# Second-Order Integrals for Systems inE2 Involving Spin

Authors: İsmet Yurduşen
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

In two-dimensional Euclidean plane, existence of second-order integrals of motion is investigated for integrable Hamiltonian systems involving spin (e.g., those systems describing interaction between two particles with spin 0 and spin 1/2) and it has been shown that no nontrivial second-order integrals of motion exist for such systems.


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## Body

## 1. Introduction

In classical mechanics, existence ofn functionally independent integrals of motion defines integrability (in Liouville sense) of a Hamiltonian system with n degrees of freedom. These integrals, including the Hamiltonian itself, must be well-defined functions on phase space and be in involution. This concept of integrability is extended to define superintegrability by requiring the existence of at least one and at most n-1 (in order to have dynamics in the system) additional integrals of motion. The total set of integrals of motion must be functionally independent; however, the additional ones are not necessarily in involution among themselves, nor with the already existing n integrals of motion (except the Hamiltonian itself). All these concepts are also introduced in quantum mechanics through well-defined linear integrals of motion operators which are supposed to be algebraically independent [1–3].In quantum mechanics, superintegrable systems are of physical interest because superintegrability entails exact solvability, meaning that the bound state energy levels can be calculated algebraically and the wave functions, expressed in terms of polynomials in the appropriate variables, possibly multiplied by an overall factor. It has been conjectured [3] that all maximally (having 2n-1 integrals of motion) superintegrable systems are also exactly solvable and this has been supported by many examples [3, 4].Systematic investigation of the superintegrable systems and their properties was initiated by the works of Smorodinsky, Winternitz, and collaborators in 1965 [1, 2]. Most of the earlier work was devoted to the quadratic superintegrability (i.e., with integrals of motion that are second-order polynomials in the momenta) and directly related with the multiseparability in two- and three-dimensional Euclidean spaces [1, 2]. Recently an extended review article has been published describing the current status of the subject [5].Superintegrability properties are also investigated for Hamiltonian systems involving particles with spin [6–11]. Systematic search for superintegrable systems with spin was initiated in [6], where the authors considered two nonrelativistic quantum particles, one with spin 1/2 and the other with spin 0. Physically, the most interesting Hamiltonian for such systems is(1)H=-ħ22mΔ+V0r→+12V1r→,σ→,L→,where {,} denotes an anticommutator and σ1, σ2, and σ3 are the usual Pauli matrices. V0(r→) and V1(r→) are scalar and interaction potentials, respectively. This Hamiltonian given in E3 would describe, for instance, a low energy (nonrelativistic) pion-nucleon interaction.In [6] first-order integrability and superintegrability were studied in E2. Articles [8, 9] were devoted to systematic search of first- and second-order superintegrability in E3.In this paper we will consider Hamiltonian (1) in E2 and investigate the existence of second-order integrals of motion in order to classify further the superintegrable systems with spin in E2. For simplicity, we will set the reduced mass m of the two-particle system equal to m=1 and use units in which the Planck constant is ħ=1. Keeping ħ in the Hamiltonian and integrals of motion does not change any of the conclusions. In particular V0 and V1 do not depend on ħ. In (1) H is a matrix operator acting on a two-component spinor and we will decompose it in terms of the 2×2 identity matrix I and σ3 (the matrix I will be dropped whenever this does not cause confusion). L is the angular momentum operator.In the next section we give the first-order integrable and superintegrable Hamiltonian systems, obtained from the analysis of the commutativity condition[H,X1]=0, where X1 is the general first-order integral of motion in E2. For details see [6]. In Section 3, we search for the existence of second-order integrals of motion in E2 for the two integrable cases obtained in Section 2. Finally, in the last section we give some conclusions.


## 2. First-Order Integrability and Superintegrability inE2

In this section let us briefly review the results obtained in [6]. Considering that the motion is constrained to Euclidean plane (i.e., assuming Ψ(r→)=Ψ(x,y) and setting p3=0, z=0), we have the following Hamiltonian:(2)H=12p12+p22+V0x,y+V1x,yσ3L3+12σ3L3V1x,ywith(3)p1=-i∂x,p2=-i∂y,L3=iy∂x-x∂y,σ3=100-1.The general first-order integral of motion to consider would be(4)X1=12F1,p1+12F2,p2+F3,where(5)Fμ=Fμ0+Fμ1σ3,μ=1,2,3.All six functions Fμν (μ=1,2,3 and ν=0,1) are real functions of x and y. Our aim is to find at least one such integral of motion from the analysis of the commutativity condition [H,X1]=0. This condition provides 12 determining equations for the 6 functions Fμν, as well as the unknown potentials V0(x,y) and V1(x,y). Six of the determining equations, which are obtained from equating the coefficients of the second-order terms to zero, give(6)F1ν=ωνy+aν,F2ν=-ωνx+bν,ν=0,1,where ων, aν, and bν are real constants and the rest of the determining equations are(7)F3ν,x=δν,1-ξ-bξV1-ωξy+aξyV1,x+ωξx-bξyV1,y,F3ν,y=δν,1-ξaξV1+ωξy+aξxV1,x-ωξx-bξxV1,y,ωνy+aνV0,x+-ωνx+bνV0,y=δν,1-ξxF3ξ,y-yF3ξ,xV1,ν,ξ=0,1.The analysis of determining equations (7) is summarized as follows.(1)Superintegrable System. There exists only one first-order superintegrable system with V1≠0:(8)H=-12Δ+12γ2x2+y2+γσ3L3,γ=const.It allows an eight-dimensional Lie algebra L of first-order integrals of motion with a basis given by(9)L±=iy∂x-x∂yI±,X±=i∂x∓γyI±,Y±=i∂y±γxI±,I±=I±σ3.The algebra L is isomorphic to the direct sum of two central extensions of the Euclidean Lie algebra e(2). Consider(10)L~e~+2⊕e~-2,e~±2=L±,X±,Y±,I±.(2)Integrable Systems. The only integrable systems with one integral of motion in addition to H that we found are given as follows.(a) Integrable system with rotationally invariant potentials:(11)V0=V0r,V1=V1r,r=x2+y2,X=ω0+ω1σ3L3,ων=const,ν=0,1.(b) Integrable system with ax-dependent interaction potential:(12)V1=V1x,V0=y22V12x+Fx,X=-i∂y-σ3∫V1xdx.The above results should be understood up to gauge transformations of the form(13)H~=U-1HU,U=eiα00e-iα,α=ακ,κ=yx,which leaves Hamiltonian (2) form invariant. However, the potentials transform accordingly. Consider(14)V~1=V1+α˙x2,V~0=V0+1+y2x212α˙2x2+α˙V1.


## 3. Second-Order Superintegrability inE2

The system obtained in Section2 with a constant interaction potential term is maximally superintegrable and hence all the higher-order integrals of motion can be expressed in terms of the first-order ones, given in (9). However, for integrable systems (11) and (12) it is worth searching for the existence of second-order integrals of motion in order to classify further the superintegrable systems with spin in E2. In this section, we investigate the existence of such second-order integrals of motion.


### 3.1. The PotentialsV0=V0(r) and V1=V1(r)

For these rotationally invariant potentials Hamiltonian (2) becomes(15)H=12p12+p22+V0+V1σ3L3,where V0 and V1 are functions of x2+y2. The general second-order integral of motion to consider would be(16)X~2=12A1,p12+A2,p1p2+12A3,p22+X1,where(17)Aμ=Aμ0+Aμ1σ3,μ=1,2,3,and X1 is the first-order integral of motion given in (4). All six functions Aμν (μ=1,2,3 and ν=0,1) are real functions of x and y. From the commutativity condition [H,X~2]=0, we search for the existence of second-order integrals of motion. The highest-order determining equations (i.e., the determining equations, obtained by equating the coefficients of the third-order terms to zero in the commutativity equation [H,X~2]=0) read(18)A1ν,x=0,A3ν,y=0,A1ν,y+2A2ν,x=0,2A2ν,y+A3ν,x=0,ν=0,1.First four of the above equations imply A1ν is a function of y only and A3ν is a function of x only. Then the last four determining equations in (18) give(19)A2ν,xx=0,A2ν,yy=0,ν=0,1,which can immediately be integrated. Hence the general second-order integral of motion can be expressed as(20)X2=G1L3p1+p1L3+G2L3p2+p2L3+G3p12-p22+2G4p1p2+X1,where Gτ (τ=1,…,4) are now constants and all the determining equations, obtained by equating the coefficients of the third-order terms to zero in the commutativity equation, are trivially satisfied. Since the potentials are rotationally invariant, the term proportional to L32 is absent in (20) (i.e., it commutes with the Hamiltonian given in (15)). Notice that the constants Gτ (τ=1,…,4) are considered as(21)Gτ=Gτ0+Gτ1σ3,τ=1,…,4,Fμ=Fμ0+Fμ1σ3,μ=1,2,3.The determining equations, obtained by equating the coefficients of the second-order terms to zero in the commutativity equation[H,X2]=0, read(22)2σ3G4-G2yV1x2+y2+2yG3x+yG4-G1x-G2yV1′x2+y2+F1,x=0,(23)2σ3G4+G1xV1x2+y2+2x-G3y+xG4+G1x+G2y·V1′x2+y2-F2,y=0,(24)2σ3G2x+G1y-2G3V1x2+y2-2G3x2+y2-2xyG1x+G2y·V1′x2+y2+F1,y+F2,x=0,where ′ denotes the derivative with respect to the argument. From (22)–(24) we obtain compatibility conditions for F1ν and F2ν (ν=0,1), which are in polar form expressed as(25)Γνr3V1,rrr+7r2V1,rr+9rV1,r+Λνr2V1,rrr+3rV1,rr-3V1,r=0,ν=0,1,where(26)Γν=G1νcos⁡θ+G2νsin⁡θ,Λν=G4νcos⁡2θ-G3νsin⁡2θ,ν=0,1.In general, (25) represents an overdetermined system of two different equations for the potentials V1, namely,(27)r3V1,rrr+7r2V1,rr+9rV1,r=0,(28)r2V1,rrr+3rV1,rr-3V1,r=0,simultaneous solutions of which give(29)V1r=-γ12r2+γ2,where γ1 and γ2 are constants. Comparing (29) with (14) we see that we can cancel the constant γ1 by a gauge transformation. Hence, we have a constant spin-orbit interaction potential that is found in Section 2. This system is maximally first-order superintegrable and thus all the higher-order integrals of motion can be expressed in terms of the first-order ones given in (9).WhenΓ0=λΓ1 and Λ0=λΛ1 with λ = constant, an exception occurs and the two equations (25) coincide. Hence, bearing in mind that V1 does not depend on θ, now (25) implies either (27) together with Λν is zero, or (28) together with Γν is zero. Thus we have the following two cases.Case 1.
Λ
ν
=
0 (ν=0,1) and(30)V1r=-12r2γ1+32γ2+3γ2log⁡r+γ3,where γ1, γ2, and γ3 are constants.Case 2.
Γ
ν
=
0 (ν=0,1) and(31)V1r=12r2χ1-12r2χ2+χ3,where χ1, χ2, and χ3 are constants.
Let us investigate these cases in detail.Case 1.
From (22) and (23) together with (24) we obtain the following forms of F1ν and F2ν(ν=0,1), which we present in polar form(32)F1ν=δν,1-ξ14G1ξcos⁡2θ+G2ξsin⁡2θ·2γ1+3γ2+4γ3r2+6γ2log⁡r+6G1ξγ2log⁡1r-G1ξr2γ3+3G2ξγ2θ+γ4νrsin⁡θ+γ5ν,(33)F2ν=δν,1-ξ14G1ξsin⁡2θ-G2ξcos⁡2θ·2γ1+3γ2+4γ3r2+6γ2log⁡r+6G2ξγ2log⁡1r-G2ξr2γ3-3G1ξγ2θ-γ4νrcos⁡θ+γ6ν,where γ4ν, γ5ν, and γ6ν are constants and (ν,ξ)=(0,1). Introducing (32) and (33) into the determining equations, obtained by equating the coefficients of the lower-order terms to zero in the commutativity equation, we obtain compatibility conditions for F3ν (ν=0,1). One has(34)16r32V0,r+rV0,rr+60γ1γ2-4γ12+63γ22-24r2γ2γ3-48r4γ32+12γ23γ2log⁡1r-log⁡r-5log⁡r-2γ1log⁡r+24γ2γ5νcos⁡θ+72G2νγ22θcos⁡θ+24γ2γ6νsin⁡θ-72G1νγ22θsin⁡θG1νcos⁡θ+G2νsin⁡θ=0.
The invariance of potential implies that the term inside the last parenthesis in (34) should be a constant, and from its form we see that it can only be the constant 0, which implies γ2=0. After setting γ2=0, V0 can be obtained from (34). However, setting γ2=0 in (30) and considering (14) we see that we can cancel the constant γ1 by a gauge transformation. Hence, again we have a constant spin-orbit interaction potential.Case 2.
For this case (22) and (23) together with (24) imply the following forms of F1ν and F2ν (ν=0,1):(35)F1ν=δν,1-ξr4χ1+χ2+2r2χ3rG3ξsin⁡θ-G4ξcos⁡θ-2r3χ13G3ξsin⁡3θ-G4ξcos⁡3θ+χ4νrsin⁡θ+χ5ν,F2ν=δν,1-ξr4χ1+χ2+2r2χ3rG3ξcos⁡θ+G4ξsin⁡θ+2r3χ13G3ξcos⁡3θ+G4ξsin⁡3θ-χ4νrcos⁡θ+χ6ν,where χ4ν, χ5ν, and χ6ν are constants and (ν,ξ)=(0,1). Introducing (35) into the determining equations, obtained by equating the coefficients of the lower-order terms to zero in the commutativity equation, we obtain compatibility conditions for F3ν (ν=0,1). One has(36)Λν17r8χ12+3χ22+36r6χ1χ3+3r3V0,r-rV0,rr-6r5χ1χ5νcos⁡θ-χ6νsin⁡θ=0.From this compatibility condition (36) we conclude that we must have either χ1=0 or χ5ν=0 and χ6ν=0.
Ifχ1=0, then, considering (14), we can annihilate the constant χ2 in (31) by a gauge transformation and hence again we have a constant spin-orbit interaction potential.
Ifχ5ν=0 and χ6ν=0, then (36) implies(37)V0r=17r6χ1272+χ228r2+32r4χ1χ2+12r2ϵ1,where ϵ1 is a constant. Upon introduction of this V0 back into the determining equations coming from first- and zeroth-order terms, this forces us to set χ1=0 in which case we are back in the previous case with a constant spin-orbit interaction potential.


### 3.2. The PotentialsV0=y2/2V12(x)+F(x) and V1=V1(x)

For these potentials, Hamiltonian (2) becomes(38)H=12p12+p22+y22V12x+Fx+V1xσ3L3+12σ3L3V1x,and the general second-order integral of motion to consider would be the one given in (16). However, by making a similar analysis given in Section 3.1, we see that the determining equations, obtained by equating the coefficients of the third-order terms to zero in the commutativity equation [H,X~2]=0, force us to write the general form of the second-order integral of motion as(39)X2=G1L3p1+p1L3+G2L3p2+p2L3+G3p12-p22+2G4p1p2+G5L32+X1,where Gτ (τ=1,…,5) are constants and X1 is the first-order integral of motion given in (4). These are considered as(40)Gτ=Gτ0+Gτ1σ3,τ=1,…,5,Fμ=Fμ0+Fμ1σ3,μ=1,2,3.Notice that a term proportional to L32 is present in (39), since it does not commute with the Hamiltonian given in (38).The determining equations, obtained by equating the coefficients of the second-order terms to zero in the commutativity equation[H,X2]=0, read (41)2σ3G4-G2yV1+yG3-2G1y+G5y2V1,x+F1,x=0,(42)2σ3G4+G1xV1+xG4+G1x-G2+G5xy·V1,x-F2,y=0,(43)2σ32G3-G2x-G1yV1+G3x-G4+3G1xy+G2+2G5xy2V1,x-F1,y-F2,x=0.From (41)–(43), we obtain compatibility conditions for F1ν and F2ν(ν=0,1),(44)12G1-G5yV1,x+4G4+2G1x-G2+2G5xyV1,xx+xG4+G1x-G2+G5xyV1,xxx=0.Since V1 is a function of x only, the coefficients of y in (44) must vanish separately (i.e., either G1=0 and G4=0 or G2=0 and G5=0). If G1=0 and G4=0, then (44) implies(45)12G5νV1,x+4G2ν+2G5νxV1,xx+xG2ν+G5νxV1,xxx=0,ν=0,1.One of these equations, say the one with ν=0, can be solved for V1 and upon introducing this solution into the other equation we obtain the following constraint on the constants:(46)G21G20=G51G50=λ1,where λ1 is a constant. On the other hand if G2=0 and G5=0, then (44) implies(47)12G1νV1,x+4G4ν+2G1νxV1,xx+xG4ν+G1νxV1,xxx=0,ν=0,1.Similarly, solving (47) for V1 gives the following constraint on the constants:(48)G11G10=G41G40=λ2,where λ2 is a constant.In both cases, the compatibility conditions forF1ν and F2ν(ν=0,1) imply the following generic form of the interaction potential:(49)V1x=-3x2α1/Gμ02+2x+λaα26x2x+λa2+α3,a=1,2,where α1, α2, α3, and λa are constants and μ is either 1 or 4, or 2 or 5.The only exception to this generic potential occurs if we haveGμ0=0, in which case the interaction potential V1 becomes(50)V1x=ζ16x2+xζ2+ζ3,where ζ1, ζ2, and ζ3 are constants.In all three cases we proceed in a similar fashion as we did for rotationally invariant potentials. More specifically, for two types of potentials (49) and (50), we find F1ν and F2ν (ν=0,1) from (41) and (42) together with (43). Then introducing these forms of F1ν and F2ν (ν=0,1) into the determining equations, obtained by equating the coefficients of the lower-order terms to zero in the commutativity equation, we obtain compatibility conditions for F3ν (ν=0,1). In order to satisfy these compatibility conditions, either ζ2 must vanish in (50) or for generic potential (49) G5 must also vanish for the case G1=G4=0 and G4 must vanish for the case G2=G5=0. Unfortunately, these equations are rather long to present here.Ifζ2=0 in (50), then, by means of gauge transformation (14), we can annihilate the constant ζ1=0 in (50) and have a constant spin-orbit interaction potential. If G5=0 in addition to G1=G4=0 or G4=0 in addition to G2=G5=0, then we repeat the analysis from the beginning and find the following results.Case 1.
G
1
=
0, G4=0, and G5=0:(51)V1x=α16x2+α3,Fx=α1272x2+x2α322,V0x,y=172α12x2+36x2α32+y2α1+6x2α32x4,where α1, α2, and α3 are constants.Case 2.
G
2
=
0, G5=0, and G4=0:(52)V1x=-α22x2+α3,Fx=α228x2+x2α322,V0x,y=18α22x2+4x2α32+4y2α3-α22x22,where again α1, α2, and α3 are constants.
Notice that the potentialsV1(x) and V0(x,y) given in (51) and (52) are exactly the same (α2→-1/3α1).We conclude that for these two cases once again we have a constant spin-orbit interaction potential (up to gauge transformation (14)) and hence all the second-order integrals of motion that are obtained from the analysis can be expressed in terms of the first-order ones given in (9).


## 3.1. The PotentialsV0=V0(r) and V1=V1(r)

For these rotationally invariant potentials Hamiltonian (2) becomes(15)H=12p12+p22+V0+V1σ3L3,where V0 and V1 are functions of x2+y2. The general second-order integral of motion to consider would be(16)X~2=12A1,p12+A2,p1p2+12A3,p22+X1,where(17)Aμ=Aμ0+Aμ1σ3,μ=1,2,3,and X1 is the first-order integral of motion given in (4). All six functions Aμν (μ=1,2,3 and ν=0,1) are real functions of x and y. From the commutativity condition [H,X~2]=0, we search for the existence of second-order integrals of motion. The highest-order determining equations (i.e., the determining equations, obtained by equating the coefficients of the third-order terms to zero in the commutativity equation [H,X~2]=0) read(18)A1ν,x=0,A3ν,y=0,A1ν,y+2A2ν,x=0,2A2ν,y+A3ν,x=0,ν=0,1.First four of the above equations imply A1ν is a function of y only and A3ν is a function of x only. Then the last four determining equations in (18) give(19)A2ν,xx=0,A2ν,yy=0,ν=0,1,which can immediately be integrated. Hence the general second-order integral of motion can be expressed as(20)X2=G1L3p1+p1L3+G2L3p2+p2L3+G3p12-p22+2G4p1p2+X1,where Gτ (τ=1,…,4) are now constants and all the determining equations, obtained by equating the coefficients of the third-order terms to zero in the commutativity equation, are trivially satisfied. Since the potentials are rotationally invariant, the term proportional to L32 is absent in (20) (i.e., it commutes with the Hamiltonian given in (15)). Notice that the constants Gτ (τ=1,…,4) are considered as(21)Gτ=Gτ0+Gτ1σ3,τ=1,…,4,Fμ=Fμ0+Fμ1σ3,μ=1,2,3.The determining equations, obtained by equating the coefficients of the second-order terms to zero in the commutativity equation[H,X2]=0, read(22)2σ3G4-G2yV1x2+y2+2yG3x+yG4-G1x-G2yV1′x2+y2+F1,x=0,(23)2σ3G4+G1xV1x2+y2+2x-G3y+xG4+G1x+G2y·V1′x2+y2-F2,y=0,(24)2σ3G2x+G1y-2G3V1x2+y2-2G3x2+y2-2xyG1x+G2y·V1′x2+y2+F1,y+F2,x=0,where ′ denotes the derivative with respect to the argument. From (22)–(24) we obtain compatibility conditions for F1ν and F2ν (ν=0,1), which are in polar form expressed as(25)Γνr3V1,rrr+7r2V1,rr+9rV1,r+Λνr2V1,rrr+3rV1,rr-3V1,r=0,ν=0,1,where(26)Γν=G1νcos⁡θ+G2νsin⁡θ,Λν=G4νcos⁡2θ-G3νsin⁡2θ,ν=0,1.In general, (25) represents an overdetermined system of two different equations for the potentials V1, namely,(27)r3V1,rrr+7r2V1,rr+9rV1,r=0,(28)r2V1,rrr+3rV1,rr-3V1,r=0,simultaneous solutions of which give(29)V1r=-γ12r2+γ2,where γ1 and γ2 are constants. Comparing (29) with (14) we see that we can cancel the constant γ1 by a gauge transformation. Hence, we have a constant spin-orbit interaction potential that is found in Section 2. This system is maximally first-order superintegrable and thus all the higher-order integrals of motion can be expressed in terms of the first-order ones given in (9).WhenΓ0=λΓ1 and Λ0=λΛ1 with λ = constant, an exception occurs and the two equations (25) coincide. Hence, bearing in mind that V1 does not depend on θ, now (25) implies either (27) together with Λν is zero, or (28) together with Γν is zero. Thus we have the following two cases.Case 1.
Λ
ν
=
0 (ν=0,1) and(30)V1r=-12r2γ1+32γ2+3γ2log⁡r+γ3,where γ1, γ2, and γ3 are constants.Case 2.
Γ
ν
=
0 (ν=0,1) and(31)V1r=12r2χ1-12r2χ2+χ3,where χ1, χ2, and χ3 are constants.
Let us investigate these cases in detail.Case 1.
From (22) and (23) together with (24) we obtain the following forms of F1ν and F2ν(ν=0,1), which we present in polar form(32)F1ν=δν,1-ξ14G1ξcos⁡2θ+G2ξsin⁡2θ·2γ1+3γ2+4γ3r2+6γ2log⁡r+6G1ξγ2log⁡1r-G1ξr2γ3+3G2ξγ2θ+γ4νrsin⁡θ+γ5ν,(33)F2ν=δν,1-ξ14G1ξsin⁡2θ-G2ξcos⁡2θ·2γ1+3γ2+4γ3r2+6γ2log⁡r+6G2ξγ2log⁡1r-G2ξr2γ3-3G1ξγ2θ-γ4νrcos⁡θ+γ6ν,where γ4ν, γ5ν, and γ6ν are constants and (ν,ξ)=(0,1). Introducing (32) and (33) into the determining equations, obtained by equating the coefficients of the lower-order terms to zero in the commutativity equation, we obtain compatibility conditions for F3ν (ν=0,1). One has(34)16r32V0,r+rV0,rr+60γ1γ2-4γ12+63γ22-24r2γ2γ3-48r4γ32+12γ23γ2log⁡1r-log⁡r-5log⁡r-2γ1log⁡r+24γ2γ5νcos⁡θ+72G2νγ22θcos⁡θ+24γ2γ6νsin⁡θ-72G1νγ22θsin⁡θG1νcos⁡θ+G2νsin⁡θ=0.
The invariance of potential implies that the term inside the last parenthesis in (34) should be a constant, and from its form we see that it can only be the constant 0, which implies γ2=0. After setting γ2=0, V0 can be obtained from (34). However, setting γ2=0 in (30) and considering (14) we see that we can cancel the constant γ1 by a gauge transformation. Hence, again we have a constant spin-orbit interaction potential.Case 2.
For this case (22) and (23) together with (24) imply the following forms of F1ν and F2ν (ν=0,1):(35)F1ν=δν,1-ξr4χ1+χ2+2r2χ3rG3ξsin⁡θ-G4ξcos⁡θ-2r3χ13G3ξsin⁡3θ-G4ξcos⁡3θ+χ4νrsin⁡θ+χ5ν,F2ν=δν,1-ξr4χ1+χ2+2r2χ3rG3ξcos⁡θ+G4ξsin⁡θ+2r3χ13G3ξcos⁡3θ+G4ξsin⁡3θ-χ4νrcos⁡θ+χ6ν,where χ4ν, χ5ν, and χ6ν are constants and (ν,ξ)=(0,1). Introducing (35) into the determining equations, obtained by equating the coefficients of the lower-order terms to zero in the commutativity equation, we obtain compatibility conditions for F3ν (ν=0,1). One has(36)Λν17r8χ12+3χ22+36r6χ1χ3+3r3V0,r-rV0,rr-6r5χ1χ5νcos⁡θ-χ6νsin⁡θ=0.From this compatibility condition (36) we conclude that we must have either χ1=0 or χ5ν=0 and χ6ν=0.
Ifχ1=0, then, considering (14), we can annihilate the constant χ2 in (31) by a gauge transformation and hence again we have a constant spin-orbit interaction potential.
Ifχ5ν=0 and χ6ν=0, then (36) implies(37)V0r=17r6χ1272+χ228r2+32r4χ1χ2+12r2ϵ1,where ϵ1 is a constant. Upon introduction of this V0 back into the determining equations coming from first- and zeroth-order terms, this forces us to set χ1=0 in which case we are back in the previous case with a constant spin-orbit interaction potential.


## 3.2. The PotentialsV0=y2/2V12(x)+F(x) and V1=V1(x)

For these potentials, Hamiltonian (2) becomes(38)H=12p12+p22+y22V12x+Fx+V1xσ3L3+12σ3L3V1x,and the general second-order integral of motion to consider would be the one given in (16). However, by making a similar analysis given in Section 3.1, we see that the determining equations, obtained by equating the coefficients of the third-order terms to zero in the commutativity equation [H,X~2]=0, force us to write the general form of the second-order integral of motion as(39)X2=G1L3p1+p1L3+G2L3p2+p2L3+G3p12-p22+2G4p1p2+G5L32+X1,where Gτ (τ=1,…,5) are constants and X1 is the first-order integral of motion given in (4). These are considered as(40)Gτ=Gτ0+Gτ1σ3,τ=1,…,5,Fμ=Fμ0+Fμ1σ3,μ=1,2,3.Notice that a term proportional to L32 is present in (39), since it does not commute with the Hamiltonian given in (38).The determining equations, obtained by equating the coefficients of the second-order terms to zero in the commutativity equation[H,X2]=0, read (41)2σ3G4-G2yV1+yG3-2G1y+G5y2V1,x+F1,x=0,(42)2σ3G4+G1xV1+xG4+G1x-G2+G5xy·V1,x-F2,y=0,(43)2σ32G3-G2x-G1yV1+G3x-G4+3G1xy+G2+2G5xy2V1,x-F1,y-F2,x=0.From (41)–(43), we obtain compatibility conditions for F1ν and F2ν(ν=0,1),(44)12G1-G5yV1,x+4G4+2G1x-G2+2G5xyV1,xx+xG4+G1x-G2+G5xyV1,xxx=0.Since V1 is a function of x only, the coefficients of y in (44) must vanish separately (i.e., either G1=0 and G4=0 or G2=0 and G5=0). If G1=0 and G4=0, then (44) implies(45)12G5νV1,x+4G2ν+2G5νxV1,xx+xG2ν+G5νxV1,xxx=0,ν=0,1.One of these equations, say the one with ν=0, can be solved for V1 and upon introducing this solution into the other equation we obtain the following constraint on the constants:(46)G21G20=G51G50=λ1,where λ1 is a constant. On the other hand if G2=0 and G5=0, then (44) implies(47)12G1νV1,x+4G4ν+2G1νxV1,xx+xG4ν+G1νxV1,xxx=0,ν=0,1.Similarly, solving (47) for V1 gives the following constraint on the constants:(48)G11G10=G41G40=λ2,where λ2 is a constant.In both cases, the compatibility conditions forF1ν and F2ν(ν=0,1) imply the following generic form of the interaction potential:(49)V1x=-3x2α1/Gμ02+2x+λaα26x2x+λa2+α3,a=1,2,where α1, α2, α3, and λa are constants and μ is either 1 or 4, or 2 or 5.The only exception to this generic potential occurs if we haveGμ0=0, in which case the interaction potential V1 becomes(50)V1x=ζ16x2+xζ2+ζ3,where ζ1, ζ2, and ζ3 are constants.In all three cases we proceed in a similar fashion as we did for rotationally invariant potentials. More specifically, for two types of potentials (49) and (50), we find F1ν and F2ν (ν=0,1) from (41) and (42) together with (43). Then introducing these forms of F1ν and F2ν (ν=0,1) into the determining equations, obtained by equating the coefficients of the lower-order terms to zero in the commutativity equation, we obtain compatibility conditions for F3ν (ν=0,1). In order to satisfy these compatibility conditions, either ζ2 must vanish in (50) or for generic potential (49) G5 must also vanish for the case G1=G4=0 and G4 must vanish for the case G2=G5=0. Unfortunately, these equations are rather long to present here.Ifζ2=0 in (50), then, by means of gauge transformation (14), we can annihilate the constant ζ1=0 in (50) and have a constant spin-orbit interaction potential. If G5=0 in addition to G1=G4=0 or G4=0 in addition to G2=G5=0, then we repeat the analysis from the beginning and find the following results.Case 1.
G
1
=
0, G4=0, and G5=0:(51)V1x=α16x2+α3,Fx=α1272x2+x2α322,V0x,y=172α12x2+36x2α32+y2α1+6x2α32x4,where α1, α2, and α3 are constants.Case 2.
G
2
=
0, G5=0, and G4=0:(52)V1x=-α22x2+α3,Fx=α228x2+x2α322,V0x,y=18α22x2+4x2α32+4y2α3-α22x22,where again α1, α2, and α3 are constants.
Notice that the potentialsV1(x) and V0(x,y) given in (51) and (52) are exactly the same (α2→-1/3α1).We conclude that for these two cases once again we have a constant spin-orbit interaction potential (up to gauge transformation (14)) and hence all the second-order integrals of motion that are obtained from the analysis can be expressed in terms of the first-order ones given in (9).


## 4. Conclusions

The main result of this paper can be given as a theorem.Theorem 1.
In Euclidean planeE2, for Hamiltonians of type (2) admitting a first-order integral, any second-order integral can necessarily be expressed as a combination of first-order integrals. Or, in particular, there exist no nontrivial second-order integrals of motion of form (16) for integrable Hamiltonian systems (15) and (38).This result, which is valid for the generic Hamiltonian systems of type (2), proves that no nontrivial generic second-order integrals of motion exist and hence carries one step further the systematic study of the classification of integrable and superintegrable Hamiltonian systems involving spin in Euclidean plane E2.In an earlier article [6] it was shown that in the presence of spin first-order integrable and superintegrable systems exist in E2. The superintegrable Hamiltonian of such systems allows the separation of variables in polar and Cartesian coordinates. Indeed, the Pauli-Schrödinger equation for them can be exactly solved. The integrable Hamiltonians also allow the separation of variables in polar and Cartesian coordinates. However, in order to solve them exactly the interaction potential V1(r) and scalar potential V0(r) have to be specified. For instance, choosing V1(r)=λ/r2 (λ = constant) and V0(r)=αr2/2 (α = constant), the radial part of the wave function of the Pauli-Schrödinger equation can be expressed in terms of Laguerre polynomials.Another way of dealing with such problems is to search for potentials admitting an additional integral of motion. Since it was shown in [6] that there exists exactly one first-order integral of motion for these integrable systems (see (11) and (12)), the additional integral of motion should be higher-order one. In this paper we search for the second-order ones and sum up our results as Theorem 1. In a future work third- and higher-order integrals of motion can be investigated for such systems.Investigation of integrability and superintegrability properties of other type of Hamiltonian systems involving spin in Euclidean plane is in progress.



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