HYPERBOLIC DOUBLE-COMPLEX LAPLACE OPERATOR. Lilia N. Apostolova

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2 Pliska Stud. Math. Bulgar. 1 (01), STUDIA MATHEMATICA BULGARICA HYPERBOLIC DOUBLE-COMPLEX LAPLACE OPERATOR Lilia N. Apostolova Dedicated to the 65 years Anniversary of Professor Petar Popivanov Abstract. In this paper is introduced the hyperbolic double-complex Laplace operator. The hyperbolic decomplexification of the hyperbolic doublecomplex Laplace operator and its characteristic set is found. The exponential eigenfunctions of the zero eigenvalue of the hyperbolic double-complex Laplace operator are found as well. 1. Hyperbolic complex and hyperbolic double-complex numbers. Let us recall two basic definitions. Definition 1. The elements of the commutative, associative algebra with zero divisors C := {x + jy = (x,y) : j = 1, x,y, R}. are called hyperbolic complex numbers. These numbers are used in geometry, mechanics, physics, etc. For more details one may consult, for instance, the book of I. Yaglom [], where hyperbolic complex numbers are called double numbers. 010 Mathematics Subject Classification: 5G5, A0, 0G5. Key words: hyperbolic double-complex Laplace operator, hyperbolic decomplexification.

3 68 Lilia N. Apostolova The addition in C is componentwise, and the multiplication is given by (x 1,y 1 ).(x,y ) := (x 1 x + y 1 y,x 1 y + y 1 x ). The multiplication by a scalar, that is a real number, is defined by λ(x,y) := (λx,λy). The numbers (x,x) and (x, x) are zero divisors, since (x y,0) = (x,y).(x, y) and therefore (x,x)(x, x) = (x x,0) = (0,0). Conjugate number of the hyperbolic complex number x + jy is called the hyperbolic complex number x jy. Definition. We call hyperbolic double-complex numbers the elements of the fourth dimensional commutative, associative hyperbolic double-complex algebra D C := {X = x 0 +jx 1 +j x +j x = Z +jw, Z = x 0 +j x, W = x 1 +j x } where j is a symbol such that j = +1 and j = j; x k R for k = 0,1,,. The numbers Z and W are hyperbolic complex numbers. The algebra D C inherites from C the componentwise addition and the multiplication with a real scalar λ R. The multiplication of two hyperbolic double-complex numbers is defined in an obvious way using the identities for the degrees of j and the distributive rule. Algebra D C has zero divisors as well. Indeed, for example, the numbers X(1 j ) and Y (1 + j ) satisfy X(1 j )Y (1 + j ) = XY (1 j ) = 0.. Holomorphic hyperbolic double-complex functions. Let us consider a function f : U DC, where U is a open subset of the algebra D C. Then f(x) = f 0 (Z,W) + jf 1 (Z,W), f 0,f 1 : U C. The function f 0 is called an even part, and the function f 1 is called an odd part of the hyperbolic double-complex function f. For the introduction of formal hyperbolic complex derivatives we use the partial derivatives with respect to the real variables x 0,x 1,x,x. We denote as follows: Z := 1 ( + j ), x 0 x W := 1 ( + j ) (1). x 1 x

4 Hyperbolic double-complex Laplace operator 69 The following examples are important for the formal calculus using so defined formal hyperbolic complex derivatives 1.) Z Z = 1,.) = 0,.) W Z W W = 1 and.) W Z = 0. We consider also the following formal derivatives () X = 1 ( Z + 1 j W ), X = 1 ( Z 1 j ). W The formal derivatives defined in () are called formal hyperbolic doublecomplex derivatives. Definition. We say that the hyperbolic double-complex function f is holomorphic hyperbolic double-complex functions if and only if f X = 1 ( f Z 1 ) f () = 0, j W Example 1. The hyperbolic double-complex function X = Z + jw is a holomorphic hyperbolic double-complex function, because (Z + jw) Z 1 j (Z + jw) W = 1 1 = 0. Exponential hyperbolic double-complex function e X is defined by the power series e X X k =. The following identity for the exponential function is fulfilled k! k=0 e X = e x 0 (cosh x + j sinhx ) ((1 + j )cosh x 1 + (j + j )sinh x 1 + (1 j )cos x 1 + (j j )sin x 1 ) ((1 + j )cosh x + (j + j)sinh x + (1 j )cos x + (j j)sin x ) = = e x 0 (cosh x + j sinhx ) [ (1 + j )(cosh x 1 + j sinh x 1 )(cosh x + j sinh x ) + + (1 j )(cos x 1 j sin x 1 )(cos x j sin x ) ].

5 70 Lilia N. Apostolova Example. The exponential hyperbolic double-complex function e X and the following composite functions e p(1+j+j +j )(Z+W), e p(1+j )(jz+w) and e q(z+jw), where p, q are hyperbolic double-complex numbers and Z, W are hyperbolic complex variables, are holomorphic hyperbolic double-complex functions. On the other hand, it is easy to check that the composite exponential hyperbolic double complex function e p(1 j+j j )(Z+W), p 0 is not a holomorphic hyperbolic double-complex function. This follows from the definition, applying the rules for computation of the formal hyperbolic complex derivatives, which are similar to those for the partial derivatives. Theorem 1 (see [1], []). The function f = f 0 +jf 1 is holomorphic hyperbolic double-complex function if and only if the Cauchy-Riemann type system () f 0 Z = f 1 W, f 0 W = j f 1 Z is satisfied by the hyperbolic complex functions f 0 and f 1.. Hyperbolic double-complex Laplace operator. The even part f 0 and the odd part f 1 of the holomorphic hyperbolic double-complex function f satisfy two second order partial differential equations with constant hyperbolic complex coefficients. By the Cauchy-Riemann type system () we derive the following system of equations for f 0 and f 1 : f 0 Z W = f 1 W, f 0 W Z = j f 1 Z, f 1 Z W = f 0 Z, f 1 W Z = j f 0 W. Therefore the functions f 0 and f 1 satisfy the equations f 1 Z = j f 1 W and, respectively, f 0 Z = j f 0 W. The second order partial differential operator with hyperbolic complex coefficient j and hyperbolic complex variables Z and W (5) hdc = Z j W =

6 Hyperbolic double-complex Laplace operator 71 = 1 ( + j ) ( j 1 + j ) = x 0 x x 1 x = 1 ( x 0 + x j x 1 x x 1 j x + j ) x 0 x is called a hyperbolic double-complex Laplace operator. The hyperbolic complex-valued function f, solution of the equation hdc f = 0 is called harmonic hyperbolic complex function. Then the even and the odd parts f 0 and f 1 of a holomorphic hyperbolic double-complex function are harmonic hyperbolic complex functions. In order to write the hyperbolic complex symbol of the hyperbolic doublecomplex Laplace operator, we replace in (5) the variable / x 0 by ξ 0, / x 1 by ξ 1, / x by ξ and / x by ξ, respectively. This way we obtain the quadratic form B(ξ 0,ξ 1,ξ,ξ ) = 1 (ξ 0 + ξ ξ 1 ξ j ξ 1 j ξ + j ξ 0 ξ ), with the following matrix Q = j 0 0 j 0 1 j j The quadratic form B(ξ 0,ξ 1,ξ,ξ ) is called a hyperbolic complex symbol of the operator hdc.. The characteristic set of the hyperbolic double-complex Laplace operator. In order to find the characteristic set Σ of the hyperbolic double-complex Laplace operator we solve the following equation of second order with hyperbolic complex coefficients (ξ 0 +j ξ ) j (ξ 1 +j ξ ) = 0 ξ 0 +ξ j ξ 1 j ξ +j ξ 0 ξ ξ 1 ξ = 0,. or the equivalent system of two real quadratic equations (6) ξ 0 + ξ ξ 1ξ = 0, ξ 1 + ξ ξ 0ξ = 0.

7 7 Lilia N. Apostolova It is true that Σ is a subset in T (R ), but it is remarkable that Σ contains two hyper-planes. Indeed, the sum of the equations in the system (6) is (7) (ξ 0 ξ ) + (ξ 1 ξ ) = 0 ξ 0 = ξ and ξ 1 = ξ, as all variables ξ 0,ξ 1,ξ,ξ are real. The difference between the first and the second equation in (6) is (8) (ξ 0 + ξ ) (ξ 1 + ξ ) = 0. which gives the equations (ξ 0 + ξ ) = ±(ξ 1 + ξ ). So we obtain parts Σ 1 and Σ of the characteristic Σ (Σ 1 Σ ) as follows Σ 1 = {(ξ 0,ξ 1,ξ,ξ ) T 0 D C \ {0} : ξ 0 = ξ 1 } Σ = {(ξ 0,ξ 1,ξ,ξ ) T 0 D C \ {0} : ξ 0 = ξ 1 } which are two transversal hyper-planes in the cotangent bundle T 0 D C of the algebra of the hyperbolic double-complex numbers D C. 5. Hyperbolic decomplexification of the hyperbolic double-complex Laplace operator. Let us present the harmonic hyperbolic complexvalued functions f 0 and f 1 by real functions g 0,g 1,g and g as follows: f 0 = g 0 + j g, f 1 = g 1 + j g. Then the functions g k, k = 0,1,, satisfy the following equation: hdc (f 0 + jf 1 ) = hdc (g 0 + jg 1 + j g + j g ) = = (g 0 + jg 1 + j g + j g ) x 0 + (g 0 + jg 1 + j g + j g ) x (g 0 + jg 1 + j g + j g ) j (g 0 + jg 1 + j g + j g ) x 1 x x 1 j (g 0 + jg 1 + j g + j g ) x + j (g 0 + jg 1 + j g + j g ) x 0 x =

8 Hyperbolic double-complex Laplace operator 7 = g 0 x 0 + g 0 x g 0 g x 1 x x g 1 x + g + x 0 x ( ) g 1 +j x + g 1 0 x g 1 g x 1 x x g 1 x + g + x 0 x +j ( g x 0 + g x ) g g 0 x 1 x x g 0 1 x + g 0 + x 0 x +j ( g x 0 + g x ) g g 1 x 1 x x g 1 1 x + g 1 = 0. x 0 x It is equivalent to the following system of two equations for two couples of realvalued functions g 0,g, and g 1,g, namely g 0 x 0 g 0 x 1 + g 0 x + g 0 x g 0 g x 1 x x g 1 x + g = 0 x 0 x g 0 g x 0 x x g 0 x + g = 0 x 1 x and the same system for the couple g 1,g. Denoting A(g) = g x + g 0 x g and B(g) = g x 1 x x 1 the above system can be rewritten as (A) A(g 0 ) B(g ) = 0 B(g 0 ) A(g ) = 0. A(g 1 ) B(g ) = 0 B(g 1 ) A(g ) = 0. + g x g x 0 x The symbol of this system is the following antisymmetric matrix ( σ(a) = ξ 0 + ξ ξ 1ξ σ(b) = ξ1 ξ + ξ 0ξ (B) σ = σ(b) = ξ 1 + ξ ξ 0ξ σ(a) = ξ 0 ξ + ξ 1ξ ), where (ξ 0,ξ 1,ξ,ξ ) T (R ) \ {0}.

9 7 Lilia N. Apostolova We see that (A B )g 0 = A(A(g 0 )) B(B(g 0 )) = A(B(g )) B(A(g )) = 0 as the operators A and B have constant coefficients. So the system (A) implies the following partial differential equation of fourth order with one unknown function: or in explicit form (A B )g k = 0 for k = 0,1,, ( g x 0 ) ( + g x g ) g x 1 x x + g 1 x g = 0 x 0 x or ( ( g g ) ( g + g ) )( ( g + g ) ( g + g ) ) = 0. x 0 x x 1 x x 0 x x 1 x for the functions g = g 0,g 1,g,g. The symbol of the operator A + B has the following form (ξ0 + ξ ξ 1 ξ ) + (ξ1 + ξ ξ 0ξ ). The determinant of the matrix (B) is the following one: ( ξ 0 + ξ detσ = det ξ 1ξ ξ1 ξ + ξ ) 0ξ ξ1 + ξ ξ 0ξ ξ0 ξ + ξ = 1ξ = (ξ 0 + ξ ξ 1 ξ ) + (ξ 1 + ξ ξ 0 ξ ) = = (ξ 0 + ξ ξ 1 ξ + ξ 1 + ξ ξ 0 ξ )(ξ 0 + ξ ξ 1 ξ ξ 1 ξ + ξ 0 ξ ) = = ( (ξ 0 ξ ) + (ξ 1 ξ ) ) ( (ξ 0 + ξ ) (ξ 1 + ξ ) ) = = ( (ξ 0 ξ ) + (ξ 1 ξ ) ) (ξ 0 + ξ + ξ 1 + ξ )(ξ 0 + ξ ξ 1 ξ ). Therefore the real solutions of the equation (9)detσ = ( (ξ 0 ξ ) + (ξ 1 ξ ) ) (ξ 0 + ξ + ξ 1 + ξ )(ξ 0 + ξ ξ 1 ξ ) = 0,

10 Hyperbolic double-complex Laplace operator 75 forms the characteristic set of the hyperbolic decomplexification of the hyperbolic double-complex Laplace operator. This way we may write the characteristic set in an explicit form Σ = Σ 1 Σ Σ where and Σ 1 = {(ξ 0,ξ 1,ξ,ξ ) : ξ 0 + ξ + ξ 1 + ξ = 0}, Σ = {(ξ 0,ξ 1,ξ,ξ ) : ξ 0 + ξ ξ 1 ξ = 0} Σ = {(ξ 0,ξ 1,ξ,ξ ) : ξ 0 = ξ, ξ 1 = ξ }, where (ξ 0,ξ 1,ξ,ξ ) T (R ) \ {0}. 6. Hyperbolic double-complex exponential eigenfuncions for the zero eigenvalue of the hyperbolic double-complex Laplace operator. Let us consider the eigenfunction problem for the hyperbolic doublecomplex Laplace operator hdc, i.e. to solve the equation (10) hdc F(Z + jw) = λf(z + jw), where F(Z + jw) : D C D C is a hyperbolic double-complex function of hyperbolic double-complex variable. Applying the method of separation of variables for the above problem we are looking for the solutions of the form F(Z + jw) = f(z).g(w), where f(z) = f 1 (Z)+jf (Z) and g(w) = g 1 (W)+jg (W), f 1,f,g 1,g : C C are hyperbolic complex valued functions of one hyperbolic complex variable. Equation (10) becomes g(w) f(z) Z jf(z) g(w) W = λf(z)g(w), where λ D C. For the zero eigenvalue λ we have and f ZZ (Z) = (m 0 + jm 1 + j m + j m )f(z) g WW (W) = j (m 0 + jm 1 + j m + j m )g(w)

11 76 Lilia N. Apostolova where m 0,m 1,m,m are real constants. Solutions of this ODEs are f(z) = e az and g(w) = e bw, where a = a 0 +ja 1 +j a +j a and b = b 0 +jb 1 +j b +j b, a k,b k R for k = 0,1,,, are hyperbolic double-complex numbers such that (11) a = j b = m 0 + jm 1 + j m + j m. If m 0 + m m 1 + m then a 0 = ε 1 m0 + m 1 + m + m + ε m0 m 1 + m m + + ε (m0 m ) + (m 1 m ) + m 0 m, a 1 = ε 1 m0 + m 1 + m + m ε m0 m 1 + m m + + ε sign(m 1 m ) (m0 m ) + (m 1 m ) (m 0 m ), a = ε 1 m0 + m 1 + m + m + ε m0 m 1 + m m ε (m0 m ) + (m 1 m ) + m 0 m, a = ε 1 m0 + m 1 + m + m ε m0 m 1 + m m ε sign(m 1 m ) (m0 m ) + (m 1 m ) (m 0 m ), where the constants ε,ε 1 and ε are equal to ±1. Analogously (b 0 + jb 1 + j b + j b ) = m + jm + j m 0 + j m 1 and b 0 = ε 1 m + m + m 0 + m 1 + ε m m + m 0 m 1 +

12 Hyperbolic double-complex Laplace operator 77 + ε sign(m 1 m ) (m m 0 ) + (m m 1 ) + m m 0, b 1 = ε 1 m + m + m 0 + m 1 ε m m + m 0 m ε (m m 0 ) + (m m 1 ) (m m 0 ), b = ε 1 m + m + m 0 + m 1 + ε m m + m 0 m 1 ε sign(m 1 m ) (m m 0 ) + (m m 1 ) + m m 0, b = ε 1 m + m + m 0 + m 1 ε m m + m 0 m 1 ε (m m 0 ) + (m m 1 ) (m m 0 ), where the constants ε,ε 1 and ε are equal to ±1. So for the eigenfunctions, corresponding to the zero eigenvalue, we obtain that F(Z + jw) = e az+bw = e a(z+j W) depends on hyperbolic double-complex parameter m = m 0 + jm 1 + j m + m and are the following (1) F ε,ε1,ε,m(z,w) = e ε 1 (1+j+j +j ) m0 +m 1 +m +m (Z+W) e ε(1 j ) Õ (m0 m ) +(m 1 m ) +m 0 m (Z+jW) e ε sign(m 1 m )(1 j ) Õ (m0 m ) +(m 1 m ) m 0 +m (jz+w) e ε (1 j+j j ) m0 m 1 +m m (Z+W).

13 78 Lilia N. Apostolova Here Z, W are hyperbolic complex numbers, ε,ε 1, ε are equal to ±1 and m 0,m 1,m and m are real numbers such that m 0 + m m 1 + m holds. Note that the numbers 1 + j + j + j, and 1 j in the formulae above are zero divisors in the algebra D C. The first three of the composite exponential hyperbolic double-complex functions in the product (1) are holomorphic hyperbolic double-complex functions and the fourth one is not a holomorphic hyperbolic double-complex function. REFERENCES [1] L. N. Apostolova, S. Dimiev, M. S. Marinov, P. Stoev. Matrix n - holomorphy. Applications of Mathematics in Engineering and Economics AMEE 10, AIP Conference Series, Conference Proceedings 19, (010), [] L. N. Apostolova, S. Dimiev, P. Stoev. Hyperbolic hypercomplex D-bar operators, hyperbolic CR-equations, and harmonicity II, Fundamental solutions for hyperholomorphic operators and hyperbolic -real geometry. Bull. Soc. Sci. Letters Lodz Ser. Rech. Deform. 60 (010), [] I. Yaglom. Complex Numbers in Geometry. Academic Press, N.Y., Institute of Mathematics and Informatics Bulgarian Academy of Science Acad. G. Bonchev Str., Bl Sofia, Bulgaria liliana@math.bas.bg