readily obtain

␸ ⬘ ␯ ⬘1 ⫹ ␰ ⬘ ␯ ⬘2 ⫽ ␳ .

共26兲

Collecting Eq. 共25兲 and Eq. 共26兲, we can construct the following matrix system:

冉

␸

␰

␸⬘

␰⬘

冊冋册冋册

␯ 1⬘ 0 ⫽ , ␳ ␯ 2⬘

共27兲

from which ( 兩 W 兩 ⫽0) we obtain


2232


冋册冉

␸ ␯ 1⬘ ⫽ ␯ ⬘2 ␸⬘

冊冋册 ⫺1

␰ ␰⬘


冉

关 ␸ ⫺ ␰␰ ⬘ ⫺1␸ ⬘ 兴 ⫺1 0 ⫽ ␳ 关 ␰ ⫺ ␸␸ ⬘ ⫺1␰ ⬘ 兴 ⫺1

关 ␸ ⬘ ⫺ ␰ ⬘ ␰ ⫺1 ␸ 兴 ⫺1 关 ␰ ⬘⫺ ␸ ⬘␸

⫺1

␰兴

⫺1

冊冋册

0 . ␳

共28兲

Then,

␯ ⬘1 ⫽ 关 ␸ ⬘ ⫺ ␰ ⬘ ␰ ⫺1 ␸ 兴 ⫺1 ␳

and ␯ 2⬘ ⫽ 关 ␰ ⬘ ⫺ ␸ ⬘ ␸ ⫺1 ␰ 兴 ⫺1 ␳ .

共29兲

To find ␯ 1 (x) and ␯ 2 (x) we have to integrate the previous equations. Example 4: Find a general solution of the nonhomogeneous quaternionic differential equation ⌿ ⬙ ⫹ j ⌿ ⬘ ⫹ 共 1⫺k 兲 ⌿⫽i x.

共30兲

Solution: The solution of the associated homogeneous equation 共see example 1兲 is ⌿ h ⫽exp关 ⫺ix 兴 q 1 ⫹exp关 ⫺ 共 i⫹ j 兲 x 兴 q 2 . The particular solution is ⌿ p ⫽exp关 ⫺ix 兴 ␯ 1 ⫹exp关 ⫺ 共 i⫹ j 兲 x 兴 ␯ 2 . Consequently, from Eqs. 共29兲 we find

␯ ⬘1 ⫽exp关 ix 兴 x k

and ␯ 2⬘ ⫽⫺exp关共 i⫹ j 兲 x 兴 x k

which after integration give

␯ 1 共 x 兲 ⫽ 共 1⫺ix 兲 exp关 ix 兴 k

and

␯ 2 共 x 兲 ⫽⫺ 21 关 1⫺ 共 i⫹ j 兲 x 兴 exp关共 i⫹ j 兲 x 兴 k.

Finally ⌿ p ⫽ 12 关共 i⫹ j 兲 x⫹k 兴 . A general solution of Eq. 共23兲 is ⌿⫽exp关 ⫺ix 兴 q 1 ⫹exp关 ⫺ 共 i⫹ j 兲 x 兴 q 2 ⫹ 21 关共 i⫹ j 兲 x⫹k 兴 . VI. CONCLUSIONS AND OUTLOOKS

The recent results on violations of quantum mechanics by quaternionic potentials1 and the possibility to get a better understanding of CP-violation phenomena within a quaternionic formulation of physical theories1,13 stimulated the study of quaternionic differential operators.2 In this paper, we have proved existence and uniqueness for quaternionic initial value problems and solved simple quaternionic differential equations by discussing the reduction of order for quaternionic homogeneous equations and by extending to the noncommutative case the method of variation of parameters and the definition of absolute value of the Wronskian functional. In view of a more complete discussion of quantum dynamical systems using quaternionic wave packets, our next research 共mathematical兲 interest will be the study of quaternionic integral transforms. The quaternionic formulation of Fourier transforms could find an immediate and interesting application in the study of delay time modifications of wave packets scattered by a quaternionic potential step. S. De Leo, G. C. Ducati, and C. C. Nishi, J. Phys. A 35, 5411 共2002兲. S. De Leo and G. C. Ducati, J. Math. Phys. 42, 2236 共2001兲. 3 W. R. Hamilton, Elements of Quaternions, 3rd ed. 共Chelsea Publishing Co., New York, 1969兲, Vol. I. 4 S. De Leo and G. C. Ducati, Int. J. Theor. Phys. 38, 2197 共1999兲. 5 S. De Leo and G. Scolarici, J. Phys. A 33, 2971 共2000兲. 1 2




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S. De Leo, G. Scolarici, and L. Solombrino, J. Math. Phys. 43, 5815 共2002兲. E. Kreyszig, Advanced Engineering Mathematics 共Wiley, New York, 1993兲, p. 165. 8 E. Study, Acta Math. 42, 1 共1920兲. 9 J. Brenner, Linear Algebr. Appl. 1, 511 共1968兲. 10 F. J. Dyson, Helv. Phys. Acta 45, 289 共1972兲. 11 H. Aslaksen, Math. Intell. 18, 57 共1996兲. 12 N. Cohen and S. De Leo, Elec. J. Lin. Alg. 7, 100 共2000兲. 13 S. Adler, Quaternonic Quantum Mechanics and Quantum Fields 共Oxford University Press, New York, 1995兲. 6 7
