Proceedings of the Komi Science Centre of the Ural Division of the Russian Academy of Sciences

Известия Коми научного центра УрО РАН

1994-5655

86221

10.19110/1994-5655-2024-5-38-45

Научные статьи

Science articles

Научные статьи

Stueckelberg particle in the uniform electric field, solutions with cylindrical symmetry

Частица Штюкельберга в электрическом поле, решения с цилиндрической симметрией

Ивашкевич

А. В.

Ivashkevich

A. V.

ivashkevich.alina@yandex.by

Саченок

П. О.

Sachenok

P. O.

Редьков

В. М.

Red'kov

V. M.

v.redkov@ifanbet.bas-net.by

Институт физики имени Б.И. Степанова Национальной академии наук Беларуси Минск Беларусь B.I. Stepanov Institute of Physics of the National Academy of Sciences of Belarus Minsk Belarus

Мозырский государственный педагогический университет имени И. П. Шамякина Мозырь Россия Mozyr State Pedagogical University named after I. P. Shamyakin Mozyr Russian Federation

Институт физики имени Б. И. Степанова Национальной академии наук Беларуси Минск Россия B. I. Stepanov Institute of Physics of the National Academy of Sciences of Belarus Minsk Russian Federation

07 08 2024

5 38 45 08 04 2024

https://komisc.editorum.ru/en/nauka/article/86221/view

В настоящей работе система 11 уравнений для массивной частицы Штюкельберга исследуется в присутствии внешнего однородного электрического поля. Применяет- ся тетрадный формализм, согласно методу Тетрода-Вейля-Фока-Иваненко. Используются цилиндрические координаты и соответствующая диагональная тетрада. Разделив переменные, получили систему дифференциальных уравнений первого порядка в частных производных по координатам (r, z). Для решения этой системы применяется метод Федорова-Гронского, согласно которому на основе 11-мерного оператора спина введены три проективных оператора, позволяющие разложить полную волновую функцию в сумму трех частей. Согласно общему методу, зависимость каждой проективной составляющей от переменной r должна определяться только одной функцией. Также используются дифференциальные ограничения первого порядка, совместимые с системой уравнений и позволяющие преобразовать все уравнения в частных производных по координатам (r, z) в обыкновенные дифференциальные уравнения по переменной z. Последняя система решена в терминах вырожденных гипергеометрических функций. Построены четыре независимые решения, в отличие от случая обычной частицы со спином 1, описываемой уравнением Даффина-Кемера, когда возможны только три решения.

In the present paper, the system of 11 equations for massive Stueckelberg particle is studied in presence of the external uniform electric field. We apply covariant formalism according to the general tetrad approach by Tetrode-Weyl-Fock-Ivanenko specified for cylindrical coordinates. After separating the variables, we derive the system of the first-order differential equations in partial derivatives with respect to coordinates (r, z). To resolve this system, we apply the Fedorov- Gronskiy method, thereby we consider the 11-dimensional spin operator and find on this base three projective operators, which permit us to expand the complete wave function in the sum of three parts. Besides, according to the general method, dependence of each projective constituent on the variable r should be determined by only one function. Also, in accordance with the general method we impose the first-order constraints which permit us to transform all differential equations in partial derivatives with respect to coordinates (r, z) into the system of 11 first-order ordinary differential equations in the variable z. The last system is solved in terms of confluent hypergeometric functions. In total, four independent types of solutions have been constructed, in contrast to the case of the ordinary spin 1 particle described by Daffin- Kemmer equation when only three types of solutions are possible.

частица Штюкельберга тетрадный формализм цилиндрическая симметрия внешнее электрическое поле разделение переменных уравнения в частных производных точные решения

Stueckelberg particle tetrad formalism cylindrical symmetry external electric field separation of the variables differential equations in partial derivatives exact solutions

1. The basic equationThe initial Stueckelberg system [1–5] of equations for amassive particle in presence of external electromagneticfields is−DaΨa − μΨ = 0,DaΨ + DbΨab − μΨa = 0,DaΨb − DbΨa − μΨab = 0,where Da = ∂a + ieAa. As the wave function, we will usethe 11-dimensional columnΦ(x) = (Ψ;Ψ0,Ψ1,Ψ2,Ψ3;Ψ01,Ψ02,Ψ03,Ψ23,Ψ31,Ψ12)t = (H,H1,H2)t,where t denotes transposition. The above system can be presentedin the block formDaGaH1 + μH = 0,ΔaDaH + KaDaH2 − μH1 = 0, (1)DaLaH1 − μH2 = 0,or differently(−DaΓa − μ)Φ = 0,Γa =0 −Ga 0Δa 0 Ka0 La 0, Φ =HH1H2. (2)All blocks were defined in [3–5]. This matrix equation forStueckelberg particle can be extended to the Riemannianspace-time in accordance with the known tetrad procedure[Γα(x)(∂∂xα +Σα(x) − ieAα)− μ]Ψ(x) = 0.(3)Local matrices Γα(x) are determined through the tetradsΓα(x) = eα(a)(x)Γa ==0 −Gaeα(a) 0Δaeα(a) 0 Kaeα(a)0 Laeα(a) 0. (4)The connection Σα(x) is defined by the formulasjab =0 0 00 jab1 00 0 jab2,Σα(x) =12jabeβ(a)(x)e(b)β;α(x), (5)Σ1(x) =12jab(1)eβa(x)e(b)β;α(x),Σ2(x) =12jab(2)eβa(x)e(b)β;α(x),where jab(1) and jab(2) designate generators for vector Ψk(x)and antisymmetric tensor Ψ[mn](x), respectively. Equation(3) may be presented with the use of the Ricci rotationcoefficients[Γc(eα(c)∂∂xα +12jabγabc − ieAc)− μ]Ψ(x) = 0.(6)In more detailed form, Eq. (6) reads−Gc(eα(c)∂α + jab(1)12γabc − ieAc)H1 − μH = 0,Δc (eα(c)∂α − ieAc)H++Kc(eα(c)∂α + jab(2)12γabc − ieAc)H2 − μH1 = 0,(7)Lc(eα(c)∂α + jab(1)12γabc − ieAc)H1 − μH2 = 0.Let us consider the Stueckelberg equation in presence of theuniform electric field. In cylindrical coordinates and correspondingdiagonal tetradxα = (t, r, ϕ, z),dS2 = dt2 − dr2 − r2dϕ2 − dz2, A0 = −Ez,the above equation takes the form (let eE ⇒ E):[Γ0(∂∂t+ iEz)+ Γ1 ∂∂r++Γ2 ∂ϕ + j12r+ Γ3 ∂∂z− μ]Ψ = 0. (8)In block form, it reads[−G0(∂∂t+ iEz)− G1 ∂∂r−−G2 1r(∂∂ϕ+ j121)− G3 ∂∂z]H1 − μH = 0,[Δ0(∂∂t+ iEz)+ Δ1 ∂∂r+ Δ2 1r∂ϕ + Δ3 ∂∂z]H++[K0(∂∂t+ iEz)+ K1 ∂∂r+ (9)+K2 ∂ϕ + j122r+ K3 ∂∂z]H2 = μH1,[L0 ∂∂t+ L1 ∂∂r+ L2 ∂ϕ + j121r+ L3 ∂∂z]H1 = μH2.In the following, it will be convenient to apply the cyclic basis,in which the third projection of the spin is diagonal (seedetails in [3–5]).Известия Коми научного центра Уральского отделения Российской академии наук № 5 (71), 2024Серия «Физико-математические науки»www.izvestia.komisc.ru392. Separation of the variablesWe apply the following substitution for the wave functionin cyclic basis¯Ψ= e−iϵteimϕ¯H(r, z)¯H1(r, z)¯H2(r, z), ¯H = h(r, z),¯H1 =h0(r, z)h1(r, z)h2(r, z)h3(r, z), ¯H2 =(Ei(r, z)Bi(r, z)). (10)Then Eqs. (9) read[+i(ϵ − Ez)G0 − G1 ddr−−G2 1r(im + j121)− ddzG3]H1 = μH,[−i(ϵ − Ez)mΔ0 + Δ1 ddr+imrΔ2 +ddzΔ3]H++[−i(ϵ − Ez) ¯K 0 + K1 ddr++K2 im + j122r+ddzK3]H2 = μH1,[−i(ϵ − Ez)L0 + L1 ddr++L2 im + j121r+ddzL3]H1 = μH2.After simple calculation, we obtain the system of 11 equations.With the use of the shortening notationsam =ddr+mr, am+1 =ddr+m + 1r,bm =ddr− mr, bm−1 =ddr− m − 1r,(11)it readsi(ϵ − Ez)h0 +ddzh2 − bm−1h1 + am+1h3 = μh,−i(ϵ − Ez)h − ddzE2 + bm−1E1 − am+1E3 = μh0,−amh + am+1B2 − ddzB3 + i(ϵ − Ez)E1 = μh1,ddzh + i(ϵ − Ez)E2 − am+1B1 − bm−1B3 = μh2,bmh + bmB2 +ddzB1 + i(ϵ − Ez)E3 = μh3,amh0 − i(ϵ − Ez)h1 = μE1,− ddzh0 − i(ϵ − Ez)h2 = μE2,−bmh0 − i(ϵ − Ez)h3 = μE3,−bmh2 +ddzh3 = μB1,bm−1h1 + am+1h3 = μB2, − ddzh1 − amh2 = μB3.3. The Fedorov-Gronskiy methodTo resolve the last system, we will implement theFedorov-Gronskiy method [6]. To this end, let us consider the11-dimensional spin operator Y = −i ¯ J12. We readily verifythat it satisfies the minimal equation Y (Y −1)(Y +1) = 0.This permits us to introduce three projective operatorsP1 =12Y (Y − 1), P2 =12Y (Y + 1),P3 = 1 − Y 2, P0 + P+1 + P−1 = 1.(12)Therefore, the complete wave function may be decomposedinto the sum of three partsΨ = Ψ0 + Ψ+1 + Ψ−1,Ψσ = PσΨ, σ = 0, +1,−1.(13)We can readily find an explicit formula of them. Besides, accordingto the Fedorov-Gronskiy method, dependence of eachprojective constituent on the variable r should be determinedby only one functionΨ1(r, z) =(0, 0, h1(z), 0, 0,E1(z),0, 0, 0, 0,B3(z))tf1(r),Ψ2(r, z) =(0, 0, 0, 0, h3(z), 0, 0,E3(z),B1(z), 0, 0)tf2(r),Ψ3(r, z) =(h1(z), h0(z), 0, h2(z),0, 0,E2(z), 0, 0,B2(z), 0)tf3(r). (14)Acting by projective operators on the above system of 11equations Pi(A11×11Ψ) = 0, we get three subsystems. Besides,in accordance with the general method, we should imposethe first-order constraints which permit us to transformall differential equations in partial derivatives with respectto coordinates (r, z) into the system of ordinary differentialequations of the variable zP1−amf3(r)h(z) + amf3(r)B2(z) − f1(r)ddzB3(z)++i(ϵ − Ez)f1(r)E1(z) = μf1(r)h1(z) ⇒amf3(r) = C1f1(r),amf3(r)h0(z) − i(ϵ − Ez)f1(r)h1(z) == μf1(r)E1(z) ⇒ amf3(r) = C1f1(r),−f1(r)ddzh1(z) − amf3(r)h2(z) == μf1(r)B3(z) ⇒ amf3(r) = C1f1(r);P240Известия Коми научного центра Уральского отделения Российской академии наук № 5 (71), 2024Серия «Физико-математические науки»www.izvestia.komisc.rubmf3(r)h(z) + bmf3(r)B2(z) + f2(r)ddzB1(z)++i(ϵ − Ez)f2(r)E3(z) = μf2(r)h3(z) ⇒bmf3(r) = C2f2(r),−bmf3(r)h0(z) − i(ϵ − Ez)f2(r)h3(z) == μf2(r)E3(z) ⇒ bmf3(r) = C2f2(r),−bmf3(r)h2(z) + f2(r)ddzh3(z) == μf2(r)B1(z) ⇒ bmf3(r) = C2f2(r);P3−i(ϵ − Ez)f3(r)h0(z) − f3(r)ddzh2(z)++bm−1f1(r)h1(z) − bm−1f1(r)h3(z) == μf3(r)h(z) ⇒ bm−1f1(r) = C3f3(r),−i(ϵ − Ez)f3(r)h(z) − f3(r)ddzE2(z)+bm−1f1(r)E1(z) − am+1f2(r)E3(z) = μf3(r)h0(z)⇒ bm−1f1(r) = C3f3(r), am+1f2(r) = C4f3(r),f3(r)ddzh(z) + i(ϵ − Ez)f3(r)E2(z)−−am+1f2(r)B1(z) − bm−1f1(r)B3(z) = μf3(r)h2(z)⇒ bm−1f1(r) = C3f3(r), am+1f2(r) = C4f3(r),−f3(r)ddzh0(z) − iϵf3(r)h2(z) = μf3(r)E2(z),bm−1f1(r)h1(z) + am+1f2(r)h3(z) = μf3(r)B2(z)⇒ bm−1f1(r) = C3f3(r), am+1f2(r) = C4f3(r).Thus, we get the following system−C1h + C1B2 − ddzB3 + i(ϵ − Ez)E1 = μh1,C1h0 −i(ϵ−Ez)h1 = μE1, − ddzh1 −C1h2 = μB3,C2h + C2B2 +ddzB1 + i(ϵ − Ez)E3 = μh3,−C2h0 − i(ϵ − Ez)h3 = μE3,−C2h2 +ddzh3 = μB1,−i(ϵ − Ez)h0 − ddzh2 + C3h1 − C3h3 = μh,−i(ϵ − Ez)h − ddzE2 + C3E1 − C4E3 = μh0,ddzh + i(ϵ − Ez)E2 − C4B1 − C3B3 = μh2,− ddzh0 −i(ϵ−Ez)h2 = μE2, C3h1 +C4h3 = μB2,and the constraintsbm−1f1(r) = C3f3(r), amf3(r) = C1f1(r),am+1f2(r) = C4f3(r), bmf3(r) = C2f2(r).(15)Eqs. (15) transform into equations for separate functionsbm−1amf3(r) = C1C3f3(r),ambm−1f1(r) = C1C3f1(r),am+1bmf3(r) = C2C4f3(r),(16)bmam+1f2(r) = C2C4f2(r).Evidently, within each pair we can assumeC3 = C1,C4 = C2.Therefore, the above differential conditions and the secondorderequations take on the formbm−1f1(r) = C1f3(r), amf3(r) = C1f1(r),am+1f2(r) = C2f3(r), bmf3(r) = C2f2(r);(17)[bm−1am − C21 ]f3(r) = 0,[ambm−1 − C21 ]f1(r) = 0,f3(r) = 0, [bmam+1 − C22 ]f2(r) = 0. (18)Explicitly, Eqs. (18) are red as(d2dr2 +1rddr− m2r2− C21)f3(r) = 0,(d2dr2 +1rddr− (m − 1)2r2− C21)f1(r) = 0,(d2dr2 +1rddr− m2r2− C22)f3(r) = 0,(d2dr2 +1rddr− (m + 1)2r2− C22)f2(r) = 0.So we get the following constraint C23 = C22 = C21 = C2,and only three different equations1(d2dr2 +1rddr− (m − 1)2r2− C2)f1(r) = 0,2(d2dr2 +1rddr− (m + 1)2r2− C2)f2(r) = 0, (19)3(d2dr2 +1rddr− m2r2− C2)f3(r) = 0.They are solved in Bessel functions. More details on the parameterC2 are given later. The meaning of parameter C2may be understood if we turn to the Klein-Fock-Gordon equationin cylindrical coordinates in presence of the uniform electricfield[d2dz2 + (ϵ − Ez)2 +d2dr2 +1rddz− m2r2− μ2]××eiϵteimϕR(r)Z(z) = 0.The variables are separated as follows[d2dz2 + (ϵ − Ez)2 − μ2 + λ]Z(z) = 0,Известия Коми научного центра Уральского отделения Российской академии наук № 5 (71), 2024Серия «Физико-математические науки»www.izvestia.komisc.ru41[d2dz2 +1rddz− m2r2− λ]R(r) = 0,so C2 = λ is the separation constant associated with thecylindrical coordinate system (see (19)).4. Solving the equations in the variable zBelow we will take into account the identitiesC1 = C2 =C3 = C. We should solve the system of equations in the variablez:−Ch + CB2 − ddzB3 + i(ϵ − Ez)E1 = μh1,Ch0 − i(ϵ − Ez)h1 = μE1, − ddzh1 − Ch2 = μB3,Ch + CB2 +ddzB1 + i(ϵ − Ez)E3 = μh3,−Ch0 − i(ϵ − Ez)h3 = μE3, −Ch2 +ddzh3 = μB1,−i(ϵ − Ez)h0 − ddzh2 + Ch1 − Ch3 = μh, (20)−i(ϵ − Ez)h − ddzE2 + CE1 − CE3 = μh0,ddzh + i(ϵ − Ez)E2 − CB1 − CB3 = μh2,− ddzh0 − i(ϵ − Ez)h2 = μE2, Ch1 + Ch3 = μB2.First, we resolve the subsystem of 6 equations−i(ϵ − Ez)h − ddzE2 + CE1 − CE3 = μh0,ddzh + i(ϵ − Ez)E2 − CB1 − CB3 = μh2,Ch0 − i(ϵ − Ez)h1 = μE1,−Ch0 − i(ϵ − Ez)h3 = μE3, (21)−Ch2 +ddzh3 = μB1, − ddzh1 − Ch2 = μB3;as algebraic one with respect to the variablesh0, h2,E1,E3,B1,B3. This results in (let dz =ddz)h0 =dzE2μ − i(Ch1 − Ch3 + hμ)(Ez − ϵ)2C2 − μ2 ,h2 =−dz(Ch1 − Ch3 + hμ) + iE2μ(Ez − ϵ)2C2 − μ2 ,E1 =12C2μ − μ3(CdzE2μ + i(h1(C − μ)(C + μ)++C(Ch3 − hμ))(Ez − ϵ)), (22)E3 =1μ3 − 2C2μ(CdzE2μ − i(C2h1 + Chμ++h3(C − μ)(C + μ))(Ez − ϵ)),B1 =1μ3 − 2C2μ(−dz(C2h1 + Chμ++h3(C − μ)(C + μ))+ iCE2μ(Ez − ϵ)),B3 =1μ3 − 2C2μ(dz (h1(C − μ)(C + μ)++C(Ch3 − hμ)) + iCE2μ(Ez − ϵ)).Now substitute these expressions into remaining 5 equations−i(ϵ − Ez)h0 − ddzh2 + Ch1 − Ch3 = μh,− ddzh0 − i(ϵ − Ez)h2 = μE2,Ch1 + Ch3 = μB2, (23)−Ch + CB2 − ddzB3 + i(ϵ − Ez)E1 = μh1,Ch + CB2 +ddzB1 + i(ϵ − Ez)E3 = μh3.As a result, we obtain1d2zhμ2C2 − μ2 +Cd2zh12C2 − μ2− Cd2zh32C2 − μ2++μ((ϵ − Ez)22C2 − μ2− 1)h +(C(ϵ − Ez)22C2 − μ2 + C)h1++Ch3((ϵ − Ez)2μ2 − 2C2− 1)= 0;2C2d2zh32C2μ − μ2 +d2zh1(μ2 − C2)μ3 − 2C2μ++Cd2zhμ2 − 2C2 + B2C − C2h3(ϵ − Ez)2μ3 − 2C2μ++(−(μ2 − C2)(ϵ − Ez)22C2μ − μ3− μ)h1++C((ϵ − Ez)2μ2 − 2C2− 1)h = 0;3C2d2zh12C2μ − μ3 +d2zh3(μ2 − C2)μ3 − 2C2μ++Cd2zh2C2 − μ2 + B2C +C2h1(ϵ − Ez)22C2μ − μ3 ++h3(−(μ2 − C2)(ϵ − Ez)22C2μ − μ3− μ)++(C(ϵ − Ez)22C2 − μ2 + C)h = 0;4[d2dz2 + (ϵ − Ez)2 − μ2 − 2C2]E2 = 0;5 − μB2 + Ch1 + Ch3 = 0.42Известия Коми научного центра Уральского отделения Российской академии наук № 5 (71), 2024Серия «Физико-математические науки»www.izvestia.komisc.ruWith the use of equation 5, from equations 2 and 3 we caneliminate the variable B2. In this way we obtain the systemof 3 equations for variables h, h1, h31 μ2(d2z− 2C2 + μ2 + (ϵ − Ez)2)h++Cμ(d2z+ 2C2 − μ2 + (ϵ − Ez)2)h1−−Cμ(d2z+ 2C2 − μ2 + (ϵ − Ez)2)h3 = 0, (24)2 Cμ(d2z+ 2C2 − μ2 + (ϵ − Ez)2)h++(μ2 − C2)(d2z+ 2C2 − μ2 + (ϵ − Ez)2)h1−−C2(d2z+ 2C2 − μ2 + (ϵ − Ez)2)h3 = 0, (25)3 Cμ(d2z+ 2C2 − μ2 + (ϵ − Ez)2)h++C2(d2z+ 2C2 − μ2 + (ϵ − Ez)2)h1−−(μ2 −C2)(d2z+2C2 −μ2 +(ϵ−Ez)2)h3 = 0. (26)The structure of these equations may be presented shortly asfollows1 A1h′′ + B1h + C1h′′1++D1h1 +M1h′′3 + N1h3 = 0,2 A2h′′ + B2h + C2h′′1++D2h1 +M2h′′3 + N2h3 = 0,3 A3h′′ + B3h + C3h′′1+D3h1 +M3h′′3 + N3h3 = 0. (27)We will combine these equations in three different ways.The first variant is(aA1 + bA2 + cA3)−1h′′ + (aB1 + bB2 + cB3)h++(aC1 + bC2 + cC3)−0h′′1++(aD1 + bD2 + cD3)h1++(aM1 + bM2 + cM3)−0h′′3++(aN1 + bN2 + cN3)h3 = 0.This results inh′′ + (aB1 + bB2 + cB3)h + (aD1 + bD2 + cD3)h1++(aN1 + bN2 + cN3)h3 = 0,where a, b, c obey the linear systemaA1 + bA2 + cA3 = 1,aC1 + bC2 + cC3 = 0, (28)aM1 + bM2 + cM3 = 0;its solution isa =1μ2 − 2C2 , b =C2C2μ − μ3 , c =C2C2μ − μ3 .The second variant is(aA1 + bA2 + cA3)−0h′′ + (aB1 + bB2 + cB3)h++(aC1 + bC2 + cC3)−1h′′1++(aD1 + bD2 + cD3)h1++(aM1 + bM2 + cM3)−0h′′3++(aN1 + bN2 + cN3)h3 = 0.This results inh′′1 + (aB1 + bB2 + cB3)h + (aD1 + bD2 + cD3)h1++(aN1 + bN2 + cN3)h3 = 0,where a, b, c obey the linear systemaA1 + bA2 + cA3 = 0,aC1 + bC2 + cC3 = 1, (29)aM1 + bM2 + cM3 = 0;its solution isa =C2C2μ − μ3 , b =1μ2 − 2C2 , c = 0.The third variant is(aA1 + bA2 + cA3)−0h′′ + (aB1 + bB2 + cB3)h++(aC1 + bC2 + cC3)−0h′′1++(aD1 + bD2 + cD3)h1++(aM1 + bM2 + cM3)−1h′′3++(aN1 + bN2 + cN3)h3 = 0.This results inh′′3 + (aB1 + bB2 + cB3)h + (aD1 + bD2 + cD3)h1++(aN1 + bN2 + cN3)h3 = 0,where a, b, c obey the linear systemaA1 + bA2 + cA3 = 0,aC1 + bC2 + cC3 = 0, (30)aM1 + bM2 + cM3 = 1;its solution isa =Cμ3 − 2C2μ, b = 0, c =12C2 − μ2 .So after this transformation we get three second-order separateequationsd2dz2 h + (2C2 + μ2 + (ϵ − Ez)2)h = 0,d2dz2 h1 + (2C2 − μ2 + (ϵ − Ez)2)h1 − 2Chμ = 0,d2dz2 h3 +(2C2 −μ2 +(ϵ−Ez)2)h3 +2Chμ = 0. (31)Let us introduce new variablesH = h1 + h3, G = h1 − h3. (32)Then instead of (31) we can obtain one separate equation[d2dz2 + 2C2 − μ2 + (ϵ − Ez)2]H = 0 (33)Известия Коми научного центра Уральского отделения Российской академии наук № 5 (71), 2024Серия «Физико-математические науки»www.izvestia.komisc.ru43and one subsystem[d2dz2 + 2C2 + μ+ + (ϵ − Ez)2]h = 0,[d2dz2 + 2C2 + μ2 + (ϵ − Ez)2]G−−2μ2G − 4μCh = 0.The last subsystem can be presented in the matrix formD(hG)= 2μ(0 02C μ)(hG),DΨ = 2μAΨ. (34)Let us find transformation which diagonalizes the mixing matrixA¯Ψ= SΨ, D¯Ψ = 2μ(SAS−1)¯Ψ, ¯Ψ =(¯h¯G).For transformation matrix S we derive the following equationsSA = A¯S, A¯ =(λ1 00 λ2),(s11 s12s21 s22)(0 02C μ)=(λ1 00 λ2)(s11 s12s21 s22),whence it followss122C = λ1s11, s12μ = λ1s12,(λ1 −2C0 (λ1 − μ))(s11s12)= 0,s222C = λ2s21, s22μ = λ2s22,(λ2 −2C0 (λ2 − μ))(s21s22)= 0.The first row is specified by relations λ1 = 0, s12 =0, s11 = 1; the second row is specified as λ2 = μ, s22 =1, s21 = 2C/μ. Thus, the needed transformation matrix SisS =(1 02Cμ 1), S−1 =(1 0−2Cμ 1). (35)Therefore, we derive three separate equations:[d2dz2 + 2C2 + μ2 + (ϵ − Ez)2]¯h= 0, (36)[d2dz2 + 2C2 − μ2 + (ϵ − Ez)2]¯G= 0, (37)[d2dz2 + 2C2 − μ2 + (ϵ − Ez)2]¯H= 0, (38)where¯h= h, ¯H = H = h1 + h3,G = h1 − h3, ¯G =2Cμh + h1 − h3.Besides we should remember the existence of the fourth independentequation for the variable E2:[d2dz2− 2C2 − μ2 + (ϵ − Ez)2]E2 = 0. (39)So, in total, four independent types of solutions exist forStueckelberg particle in the external uniform electric field,in contrast to the ordinary spin 1 particle described by theDaffin-Kemmer equation when only three independent solutionsare possible. All four equations (36)–(39) have the samemathematical structure. In the papers [7, 8], solutions forequation of the form (39) were constructed in terms of theconfluent hypergeometric functions.The authors declare no conflict of interest.

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