2010 Mathematics Subject Classification: 35G35, 32A30, 30G35.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

N. Apostolova, Lilia

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Pliska Stud. Math. Bulgar. 21 (2012), 267–278STUDIA MATHEMATICABULGARICAHYPERBOLIC DOUBLE-COMPLEX LAPLACE OPERATORLilia N. ApostolovaDedicated to the 65 years Anniversary of Professor Petar PopivanovAbstract. In this paper is introduced the hyperbolic double-complexLaplace operator. The hyperbolic decomplexification of the hyperbolic double-complex Laplace operator and its characteristic set is found. The exponen-tial eigenfunctions of the zero eigenvalue of the hyperbolic double-complexLaplace operator are found as well.1. Hyperbolic complex and hyperbolic double-complex num-bers. Let us recall two basic definitions.Definition 1. The elements of the commutative, associative algebra with zerodivisorsC˜ := {x+ jy = (x, y) : j2 = 1, x, y,∈ R}.are called hyperbolic complex numbers.These numbers are used in geometry, mechanics, physics, etc. For moredetails one may consult, for instance, the book of I. Yaglom [3], where hyperboliccomplex numbers are called double numbers.2010 Mathematics Subject Classification: 35G35, 32A30, 30G35.Key words: hyperbolic double-complex Laplace operator, hyperbolic decomplexification.268 Lilia N. ApostolovaThe addition in C˜ is componentwise, and the multiplication is given by(x1, y1).(x2, y2) := (x1x2 + y1y2, x1y2 + y1x2).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 (x2 − y2, 0) =(x, y).(x,−y) and therefore (x, x)(x,−x) = (x2−x2, 0) = (0, 0). Conjugate num-ber of the hyperbolic complex number x + jy is called the hyperbolic complexnumber x− jy.Definition 2. We call hyperbolic double-complex numbers the elements ofthe fourth dimensional commutative, associative hyperbolic double-complex alge-braDC˜2 := {X = x0+ jx1+ j2x2+ j3x3 = Z+ jW, Z = x0+ j2x2, W = x1+ j2x3}where j is a symbol such that j4 = +1 and j2 = j; xk ∈ R for k = 0, 1, 2, 3.The numbers Z and W are hyperbolic complex numbers. The algebra DC˜2inherites from C˜ the componentwise addition and the multiplication with a realscalar λ ∈ R. The multiplication of two hyperbolic double-complex numbersis defined in an obvious way using the identities for the degrees of j and thedistributive rule.Algebra DC˜2 has zero divisors as well. Indeed, for example, the numbersX(1− j2) and Y (1 + j2) satisfy X(1 − j2)Y (1 + j2) = XY (1− j4) = 0.2. Holomorphic hyperbolic double-complex functions. Let usconsider a function f : U → DC˜2, where U is a open subset of the algebraDC˜2. Then f(X) = f0(Z,W ) + jf1(Z,W ), f0, f1 : U → C˜. The function f0 iscalled an even part, and the function f1 is called an odd part of the hyperbolicdouble-complex function f .For the introduction of formal hyperbolic complex derivatives we use the par-tial derivatives with respect to the real variables x0, x1, x2, x3. We denote asfollows:∂∂Z:=12(∂∂x0+ j2∂∂x2),∂∂W:=12(∂∂x1+ j2∂∂x3).(1)Hyperbolic double-complex Laplace operator 269The following examples are important for the formal calculus using so definedformal hyperbolic complex derivatives 1.)∂Z∂Z= 1, 2.)∂Z∂W= 0, 3.)∂W∂W= 1 and4.)∂W∂Z= 0.We consider also the following formal derivatives∂∂X=12(∂∂Z+1j∂∂W),∂∂X∗=12(∂∂Z− 1j∂∂W).(2)The formal derivatives defined in (2) are called formal hyperbolic double-complex derivatives.Definition 3. We say that the hyperbolic double-complex function f is holo-morphic hyperbolic double-complex functions if and only if∂f∂X∗=12(∂f∂Z− 1j∂f∂W)= 0,(3)Example 1. The hyperbolic double-complex function X = Z + jW is aholomorphic hyperbolic double-complex function, because∂(Z + jW )∂Z− 1j∂(Z + jW )∂W= 1− 1 = 0.Exponential hyperbolic double-complex function eX is defined by the powerseries eX =∞∑k=0Xkk!. The following identity for the exponential function is fulfilledeX = ex0(coshx2 + j2 sinhx2)××((1 + j2) cosh x1 + (j + j3) sinhx1 + (1− j2) cos x1 + (j − j3) sinx1)××((1 + j2) cosh x3 + (j3 + j) sinhx3 + (1− j2) cos x3 + (j3 − j) sin x3) == ex0(cosh x2 + j2 sinhx2)[(1 + j2)(cosh x1 + j sinhx1)(cosh x3 + j sinhx3) ++ (1− j2)(cos x1 − j sinx1)(cos x3 − j sinx3)].270 Lilia N. ApostolovaExample 2. The exponential hyperbolic double-complex function eX andthe following composite functionsep(1+j+j2+j3)(Z+W ), ep(1+j2)(jZ+W ) and eq(Z+jW ),where p, q are hyperbolic double-complex numbers and Z, W are hyperboliccomplex variables, are holomorphic hyperbolic double-complex functions.On the other hand, it is easy to check that the composite exponential hyper-bolic double complex functionep(1−j+j2−j3)(Z+W ), p 6= 0is not a holomorphic hyperbolic double-complex function. This follows from thedefinition, applying the rules for computation of the formal hyperbolic complexderivatives, which are similar to those for the partial derivatives.Theorem 1 (see [1], [2]). The function f = f0+ jf1 is holomorphic hyper-bolic double-complex function if and only if the Cauchy-Riemann type system∂f0∂Z=∂f1∂W,∂f0∂W= j2∂f1∂Z(4)is satisfied by the hyperbolic complex functions f0 and f1.3. Hyperbolic double-complex Laplace operator. The even part f0and the odd part f1 of the holomorphic hyperbolic double-complex function fsatisfy two second order partial differential equations with constant hyperboliccomplex coefficients. By the Cauchy-Riemann type system (4) we derive thefollowing system of equations for f0 and f1:∂2f0∂Z∂W=∂2f1∂W 2,∂2f0∂W∂Z= j2∂2f1∂Z2,∂2f1∂Z∂W=∂2f0∂Z2,∂2f1∂W∂Z= j2∂2f0∂W 2.Therefore the functions f0 and f1 satisfy the equations∂2f1∂Z2= j∂2f1∂W 2and,respectively,∂2f0∂Z2= j∂2f0∂W 2. The second order partial differential operator withhyperbolic complex coefficient j and hyperbolic complex variables Z and W∆hdc =∂2∂Z2− j ∂2∂W 2=(5)Hyperbolic double-complex Laplace operator 271=14(∂∂x0+ j2∂∂x2)2− j2 14(∂∂x1+ j2∂∂x3)2==14(∂2∂x20+∂2∂x22− 2 ∂2∂x1∂x3− j2 ∂2∂x21− j2 ∂2∂x23+ 2j2∂2∂x0∂x2)is called a hyperbolic double-complex Laplace operator.The hyperbolic complex-valued function f , solution of the equation ∆hdcf = 0is called harmonic hyperbolic complex function. Then the even and the odd partsf0 and f1 of a holomorphic hyperbolic double-complex function are harmonichyperbolic complex functions.In order to write the hyperbolic complex symbol of the hyperbolic double-complex Laplace operator, we replace in (5) the variable ∂/∂x0 by ξ0, ∂/∂x1 byξ1, ∂/∂x2 by ξ2 and ∂/∂x3 by ξ3, respectively.This way we obtain the quadratic formB(ξ0, ξ1, ξ2, ξ3) =14(ξ20 + ξ22 − 2ξ1ξ3 − j2ξ21 − j2ξ23 + 2j2ξ0ξ2),with the following matrixQ =141 0 j2 00 −j2 0 −1j2 0 1 00 −1 0 −j2 .The quadratic form B(ξ0, ξ1, ξ2, ξ3) is called a hyperbolic complex symbol of theoperator ∆hdc.4. The characteristic set of the hyperbolic double-complex La-place operator. In order to find the characteristic set Σ of the hyperbolicdouble-complex Laplace operator we solve the following equation of second orderwith hyperbolic complex coefficients(ξ0+ j2ξ2)2− j2(ξ1+ j2ξ3)2 = 0 ⇐⇒ ξ20+ξ22− j2ξ21− j2ξ23+2j2ξ0ξ2−2ξ1ξ3 = 0,or the equivalent system of two real quadratic equationsξ20 + ξ22 − 2ξ1ξ3 = 0,(6)ξ21 + ξ23 − 2ξ0ξ2 = 0.272 Lilia N. ApostolovaIt is true that Σ is a subset in T ∗(R4), but it is remarkable that Σ containstwo hyper-planes. Indeed, the sum of the equations in the system (6) is(ξ0 − ξ2)2 + (ξ1 − ξ3)2 = 0 ⇐⇒ ξ0 = ξ2 and ξ1 = ξ3,(7)as all variables ξ0, ξ1, ξ2, ξ3 are real.The difference between the first and the second equation in (6) is(ξ0 + ξ2)2 − (ξ1 + ξ3)2 = 0.(8)which gives the equations (ξ0 + ξ2) = ±(ξ1 + ξ3).So we obtain parts Σ1 and Σ2 of the characteristic Σ ⊃ (Σ1 ∪ Σ2) as followsΣ1 = {(ξ0, ξ1, ξ2, ξ3) ∈ T ∗0DC˜2 \ {0} : ξ0 = ξ1}Σ2 = {(ξ0, ξ1, ξ2, ξ3) ∈ T ∗0DC˜2 \ {0} : ξ0 = −ξ1}which are two transversal hyper-planes in the cotangent bundle T ∗0DC˜ of thealgebra of the hyperbolic double-complex numbers DC˜.5. Hyperbolic decomplexification of the hyperbolic double-com-plex Laplace operator. Let us present the harmonic hyperbolic complex-valued functions f0 and f1 by real functions g0, g1, g2 and g3 as follows: f0 =g0 + j2g2, f1 = g1 + j2g3. Then the functions gk, k = 0, 1, 2, 3 satisfy the follow-ing equation:4∆hdc(f0 + jf1) = 4∆hdc(g0 + jg1 + j2g2 + j3g3) ==∂2(g0 + jg1 + j2g2 + j3g3)∂x20+∂2(g0 + jg1 + j2g2 + j3g3)∂x22−−2∂2(g0 + jg1 + j2g2 + j3g3)∂x1∂x3− j2 ∂2(g0 + jg1 + j2g2 + j3g3)∂x21−−j2 ∂2(g0 + jg1 + j2g2 + j3g3)∂x23+ 2j2∂2(g0 + jg1 + j2g2 + j3g3)∂x0∂x2=Hyperbolic double-complex Laplace operator 273=∂2g0∂x20+∂2g0∂x22− 2 ∂2g0∂x1∂x3− ∂2g2∂x21− ∂2g2∂x23+ 2∂2g2∂x0∂x2++j(∂2g1∂x20+∂2g1∂x22− 2 ∂2g1∂x1∂x3− ∂2g3∂x21− ∂2g3∂x23+ 2∂2g3∂x0∂x2)++j2(∂2g2∂x20+∂2g2∂x22− 2 ∂2g2∂x1∂x3− ∂2g0∂x21− ∂2g0∂x23+ 2∂2g0∂x0∂x2)++j3(∂2g3∂x20+∂2g3∂x22− 2 ∂2g3∂x1∂x3− ∂2g1∂x21− ∂2g1∂x23+ 2∂2g1∂x0∂x2)= 0.It is equivalent to the following system of two equations for two couples of real-valued functions g0, g2, and g1, g3, namely∣∣∣∣∣∣∣∣∣∂2g0∂x20+∂2g0∂x22− 2 ∂2g0∂x1∂x3− ∂2g2∂x21− ∂2g2∂x23+ 2∂2g2∂x0∂x2= 0∂2g0∂x21+∂2g0∂x23− 2 ∂2g0∂x0∂x2− ∂2g2∂x20− ∂2g2∂x22+ 2∂2g2∂x1∂x3= 0and the same system for the couple g1, g3.Denoting A(g) =∂2g∂x20+∂2g∂x22− 2 ∂2g∂x1∂x3and B(g) =∂2g∂x21+∂2g∂x23− 2 ∂2g∂x0∂x2the above system can be rewritten as(A)∣∣∣∣∣A(g0)−B(g2) = 0B(g0)−A(g2) = 0.∣∣∣∣∣A(g1)−B(g3) = 0B(g1)−A(g3) = 0.The symbol of this system is the following antisymmetric matrix(B) σ =(σ(A) = ξ20 + ξ22 − 2ξ1ξ3 −σ(B) = −ξ21 − ξ23 + 2ξ0ξ2σ(B) = ξ21 + ξ23 − 2ξ0ξ2 −σ(A) = −ξ20 − ξ22 + 2ξ1ξ3),where (ξ0, ξ1, ξ2, ξ3) ∈ T ∗(R4) \ {0}.274 Lilia N. ApostolovaWe see that (A2−B2)g0 = A(A(g0))−B(B(g0)) = A(B(g2))−B(A(g2)) = 0as the operators A and B have constant coefficients. So the system (A) implies thefollowing partial differential equation of fourth order with one unknown function:(A2 −B2)gk = 0 for k = 0, 1, 2, 3or in explicit form(∂2g∂x20+∂2g∂x22− 2 ∂2g∂x1∂x3)2−(∂2g∂x21+∂2g∂x23− 2 ∂2g∂x0∂x2)2= 0or((∂g∂x0− ∂g∂x2)2+(∂g∂x1− ∂g∂x3)2)(( ∂g∂x0+∂g∂x2)2−(∂g∂x1+∂g∂x3)2)= 0.for the functions g = g0, g1, g2, g3.The symbol of the operator −A2 + B2 has the following form −(ξ20 + ξ22 −2ξ1ξ3)2 + (ξ21 + ξ23 − 2ξ0ξ2)2. The determinant of the matrix (B) is the followingone:detσ = det(ξ20 + ξ22 − 2ξ1ξ3 −ξ21 − ξ23 + 2ξ0ξ2ξ21 + ξ23 − 2ξ0ξ2 −ξ20 − ξ22 + 2ξ1ξ3)== −(ξ20 + ξ22 − 2ξ1ξ3)2 + (ξ21 + ξ23 − 2ξ0ξ2)2 == −(ξ20 + ξ22 − 2ξ1ξ3 + ξ21 + ξ23 − 2ξ0ξ2)(ξ20 + ξ22 − 2ξ1ξ3 − ξ21 − ξ23 + 2ξ0ξ2) ==((ξ0 − ξ2)2 + (ξ1 − ξ3)2) ((ξ0 + ξ2)2 − (ξ1 + ξ3)2)==((ξ0 − ξ2)2 + (ξ1 − ξ3)2)(ξ0 + ξ2 + ξ1 + ξ3) (ξ0 + ξ2 − ξ1 − ξ3) .Therefore the real solutions of the equationdetσ =((ξ0 − ξ2)2 + (ξ1 − ξ3)2)(ξ0 + ξ2 + ξ1 + ξ3) (ξ0 + ξ2 − ξ1 − ξ3) = 0,(9)Hyperbolic double-complex Laplace operator 275forms the characteristic set of the hyperbolic decomplexification of the hyperbolicdouble-complex Laplace operator.This way we may write the characteristic set in an explicit form Σ = Σ1 ∪Σ3 ∪ Σ3 whereΣ1 = {(ξ0, ξ1, ξ2, ξ3) : ξ0 + ξ2 + ξ1 + ξ3 = 0},Σ2 = {(ξ0, ξ1, ξ2, ξ3) : ξ0 + ξ2 − ξ1 − ξ3 = 0}andΣ3 = {(ξ0, ξ1, ξ2, ξ3) : ξ0 = ξ2, ξ1 = ξ3}, where (ξ0, ξ1, ξ2, ξ3) ∈ T ∗(R4) \ {0}.6. Hyperbolic double-complex exponential eigenfuncions forthe zero eigenvalue of the hyperbolic double-complex Laplace ope-rator. Let us consider the eigenfunction problem for the hyperbolic double-complex Laplace operator ∆hdc, i.e. to solve the equation∆hdcF (Z + jW ) = λF (Z + jW ),(10)where F (Z + jW ) : DC˜2 → DC˜2 is a hyperbolic double-complex function ofhyperbolic double-complex variable.Applying the method of separation of variables for the above problem we arelooking for the solutions of the form F (Z + jW ) = f(Z).g(W ), where f(Z) =f1(Z)+jf2(Z) and g(W ) = g1(W )+jg2(W ), f1, f2, g1, g2 : C˜→ C˜ are hyperboliccomplex valued functions of one hyperbolic complex variable. Equation (10)becomesg(W )∂2f(Z)∂Z2− jf(Z)∂2g(W )∂W 2= λf(Z)g(W ),where λ ∈ DC˜2. For the zero eigenvalue λ we havef ′′ZZ(Z) = (m0 + jm1 + j2m2 + j3m3)f(Z)andg′′WW (W ) = j2(m0 + jm1 + j2m2 + j3m3)g(W )276 Lilia N. Apostolovawhere m0,m1,m2,m3 are real constants. Solutions of this ODEs are f(Z) = eaZand g(W ) = ebW , where a = a0+ ja1+ j2a2+ j3a3 and b = b0+ jb1+ j2b2+ j3b3,ak, bk ∈ R for k = 0, 1, 2, 3, are hyperbolic double-complex numbers such thata2 = j2b2 = m0 + jm1 + j2m2 + j3m3.(11)If m0 +m2 ≥ |m1 +m3| thena0 =ε14√m0 +m1 +m2 +m3 +ε24√m0 −m1 +m2 −m3++ε2√2√√(m0 −m2)2 + (m1 −m3)2 +m0 −m2,a1 =ε14√m0 +m1 +m2 +m3 − ε24√m0 −m1 +m2 −m3++ε sign(m1 −m3)2√2√√(m0 −m2)2 + (m1 −m3)2 − (m0 −m2),a2 =ε14√m0 +m1 +m2 +m3 +ε24√m0 −m1 +m2 −m3−− ε2√2√√(m0 −m2)2 + (m1 −m3)2 +m0 −m2,a3 =ε14√m0 +m1 +m2 +m3 − ε24√m0 −m1 +m2 −m3−−ε sign(m1 −m3)2√2√√(m0 −m2)2 + (m1 −m3)2 − (m0 −m2),where the constants ε, ε1 and ε2 are equal to ±1.Analogously (b0 + jb1 + j2b2 + j3b3)2 = m2 + jm3 + j2m0 + j3m1 andb0 =ε14√m2 +m3 +m0 +m1 +ε24√m2 −m3 +m0 −m1+Hyperbolic double-complex Laplace operator 277+ε sign(m1 −m3)2√2√√(m2 −m0)2 + (m3 −m1)2 +m2 −m0,b1 =ε14√m2 +m3 +m0 +m1 − ε24√m2 −m3 +m0 −m1++ε2√2√√(m2 −m0)2 + (m3 −m1)2 − (m2 −m0),b2 =ε14√m2 +m3 +m0 +m1 +ε24√m2 −m3 +m0 −m1−−ε sign(m1 −m3)2√2√√(m2 −m0)2 + (m3 −m1)2 +m2 −m0,b3 =ε14√m2 +m3 +m0 +m1 − ε24√m2 −m3 +m0 −m1−− ε2√2√√(m2 −m0)2 + (m3 −m1)2 − (m2 −m0),where the constants ε, ε1 and ε2 are equal to ±1.So for the eigenfunctions, corresponding to the zero eigenvalue, we obtainthat F (Z + jW ) = eaZ+bW = ea(Z+j2W ) depends on hyperbolic double-complexparameter m = m0 + jm1 + j2m2 +m3 and are the followingFε,ε1,ε2,m(Z,W ) = eε1(1+j+j2+j3)4√m0+m1+m2+m3(Z+W )×(12)× eε(1−j2)2√2q√(m0−m2)2+(m1−m3)2+m0−m2(Z+jW )×× eε sign(m1−m3)(1−j2)2√2q√(m0−m2)2+(m1−m3)2−m0+m2(jZ+W )×× e ε2(1−j+j2−j3)4√m0−m1+m2−m3(Z+W ).278 Lilia N. ApostolovaHere Z, W are hyperbolic complex numbers, ε, ε1, ε2 are equal to ±1 andm0,m1,m2 and m3 are real numbers such that m0 +m2 ≥ |m1 +m3| holds.Note that the numbers 1 + j + j2 + j3, and 1− j2 in the formulae above arezero divisors in the algebra DC˜2.The first three of the composite exponential hyperbolic double-complex func-tions in the product (12) are holomorphic hyperbolic double-complex functionsand the fourth one is not a holomorphic hyperbolic double-complex function.REFERENCES[1] L. N. Apostolova, S. Dimiev, M. S. Marinov, P. Stoev. Matrix 2n-holomorphy. Applications of Mathematics in Engineering and Economics –AMEE’10, AIP Conference Series, Conference Proceedings 1293, (2010), 165–176.[2] L. N. Apostolova, S. Dimiev, P. Stoev. Hyperbolic hypercomplex D-baroperators, hyperbolic CR-equations, and harmonicity II, Fundamental solu-tions for hyperholomorphic operators and hyperbolic 4-real geometry. Bull.Soc. Sci. Letters Lodz Ser. Rech. Deform. 60 (2010), 61–72.[3] I. Yaglom. Complex Numbers in Geometry. Academic Press, N.Y., 1968.Institute of Mathematics and InformaticsBulgarian Academy of ScienceAcad. G. Bonchev Str., Bl. 81113 Sofia, Bulgariae-mail: liliana@math.bas.bg

Hyperbolic Double-Complex Laplace Operator

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Hyperbolic Double-Complex Laplace Operator

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