IntroductionTo treat Maxwell equations we make use of complex representationof them according to the known approach by Majorana-Oppenheimer [1–11], also see [12, 13] and referencestherein for extending this approach to curved space-timemodels.The situation is specified in quasi-Cartesian coordinatesin Lobachevsky space, they are appropriate for modeling amedium nonuniform along the axis z. Exact solutions of thecovariant Maxwell equations in complex E +iB form of Majorana-Oppenheimer have been constructed. The problem reducesto a second order differential equation for a certain primaryfunction which can be associated with the one-dimensionalSchrödinger problem for a particle in external potentialfield U(z) = U0e2z. In quantum mechanics, curved geometryacts as an effective potential barrier with reflection coefficientR = 1; in electrodynamic context results are similar:Lobachevsky geometry simulates a medium that effectivelyacts as an ideal mirror. Penetration of the electromagneticfield into the effective medium along the axis z depends onthe parameters of the electromagnetic waves ω, k21 +k22 andthe curvature radius ρ of the used Lobachevsky space. Thesegeneralized quasi-plane solutions f(t, x, y, z) = E + iBand the relevant system of equations are transformed into thereal form, which permit us to relate geometry characteristicswith expressions for effective tensors of electric and magneticpermittivities.1. Cartesian coordinates in Lobachevsky spaceWe will apply the coordinate system in Lobachevsky spacedS2 = dt2 − e−2z(dx2 + dy2) − dz2,dV = e−2zdxdydz. (1)It is helpful to have at hand some details of the parametrizationof the modelH3 by coordinates (x, y, z). It is known thatthis model can be identified with a branch of hyperboloid in4-dimension flat spaceu20− u21− u22− u23= ρ2, u0 = +√ρ2 + u2.Coordinates (x, y, z) are referred to ua by relationsu0 =12[(ez + e−z) + (x2 + y2)e−z], u1 = xe−z,u2 = ye−z, u3 =12[(ez − e−z) + (x2 + y2)e−z].We will employ the Poincare realization for Lobachevskyspace as the inside part of the 3-sphereqi =uiu0=√ uiρ2 + u21+ u22+ u23, qiqi < 1.Quasi-Cartesian coordinates (x, y, z) are referred to qi asfollowsq1 =2xx2 + y2 + e2z + 1,q2 =2yx2 + y2 + e2z + 1,q3 =x2 + y2 + e2z − 1x2 + y2 + e2z + 1. (2)Inverses to (2) relations arex =q11 − q3, y =q21 − q3, ez =√1 − q21 − q3. (3)In particular, note that on the axis q1 = 0, q2 = 0, q ∈(−1, +1) relations (3) assume the following parametrizationof the axis zx = 0, y = 0, ez =√1 + q31 − q3,so thatq3 → +1, ez → +∞, z → +∞;q3 → −1, ez → +0, z → −∞.Solutions of the Maxwell equations, constructed in the followingway, can be of interest for description of electromagneticwaves in special media because Lobachevsky geometrysimulates effectively a special medium [12, 13], inhomogeneousalong the axis z. Effective electric permittivity tensorϵik(x) is given byϵik(x) = −√−gg00(x)gik(x) =1 0 00 1 00 0 e−2z,whereas the effective magnetic permittivity tensor is(μ−1)ik(x) =√−gg22g33 0 00 g33g11 00 0 g11g22 ==1 0 00 1 00 0 e2z.The constitutive relations readDi = ϵ0ϵikEk, Bi = μ0μikHk;two tensors coincide ϵik(x) = (μ−1)ik(x).2. Maxwell equations in complex form, separationof the variablesIn the coordinates (1), we will use the tetradeβ(α) =1 0 0 00 ez 0 00 0 ez 00 0 0 1,e(α)β =1 0 0 00 −e−z 0 00 0 −e−z 00 0 0 −1.Известия Коми научного центра Уральского отделения Российской академии наук № 5 (71), 2024Серия «Физико-математические науки»www.izvestia.komisc.ru59In this tetrad, the matrix equation (see notations in [12, 13])has the form(−i∂t + α1ez∂x + α2ez∂y + α3∂z−−α1s2 + α2s1)(0E + iB)= 0. (4)Let us apply the substitution(0E + iB)= e−iωteik1xeik2y(0f(z)),ei(k1x+k2y−ωt) = eiφ.Eq. (4) gives(−ω + α1ezik1 + α2ezik2 + α3 ddz−−α1s2 + α2s1)(0, f1(z), f2(z), f3(z))t = 0.Here ()t stands for transposition. After calculation withthe use of explicit expressions for all involved matrices(see [12, 13]), we derive the first-order system for functionsf1(z), f2(z), f3(z)ik1ezf1 + ik2ezf2 +(ddz− 2)f3 = 0,−ωf1 −(ddz− 1)f2 + ik2ezf3 = 0,−ωf2 +(ddz− 1)f1 − ik1ezf3 = 0,−ωf3 − ik2ezf1 + ik1ezf2 = 0.Allowing for three last equations in the first one, we getthe identity 0 = 0. So, there exist only three independentequations (we will simplify notations: k1 = a, k2 = b)ωf3 = −ibezf1 + iaezf2,ωf1 = −(ddz− 1)f2 + ibezf3,ωf2 =(ddz− 1)f1 − iaezf3. (5)With substitutions f1 = ezF1(z), f2 = ezF2(z) from Eqs.(5) we getωf3 = −ibe2zF1 + iae2zF2, ωF1 = − ddzF2 + ibf3,ωF2 =ddzF1 − iaf3. (6)There exists a particular case readily treatable, whena = 0, b = 0, f3 = 0:ωF1 = − ddzF2, ωF2 =ddzF1,that isF1(z) = e±iωz, F2(z) = ±ie±iωz,which leads to the following plane wave solutionsΦ± =(0E + iB)= e−iωtez (0, e±iωz,±ie±iωz, 0)t,whence we getE+1 + iB+1 = cos(ωt − ωz) − i sin(ωt − ωz),E+2 + iB+2 = sin(ωt − ωz) + i cos(ωt − ωz),andE−1 + iB−1 = cos(ωt + ωz) − i sin(ωt + ωz),E−2 + iB−2 = −sin(ωt + ωz) − i cos(ωt + ωz).Let us present this solution in the real formE+1 = cos(ωt − ωz), E+2 = sin(ωt − ωz), E+3 = 0,B+1 = −sin(ωt − ωz), B+2 = cos(ωt − ωz), B+3 = 0andE−1 = cos(ωt + ωz), E−2 = −sin(ωt + ωz), E−3 = 0,B−1 = −sin(ωt+ωz), B−2 = −cos(ωt+ωz), B−3 = 0.In turn, from complex-valued identities (in this case, we haveφ = −ωt)E + iB = eiφf(z) = eiφ(F(z) + iG(z)) == (cos φ + i sin φ)(F(z) + iG(z)),F∗ = F, G∗ = G, φ = k1x + k2y − ωtwe derive expressions for real vectors E and B:E = cos φF(z) − sinφG(z),B = sin φF(z) + cosφG(z), φ = −ωt.Let us turn back to the general system (6); with the help ofthe first equation we eliminate the variable f3, so producingthe system of linked equations for F1 and F2 (ddz+abe2zω)F2 =b2e2z − ω2ωF1,(ddz− abe2zω)F1 =ω2 − a2e2zωF2. (7)In the new variable Z, ez =√ωZ two last equations arewritten asZ(ddZ+ abZ)F2 = (b2Z2 − ω)F1,Z(ddZ− abZ)F1 = −(a2Z2 − ω)F2. (8)This system can be solved straightforwardly in terms ofthe Heun confluent functions. Indeed, from (8) it follows asecond order differential equation for F1d2F1dZ2− a2Z2 + ωZ(a2Z2 − ω)dF1dZ++(ω2Z2 +2abωa2Z2 − ω− (a2 + b2)ω)F1 = 0,60Известия Коми научного центра Уральского отделения Российской академии наук № 5 (71), 2024Серия «Физико-математические науки»www.izvestia.komisc.ruwhere we note the presence of an additional singular pointZ = ±√ωa−1. In the new variable y = a2Z2ω−1, wearrive at the equationd2F1dy2 +(1y− 1y − 1)dF1dy++(ω24y2− 2abω + (a2 + b2)ω24a2y+bω2a(y − 1))F1 = 0.With the use of the substitution F1 = ycg1(y), c =±iω/2, further we derived2g1dy2 +(2c + 1y− 1y − 1)dg1dy++(2c − ω2/2 − bω/a − b2ω2/(2a2)2y++−2c + bω/a2(y − 1))g1 = 0,which can be identified with the confluent Heun equation. Belowwe will develop a method that makes possible to constructsolutions of the system (7) in terms of more simpleBessel functions.3. Solutions in terms of the Bessel functionsLet us perform a linear transformation over the system(7):F1 = αG1 + βG2, F2 = mG1 + nG2;G1 = nF1 − βF2, G2 = −mF1 + αF2; (9)suppose the constraint αn−βm = 1. Combining equationsfrom (7), we getnZ(ddZ− abZ)F1 − βZ(ddZ+ abZ)F2 == −n(a2Z2 − ω)F2 − β(b2Z2 − ω)F1,−mZ(ddZ− abZ)F1 + αZ(ddZ+ abZ)F2 == m(a2Z2 − ω)F2 + α(b2Z2 − ω)F1,whence it followsZddZG1 − Z2ab(nF1 + βF2) == −Z2(na2F2 + βb2F1) + ω(nF2 + βF1),ZddZG2 + Z2ab(mF1 + αF2) == Z2(ma2F2 + αb2F1) − ω(mF2 + αF1). (10)Taking into account (9), we reduce Eqs. (10) to other form[ZddZ− Z2ab(nα + mβ) + Z2(a2mn + b2αβ)−−ω(nm+αβ)]G1 =[−Z2(an−bβ)2+ω(n2+β2]G2,[ZddZ+ Z2ab(nα + mβ) − Z2(a2mn + b2αβ)++ω(nm+αβ)]G2 =[Z2(am−bα)2−ω(m2+α2)]G1.Let us impose additional restrictions:the first one isan − bβ = 0, that is βn=ab,[ZddZ− Z2ab(nα + mβ) + Z2(a2mn + b2αβ)−−ω(nm + αβ)]G1 = ω(n2 + β2)G2,[ZddZ+ Z2ab(nα + mβ) − Z2(a2mn + b2αβ)++ω(nm+αβ)]G2 =[Z2(am−bα)2−ω(m2+α2)]G1;(11)the second one isam − bα = 0, that is αm=ab,[ZddZ− Z2ab(nα + mβ) + Z2(a2mn + b2αβ)−−ω(nm+αβ)]G1 =[−Z2(an−bβ)2+ω(n2+β2)]G2,[ZddZ+ Z2ab(nα + mβ) − Z2(a2mn + b2αβ)++ω(nm + αβ)]G2 = −ω(m2 + α2)G1.These two possibilities are equivalent to each other, fordefiniteness we will use the variant (11). It can be presentedin more symmetrical formF1 = αG1 + βG2 =√ ba2 + b2G1 +√ aa2 + b2G2,F2 = mG1 + nG2 = −√ aa2 + b2G1 +√ ba2 + b2G2;(12)at this Eqs. (6) lead toZddZG1 = ωG2,ZddZG2 =[Z2(a2 + b2) − ω]G1. (13)From (13) we derive a second order equation for G1:(Z2 d2dZ2 + ZddZ+ ω2 − ω(a2 + b2)Z2)G1 = 0.(14)It is convenient to transform this equation into the initial variablez, then it readsez =√ωZ,Известия Коми научного центра Уральского отделения Российской академии наук № 5 (71), 2024Серия «Физико-математические науки»www.izvestia.komisc.ru61(Z2 d2dZ2 + ω2 − (a2 + b2)e2z)G1 = 0. (15)It can be associated with the Schrödinger equation(d2dz2 + m − U(z))φ(z) = 0 (16)with the potential function U(z) = (a2 + b2)e2z, the correspondingeffective force acts on the left Fz = −2(a2 +b2)e2z. The situation described by Eq. (15) can be illustratedin Fig. 1.zU(z)ǫ = ω2Figure 1. Effective potential curve.Рисунок 1. Эффективная потенциальная кривая.Therefore, we should expect the properties of the electromagneticsolutions similar to those existing in the relevantquantum-mechanical problem. Note that when a = k1 =0, b = k2 = 0, this force vanishes. In accordance with(16), an equation below ω2 = U(z) = (a2 + b2)e2z determinesa critical point z0 in which behavior of the functionG1(x) must change dramatically. To such a point z0,x0 = i√a2 + b2ez0 = iω. Expression for the turning pointz0 is given by the formulaz0 = ρ lnωρ√k21 + k22.The last relation is written in the usual units. The ρ is a curvatureradius of Lobachevsky space, it is a free parameter ofthe model description.The primary variable G1(x) determine all remainingones. Let us turn back to Eq. (14). In the variable x =i√ω(a2 + b2)Z = i√a2 + b2ez it takes the Bessel form(d2x2 +1xdx+ 1 +ω2x2)G1 = 0. (17)The first-order system (13), being transformed to the variablex, readsxdxG1 = ωG2, xdxG2 = −ω2 + x2ωG1.The second function is determined by relationG2 =1ωxddxG2 =1ωddzG1.In turn, taking into account the transformation (12), we get(see (6))f3 =e2zω(−ibF1 + iaF2) =√a2 + b2iωe2zG1(z).Let us write down the final expressions for obtained solutionsE(z) + iB(z) = (cos φ + i sin φ)f(z),φ = ax + by − iωt,wheref1(z) = ezF1(z) == ez(√ ba2 + b2G1 +√ aa2 + b2G2),f2(z) = ezF2(z) == ez(−√ aa2 + b2G1 +√ aa2 + b2G2),f3(z) = −i√a2 + b2ωe2zG1(z),where G1(z) is the solution to equation (17),G2(z) =1ωddzG1(z), x = i√a2 + b2ez.ConclusionIn the frames of the quantum mechanics, Lobachevskygeometry acts as an effective potential barrier with reflectioncoefficient R = 1. In electrodynamic context, resultsare similar: this geometry simulates a medium that effectivelyacts as an ideal mirror distributed in space. Penetrationof the electromagnetic field into the effective medium alongthe axis z depends on the parameters of an electromagneticwaves ω, k21 + k22 and the curvature radius ρ of the usedLobachevsky model. The generalized quasi-plane wave solutionsf(t, x, y, z) = E + iB and the relevant system ofequations are transformed into the real form, which permit usto relate geometry characteristics with expressions for effectivetensors of electric and magnetic permittivities.The authors declare no conflict of interest.