Three-Dimensional Solutions of Supersymmetrical Intertwining Relations and Pairs of Isospectral Hamiltonians
aa r X i v : . [ h e p - t h ] F e b THREE-DIMENSIONAL SOLUTIONS OF SUPERSYMMETRICALINTERTWINING RELATIONS AND PAIRS OF ISOSPECTRALHAMILTONIANS
F. Cannata ,a , M.V. Ioffe ,b , D.N. Nishnianidze , ,c INFN, Via Irnerio 46, 40126 Bologna, Italy. Department of Theoretical Physics, Sankt-Petersburg State University,198504 Sankt-Petersburg, Russia Akaki Tsereteli State University, 4600 Kutaisi, Georgia
The general solution of SUSY intertwining relations for three-dimensionalSchr¨odinger operators is built using the class of second order superchargeswith nondegenerate constant metric. This solution includes several mod-els with arbitrary parameters. We are interested only in quantum systemswhich are not amenable to separation of variables, i.e. can not be reduced tolower dimensional problems. All constructed Hamiltonians are partially in-tegrable - each of them commutes with a symmetry operator of fourth orderin momenta. The same models can be considered also for complex values ofparameters leading to a class of non-Hermitian isospectral Hamiltonians.
The method of supersymmetrical (SUSY) intertwining relations appeared naturally inthe framework of a supersymmetrical approach to nonrelativistic Quantum Mechanics [1].Briefly speaking in the most general formulation the method of SUSY intertwining relations a E-mail: [email protected] b E-mail: m.ioff[email protected] c E-mail: [email protected] d H Q + = Q + H ; (1)where H , is a pair of Schr¨odinger Hamiltonians, and the intertwining operator Q + is calledthe supercharge. This relation and its Hermitian conjugate with supercharge Q − = ( Q + ) † lead to the isospectrality of Hamiltonians H , up to possible zero modes of supercharges.The bound state eigenfunctions of H , are related (up to normalization factors) by thesupercharges: H , Ψ (1) , (2) n ( ~x ) = E n Ψ (1) , (2) n ( ~x ); H , = − ∆ (3) + V , ( ~x ); (2)Ψ (2) n ( ~x ) = Q − Ψ (1) n ( ~x ); Ψ (1) n ( ~x ) = Q + Ψ (2) n ( ~x ); ~x = ( x , x , x ); n = 0 , , , ... (3)If Q + has some zero modes, and they coincide with the wave functions of H (or corre-spondingly, Q − has some zero modes coinciding with the wave functions of H ), these wavefunctions are annihilated according to (2) and have no analogous states in the spectra of thepartner Hamiltonian.This scheme works independently on the nature and specific properties of operators H , .In particular, it was realized in one-dimensional space with Q ± of first order in derivativesboth for scalar [1] and matrix [3] potentials in the stationary formulation of Schr¨odinger equa-tion. The case of nonstationary Schr¨odinger equation was studied in [4]. The SUSY inter-twining relations with second order supercharges Q ± were introduced first in one-dimensionalcase in [5], including new irreducible supercharges which cannot be factorized with two firstorder multipliers.A very useful role was played by second order supercharges in the case of two-dimensionalSUSY intertwining relations [6]. A class of solutions of such intertwining relations wasobtained. Each of these Hamiltonians is completely integrable, commuting with its nontrivial d From a pure mathematical point of view, the problem can be formulated as follows. We look for twodifferent factorizations of the operator of fourth order in derivatives (1) such that H , are of Schr¨odingerform, and Q + is the same intertwining second order multiplier. R = Q + Q − or R = Q − Q + . Sometimes,besides being completely integrable the model is partially solvable: part of its spectrum andcorresponding wave functions are constructed by special methods like shape invariance andSUSY separation of variables [7], [6]. In the case of two-dimensional generalization of Morsepotential the exact solvability of the model was proven [8]. It is necessary to stress that onlymodels, which do not allow for conventional separation of variables, were considered in thepapers on two-dimensional Quantum Mechanics mentioned above.Some attempts were made to generalize the method of SUSY intertwining relations to thephysically most interesting case of three-dimensional space. First, the direct generalizationof formalism with the first order supercharges was constructed for arbitrary dimensionalityof space in [2]. This approach has a specific property: the intertwining relations link scalarHamiltonians with a chain of matrix Schr¨odinger-like operators. These Hamiltonians can beinterpreted as operators describing quantum systems with some internal degrees of freedom(for example, spin [9]), but rather frequently this property seems to be inconvenient. Inter-twining of two scalar Hamiltonians by first order supercharges was shown [10] to offer onlysolutions with conventional separation of variables. The only alternative generalization [11],to our knowledge, is analogous to the idea elaborated earlier in two-dimensional case [6]: touse the intertwining relations between pairs of scalar Hamiltonians but with second ordersupercharges. In paper [11] the particular solutions of such intertwining relations were foundfor the special form of second order supercharges ∂ ∂ x − ∂ ∂ x + ... .In the present paper we will consider the most general form of supercharges Q ± with constant metric g ik , i.e. Q ± = g ik ∂∂x i ∂∂x k + ..., where the summation over repeated indices i, k = 1 , , g ik excludingthe degenerate (up to rotations) case of g ik = (1 , , As announced in Introduction, we shall study the intertwining (1) with the most generalsupercharges of second order in derivatives with constant metric (highest order coefficients) g ik : Q + = g ik ∂ i ∂ k + C i ( ~x ) ∂ i + B ( ~x ); ∂ i ≡ ∂∂x i . (4)By space rotations the matrix g ik can be diagonalized to g ik = g ii δ ik . At first, we will beinterested in situations when all three diagonal elements g ii after such rotation do not vanish.The case g ii = (1 , − ,
0) was considered in [11], the case g ii = (1 , ,
0) will be consideredlater in this Section, the case g ii = (1 , a, a = ± g ii = (1 , ,
0) only some particular solutions will be given in thevery end of this Section.Thus, let us consider now the supercharges Q ± , which by a suitable normalization canbe reduced to two classes of g ik :( A ) g ii = (1 , , d ); d = 0 d = 1 (5)( B ) g ii = (1 , d , d ); d = 0 , d = 0 , d = d . (6)The intertwining relations (1) for constant g ik can be rewritten as a system of 6+3+1 = 10differential equations, some of them nonlinear: ∂ i C k ( ~x ) + ∂ k C i ( ~x ) = 2 V ( ~x ) g ik ; (7)∆ (3) C i ( ~x ) + 2 ∂ i B ( ~x ) + 2 g ik ∂ k V = 2 V ( ~x ) C i ( ~x ); (8)∆ (3) B ( ~x ) + g ik ∂ i ∂ k V ( ~x ) + C i ( ~x ) ∂ i V ( ~x ) = 2 V ( ~x ) B ( ~x ); (9) V ( ~x ) − V ( ~x ) ≡ V ( ~x ) . (10)4he six equations (7) and the defining Eq.(10) can be combined: ∂ C ( ~x ) = d − ∂ C ( ~x ) = d − ∂ C ( ~x ) = V ( ~x ); (11) ∂ i C k ( ~x ) + ∂ k C i ( ~x ) = 0 . ; i = k. (12)After further differentiation of (12) in respect to x j with j = i, j = k we obtain: C ( ~x ) = f ( x , x ) + f ( x , x ); C ( ~x ) = [ g ( x , x ) + g ( x , x )] d ; C ( ~x ) = [ h ( x , x ) + h ( x , x )] d , where f, g, h are auxiliary functions specified in the following.By substitution of these relations into (11), we get the functional-differential equations,which are integrated in a general form: g ( x , x ) = x G ′ ( x ) + G ( x ) + K ( x ); f ( x , x ) = x G ′ ( x ) + G ( x ) + M ( x ); f ( x , x ) = ˜ G ( x ) + x ˜ G ′ ( x ) + K ′ ( x ) − G ( x ); g ( x , x ) = ˜ G ( x ) + x ˜ G ′ ( x ) + K ′ ( x ) − G ( x ); h ( x , x ) = x ˜ G ′ ( x ) + ˜ L ( x ) + M ( x ); h ( x , x ) = x ˜ G ′ ( x ) + L ( x ) + G ( x ) − M ( x ) . Therefore, we have the expressions for coefficient functions C i ( ~x ) of the supercharges interms of the eleven arbitrary functions above G i ( x i ) , ˜ G ( x ) , ˜ G ( x ) .... If these expressions areinserted into equations (12), the eleven functions above will be specified being polynomialsmaximally of second order in space coordinates. Thus the general solution of (11) takes theform: C ( ~x ) = a ( x − d x − d x ) + 2 a x x + 2 a x x + bx − b x − b x + κ ; C ( ~x ) = a ( − x + d x − d x ) + 2 a d x x + 2 a d x x + bd x + b x − b x + κ ;(13) C ( ~x ) = a ( − x − d x + d x ) + 2 a d x x + 2 a d x x + bd x + b x + b x + κ ;5nd the difference between potentials V , is given now by: V ( ~x ) = 2( a x + a x + a x ) + b. (14)The next step consists in solving the three equations (8), where the first Laplacian termsare constants defined from (13): ∂ B ( ~x ) + ∂ V ( ~x ) = V ( ~x ) C ( ~x ) − c ; c ≡ a (1 − d − d ); ∂ B ( ~x ) + d ∂ V ( ~x ) = V ( ~x ) C ( ~x ) − c ; c ≡ a ( d − − d ); (15) ∂ B ( ~x ) + d ∂ V ( ~x ) = V ( ~x ) C ( ~x ) − c ; c ≡ a ( d − − d ) . Now we will restrict ourselves to the case of metric (5), i.e. d = 1 , d ≡ d = 0 , . Then,after the simple manipulations (derivatives and linear combinations) with (15) we obtain thenecessary conditions: a = a = b = 0 . (16)As for the constant a , it is convenient to consider separately two options: ( i ) a = 0 and( ii ) a = 0 . In the case ( i ) a = 0 a suitable translation of space coordinates allows to cancel thelinear terms in (13): C ( ~x ) = 2 a x x + e ; C ( ~x ) = 2 a x x + e ; C ( ~x ) = a ( dx − x − x ) + e ; V ( ~x ) = 2 a x , (17)where e i are new arbitrary constants. Taking into account the numerical values of theconstants, Eq.(16), we obtain for the considered case ( d = 1; a = 0) the general formulas: V ( ~x ) = a d d − x − d − a x ( x + x ) − a e x + c x ++ 2 a x ( e x + e x ) + q ( x ) − F ( x , x )]; (18) B ( ~x ) = − a d d − x + 1 d − d + 1) a x ( x + x ) − a e x + c x ++ 2 da x ( e x + e x ) + + dq ( x ) − F ( x , x )] , (19)6here q ( x ) and F ( x , x ) are arbitrary functions, which will be defined from the lastintertwining relations Eq.(9).We use the divergence of vector equation (8) to transform (9) to: − (1 + d (3) V ( ~x ) + C i ( ~x ) ∂ i V ( ~x ) + (2 + d ) V ( ~x ) + C i ( ~x ) ∂ i V ( ~x ) − V ( ~x ) B ( ~x ) = 0 . (20)Since the l.h.s. can be written as a polynomial in x , we put the coefficients to zero. Insert-ing explicit expressions (17), (18), (19) into (20) and after rather long but straightforwardalgebra, we derive that the solution exists only for the metric (1 , , − , i.e. for d = − . Inthis case the function q is linear (with arbitrary constant α ): q ( x ) = − a x + α . (21)As for the function F , it is defined by two differential equations: − F ( x , x ) + α ) − ( x ∂ + x ∂ ) F ( x , x ) + ( e + e ) + ( e − a ρ )(3 a ρ − e ) = 0; (22)( e ∂ + e ∂ ) F ( x , x ) = 2 a ( e − a ρ )( e x + e x ) , (23)where we introduced the radial coordinate in the plane ρ ≡ x + x . In the particular case of constants e = e = 0 the equation (23) is trivially satisfied, andthe general solution for F can be represented in polar coordinates ρ, φ as: F ( x , x ) = − a ρ + a e ρ − f ( φ ) ρ −
12 (2 α + e ) . (24)Thus we obtain the expressions for the partner potentials: V , ( ~x ) = a x + ρ ) + 3 a x ρ − a e x + ρ ) ± a x + f ( φ ) ρ + Const, (25)which are anharmonic oscillators of fourth order with additional 1 /ρ term.In the particular case e e = 0 it is convenient to introduce the new space coordinates: y = x e + x e ; y = x e − x e , (26)7o that in (22), (23) e ∂ + e ∂ = 2 ∂ y ; x ∂ + x ∂ = y ∂ y + y ∂ y . (27)Then the equation (23) can be solved in a general form: F ( x , x ) = − a ( e + e ) y − a ( e − e )8 y y ( y + y ) −− a (3 e + 3 e − e e )16 y y + a e e + e ) y + 2( e − e ) y y ] + p ( y ) , (28)with a function p ( y ) defined from (22): p ( y ) = a e ( e + e )4 y − e − e − e + 2 α κy − a ( e + e ) y . (29)Now, inserting this function into (28) and going back to the variables x i , we obtain the finalexpression for potentials V , : V , ( ~x ) = a x + ρ ) + 3 a x ρ − a e x + ρ ) + a e ( e x + e x ) −− κe e e x − e x ) ± a x + Const. (30)The last option for the values of e , e we have to consider is the case with one of themvanishing and the other not. For definiteness, let us take e = 0 . Then the general solutionof (23) is: F ( x , x ) = a x (2 e − a x − a x ) + q ( x ) , (31)with an arbitrary function q ( x ) . Using it in (22), we obtain a first order differential equationfor q ( x ) , which admits the general solution: q ( x ) = − a x + a e x + 2 µx + 12 ( e − e − α ) (32)( µ is an arbitrary integration constant). The resulting potential has the form: V , ( ~x ) = a x + ρ ) + 3 a x ρ − a e x + ρ ) + a e x x ± a x − µx + Const (33)8n the case ( ii ) a = 0 equations (13), (16) together with a suitable shift of space coordi-nates allow to conclude that the coefficient functions C i ( ~x ) become linear functions: C ( ~x ) = bx − b x ; C ( ~x ) = bx − b x ; C ( ~x ) = bdx + b x + b x ; V ( ~x ) = b (34)with arbitrary real constants b, b , b . Then from the explicit formulas for V , we find thatboth partner Hamiltonians H , being nonisotropic harmonic oscillator are amenable to sep-aration of variables.But there is just one solution for the case a = 0 which does not allow separationof variables. In order to obtain it, we have to extend the original assumption about theconstants in (34). Indeed, if we allow b and b to be complex numbers with b = ib , aftersimple calculations we obtain: V ( ~x ) = b x + x + x ) − bb ( x + ix ) x + b x + ix ) ++ n ( x + ix ) + ω ( x + ix ) + Const, (35) V ( ~x ) = V ( ~x ) + 2 b, (36)with arbitrary constants n, ω. Both potentials are second order polynomials with complex coefficients e and they differfrom each other by the arbitrary constant b. Such pair of intertwined Hamiltonians realize thesimplest kind of shape invariance [14], [7]. In this context, it means that if the Hamiltonian H has the wave function Ψ (1) k with eigenvalue E k (see Eq.(2)), then the function Ψ (2) k = Q − Ψ (1) k is also wave function of the same Hamiltonian H with eigenvalue E k + 2 b. Therefore,an arbitrary eigenvalue E in the spectra of H is accompanied by the tower of eigenstateswith energies E + 2 bk . The same is obviously true for H . We must stress that this propertyof the spectra are formulated up to possible zero modes of supercharges Q ± . These zeromodes can, in principle, truncate the tower described above.An additional property of the spectrum of model (35) is provided by its symmetry. In-deed, for real values of parameter n both Hamiltonians H , are invariant under the combined e The complex coefficients appear in the supercharges Q ± also. See detail discussion of one-dimensionalcase in [12] and of two-dimensional case in [13]. P , i.e. x → − x , and time reflection T. Due to this P T -symmetry, the spectrumof Hamiltonians consists of [15], [16] real eigenvalues E = E ⋆ and complex conjugate pairs E, E ⋆ . One can notice that all models above which were supposed to have real potentials canbe easily complexified also by chosing complex values of parameters in (25), (30), (33).Although no property of shape invariance is observed for these potentials, some invariancesunder combined reflections can be realized with suitable choice of complex parameters. Forexample, P T − symmetry of potential (30) for pure imaginary e and other parameters real.Let us consider now the case when the metric can be reduced to( C ) g ik = (1 , , . (37)Then the general solution of Eq.(7) is: C = − x (2 αx + α ) − νx + ν ); C = − x (2 αx + α ) − νx + ν ); C = α ( x + x ) + α x + α x + β ; V = − αx + ν ) , (38)with arbitrary constants α, α , , ν, ν , . For α = 0 , it can be transformed by translations to: C = − αx x + ν ) , C = − αx x + ν ); C = α ( x + x ) + β ; V = − αx . The first two equations of Eq.(8) lead to: B + V = 2 αx [ α ( x + x ) x + 2( ν x + ν x )] + f ( x ) , (39)with an arbitrary function f ( x ) . The last equation of Eq.(8) gives B ( ~x ): B ( ~x ) = − αx [ α ( x + x ) + β ] − αx + F ( x , x ) , (40)where F ( x , x ) is also an arbitrary function, and β is a constant. Therefore the potential V ( ~x ) is obtained: V ( ~x ) = 3 α x ρ + αβx + 2 α (2 ν x + 2 ν x + 1) x + f − F . f ( x ) , F ( x , x ) are defined by Eq.(9) using also Eq.(8): f ( x ) = 4 α x + β x + β x + β x + β ; F ( x , x ) = − α ρ − β ρ − f ( φ ) ρ , with arbitrary periodical function f ( φ ) of polar coordinate φ. The constants above must bechosen vanishing: ν = ν = β = β = 0. Finally, the partner potentials V , ( ~x ) are obtainedas: V , = α (3 x ρ + 4 x + 12 ρ ) + β ( x + 12 ρ ) + f ( φ ) ρ ∓ αx . (41)In the case α = 0 , Eq.(38) gives after translation: C = − ( α x + 2 ν x ); C = − ( α x + 2 νx ); C = α x + α x ; V = − ν, (42)and, using (8): B = − ν ( α x + α x ) x + N ( x , x ); (43) V = 2 ν ρ + 4 ν ( α x + α x ) + n ( x ) − N ( x , x ) , (44)with arbitrary functions n ( x ) , N ( x , x ) . The analysis of Eqs.(42)-(44) together withEqs.(8), (9) shows that the only solution with real parameters corresponds to models al-lowing standard separation of variables, which are not studied in this paper.The only constant metric g ik in Q ± , which does not allow to find the general solution,is (up to rotations) the degenerate one - g ii = (1 , , . In this case only particular solutionswere found: V = 19 ( αx + x ) x − / − c αx + x ) x / + 16 c α ) ( αx + x ) −− a α ( αx + x ) + c x + 1 + α x / + 3 a c x / + 736 x − + Φ( αx − x ) ,V = 29 ( αx + x ) x − / + c V = x [3( αx + x ) − b ( αx + x ) − b + c )] + 1 + α x − ωx − +11 + α [4( αx + x ) − b ( αx + x ) − c ( αx + x ) ++ b (4 b + c )( αx + x )] + Φ( αx − x ) ,V = − αx + x ) + b, where Φ is an arbitrary function of its argument. The aim of the present paper was to provide an exhaustive analysis of intertwining re-lations for second order supercharges with constant metric in three-dimensional case. Inthis space the intertwining relations are rather restrictive: only specific classes of solutionsexist. Namely, four different classes of real potentials were built in the framework of generalsolution of intertwining relations with nondegenerate metric - (25), (30), (33), (41). All ofthem have the form of anharmonic oscillator of fourth order in coordinates plus the termwith 1 /x − dependence. In addition, particular solutions were obtained for degenerate metric(1 , , . For all these models pairs of almost (up to zero modes of Q ± ) isospectral Hamil-tonians were obtained. Each model is at least partially integrable, since the Hamiltonianscommute with the corresponding symmetry operators of fourth order in momenta.Although the intertwining relations themselves do not provide, in general, the wave func-tions and energy eigenvalues, in two-dimensional space the specific SUSY methods - SUSYseparation of variables and shape invariance [7], [6], [8] - allowed to find a part or eventhe whole spectrum. In contrast to this situation, it seems to be impossible to apply thesemethods in three-dimensional space: there is no shape invariance property and there are notools to find zero modes of Q ± explicitly.It was also useful to consider the obtained results from the point of view of QuantumMechanics with non-Hermitian Hamiltonians [12], [13], [15], [16]. In this case, the potentials12ith complex values of parameters obey some discrete symmetries with antilinear operators,leading to corresponding specific properties of complex spectra. Acknowledgments
The work was partially supported by INFN and University of Bologna (M.V.I. and D.N.N.),by Russian grant RNP 2.1.1/1575 (M.V.I.) and grant ATSU/09317 (D.N.N.).
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