aa r X i v : . [ m a t h - ph ] M a y Matrix superpotentials A.G. Nikitin a and Yuri Karadzhov aa Institute of Mathematics, National Academy of Sciences of Ukraine,3 Tereshchenkivs’ka Street, Kyiv-4, Ukraine, 01601
Abstract
We present a collection of matrix valued shape invariant potentials whichgive rise to new exactly solvable problems of SUSY quantum mechanics. Itincludes all irreducible matrix superpotentials of the generic form W = kQ + k R + P where k is a variable parameter, Q is the unit matrix multiplied by areal valued function of independent variable x , and P , R are hermitian matri-ces depending on x . In particular we recover the Pron’ko-Stroganov ”matrixCoulomb potential” and all known scalar shape invariant potentials of SUSYquantum mechanics.In addition, five new shape invariant potentials are presented. Three ofthem admit a dual shape invariance, i.e., the related hamiltonians can be fac-torized using two non-equivalent superpotentials. We find discrete spectrumand eigenvectors for the corresponding Schr¨odinger equations and prove thatthese eigenvectors are normalizable. E-mail: [email protected] , [email protected] Introduction
Invented by E. Witten [1] as a toy model supersymmetric quantum mechanics (SSQM)became a fundamental field including many interesting external and internal prob-lems. In particular the SSQM presents powerful tools for explicit solution of quan-tum mechanical problems using the shape invariance approach [2]. Unfortunately,the number of problems satisfying the shape invariance condition is rather restricted.However, such problems include practically all cases when the related Schr¨odingerequation is exactly solvable and has an explicitly presentable potential. Well knownexceptions are exactly solvable Schr¨odinger equations with Natanzon potentials [3]which are formulated in terms of implicit functions. The list of shape invariant po-tentials depending on one variable can be found in [4].An interesting example of QM problem which admits a shape invariant supersym-metric formulation was discovered by Pron’ko and Stroganov [5] who studied a motionof a neutral non-relativistic fermion which interacts anomalously with the magneticfield generated by a thin current carrying wire.The supersymmetric approach to the Pron’ko-Stroganov (PS) problem was firstapplied in paper [6] with using the momentum representation. In paper [7] this prob-lem was solved using its shape invariance in the coordinate representation. Recentlya relativistic generalization of the PS problem was proposed [8] which can also beintegrated using its supersymmetry with shape invariance.The specificity of the PS problem is that it is formulated using a matrix super-potential while in the standard SSQM the superpotential is simply a scalar function.Matrix superpotentials themselves were discussed in many papers, see, e.g., [9], [10],[11], [13] but this discussion was actually reduced to analysis of particular examples.In papers [14] such superpotentials were used for analysis of motion of a spin par-ticle in superposed magnetic and scalar fields. In paper [11] a certain class of suchsuperpotentials was described which however was ad hoc restricted to 2 × a priori suppositions about the dimension of the involved matri-ces but restrict ourselves to linear and inverse dependence of the superpotentials onvariable parameters. Trough our approach however the problem of classification ofindecomposable matrix potentials which are shape invariant appears to be completelysolvable. We present a classification of such potentials and discuss the correspondingexactly solvable problems for coupled systems of Schr¨odinger equations. In partic-1lar the discrete energy spectra and exact solutions for these models are found andnormalizability of the ground and exited states is proven. Solutions corresponding tocontinuous spectra are not considered here.Three out of five found hamiltonians admit alternative factorizations with usingdifferent superpotentials. The corresponding potentials are shape invariant w.r.t.shifts of two different parameters. Such dual shape invariance results in existence oftwo alternative spectra branches. Moreover, for some values of free parameters boththese branches can be realized. The PS problem was discussed in numerous papers, see, e.g., [5]-[7]. Thus we willnot present its physical motivations and calculation details but start with the corre-sponding equation for radial functions [7]ˆ H κ ψ = E κ ψ (1)where ˆ H κ is a Hamiltonian with a matrix potential, E κ and ψ are its eigenvalueand eigenfunction correspondingly, moreover, ψ is a two-component spinor. Up tonormalization of the radial variable x the Hamiltonian ˆ H κ can be represented asˆ H κ = − ∂ ∂x + κ ( κ − σ ) 1 x + σ x (2)where σ and σ are Pauli matrices and κ is a natural number. In addition, solutionsof equation (1) must be normalizable and vanish at x = 0.Hamiltonian ˆ H κ can be factorized asˆ H κ = a + κ a − κ + c κ (3)where a − κ = ∂∂x + W κ , a + κ = − ∂∂x + W κ , c κ = − κ + 1) and W is a matrix superpotential W κ = 12 x σ − κ + 1 σ − κ + 12 x . (4)Another nice property of Hamiltonian ˆ H κ is that its superpartner ˆ H + κ is equal toˆ H κ +1 , namelyˆ H + κ = a − κ a + κ + c κ = − ∂ ∂x + ( κ + 1)( κ + 1 − σ ) 1 x + σ x = ˆ H κ +1 Thus equation (1) admits supersymmetry with shape invariance and can be solvedusing the standard technique of SSQM [4].2
Generic matrix shape invariant potentials
Following a natural desire to find other matrix potentials which are form invariantwe consider equation (1) with H k = − ∂ ∂x + V k ( x ) , (5)where V k ( x ) is an n × n dimensional matrix potential depending on variable x andparameter k .Suppose that the Hamiltonian accepts factorization (3) where W k ( x ) is a super-potential. Our goal is to find such superpotentials which generate form invariantpotentials V k ( x ).Assume W k ( x ) is Hermitian. Then the corresponding potential V k ( x ) and itssuperpartner V + k ( x ), i.e., V k ( x ) = − ∂W k ∂x + W k and V + k ( x ) = ∂W k ∂x + W k (6)are Hermitian too.Suppose also that the Hamiltonian be shape invariant, i.e., H + k = H k + α + C k , (7)thus V + k = V k + α + C k or W k + W ′ k = W k + α − W ′ k + α + C k (8)were C k and α are constants.Let us state the problem of classification of shape invariant superpotentials, i.e., n × n matrices whose elements are functions of x, k satisfying conditions (8). In thefollowing section we present such classification for a special class of superpotentialswhose dependence on k is defined by terms proportional to k and k only. To generalize (4) we consider superpotentials of the following special form W k = kQ + 1 k R + P, (9)where P , R and Q are n × n Hermitian matrices depending on x . Moreover, wesuppose that Q = Q ( x ) is proportional to the unit matrix. This supposition can bemotivated by two reasons: 3 our goal is to generalize superpotential (4) in which the term linear in k isproportional to the unit matrix; • restricting ourselves to such Q it is possible to make a complete classification ofthe corresponding superpotentials (9) satisfying shape invariance condition (8).We do not make any a priori supposition about possible values of the continuousindependent variable x . However we suppose that relations (8) are valid also for k → k ′ where k ′ = k + α, k + 2 α, ..., k + nα and n is a natural number which issufficiently large to make the following speculations.It is reasonable to restrict ourselves to the case when the matrices P and R cannot be simultaneously transformed to a block diagonal form since if such (unitary)transformation is admissible, the related superpotentials are completely reducible.Thus we suppose that the pair of matrices < P, R > is irreducible. Let us show thatin this case it is sufficient to consider 1 × × α = 0 we conclude that the corresponding W k shouldbe linear in x provided relation (8) is satisfied: W k = 12 C k x + M k (10)where C k is a constant multiplied by the unit matrix and M k is a constant hermitianmatrix which can be diagonalized. In this way we obtain a direct sum of shifted onedimensional oscillators whose irreducible components can be represented in form (10)where C k and M k are constants.Let α = 0. Substituting (9) into (8) and multiplying the obtained expression by k ( k + α ) we obtain AB ( Q ′ − αQ ) + 2 B ( P ′ − αQP ) + αB { R, P } + ABR ′ + αAR = B C k (11)where { R, P } = RP + P R is anticommutator of matrices P and Q , A = 2 k + α , B = k ( k + α ) and the prime denotes derivative w.r.t. x .All terms in the l.h.s. of equations (11) are polynomials in discrete variable k . Inorder for this equation be consistent, its r.h.s. (which includes an arbitrary element C k ) should also be a polynomial of the same order whose general form is B C k = ναAB − µB − αλB + ρAB + αω A (12)where the Greek letters denote arbitrary parameters. Substituting (12) into (11) andequating coefficients for linearly independent terms we obtain the following system: Q α − Q ′ + να = 0 , (13) P ′ − αQP + µ = 0 , (14) { R, P } + λ = 0 (15) R ′ = ρ, R = ω . (16)4t follows from (16) that ρ = 0 and R is a constant matrix whose square isproportional to the unit one.If R is proportional to the unit matrix I or is the zero matrix (in the last case ω = 0) the corresponding superpotential (9) is reducible. Let ω = 0 and R = ± ωI then the general form of P satisfying (15) is P = λ ω R + ˜ P (17)where ˜ P is a matrix which anticommutes with R .A straightforward analysis of equation (14) shows that it is easily integrable, butto obtain non-trivial Q it is necessary to set µ = λ = 0. Indeed, without loss ofgenerality hermitian matrix R whose square is proportional to the unit matrix canbe chosen in the diagonal form: R = ω (cid:18) I m × m m × s s × m − I s × s (cid:19) , m + s = n (18)where ω = 0 is a constant, I ... and 0 ... are the unit and zero matrices whose dimensionis indicated in subindices, and without loss of generality we suppose that s ≥ m .The corresponding matrix ˜ P satisfying (25) has the following generic form:˜ P = (cid:18) m × m M m × s M † s × m s × s (cid:19) (19)where M m × s is an arbitrary matrix of dimension m × s . Substituting (17)–(19) into(14) we obtain the following equations:˜ P ′ = 2 αQ ˜ P , (20) (cid:18) λω R + µI n × n (cid:19) Q = 0 . (21)Analyzing equation (21) we conclude that for λ + µ = 0 the matrix in bracketsis invertible and so we have to set Q = 0. If Q is nontrivial we have to set λ = µ = 0.As a result the system (14)–(16) is reduced to the following form Q α − Q ′ + να = 0 , (22) P = ˜ P exp (cid:18) α Z Qdx (cid:19) , (23) { ˜ P , R } = 0 , R = ω , (24) C k = αω (2 k + α ) k ( k + α ) + να (2 k + α ) (25)where both R and ˜ P are constant matrices.5hus the problem of classification of matrix valued shape invariant potentials (9)is reduced to solving the first order differential equation (22) for function Q and thealgebraic problem (24) for hermitian matrices R and ˜ P .Let us show that hermitian n × n matrices ˜ P and R which satisfy conditions (24)can be simultaneously transformed to a block diagonal form. Moreover, irreduciblematrices satisfying (24) are nothing but the 2 × × R ˜ P = 0. Starting with (18) and (19)and applying a unitary transformation R → R ′ = U RU † , ˜ P → ˜ P ′ = U ˜ P U † ,U = (cid:18) u m × m m × s s × m u s × s (cid:19) where u m × m and u s × s are unitary submatrices, we obtain˜ P ′ = (cid:18) m × m M ′ m × s M ′† s × m s × s (cid:19) , R ′ = R (26)with M ′ m × s = u m × m M m × s u † s × s . (27)Transformation (27) can be used to simplify submatrix M m × s . In particular thissubmatrix can be reduced to the following form M ′ m × s = (cid:16) f M m × m m × ( s − m ) (cid:17) (28)where f M m × m is a diagonal matrix: f M m × m = diag ( µ , µ , · · · , µ m ) (29)where µ , µ , ... are real parameters. Without loss of generality we suppose that thereare r nonzero parameters µ , µ , ...µ r with 0 ≤ r ≤ m being the rank of matrix M .Notice that transformation (27)–(29) for rectangular matrices M is called singularvalue decomposition . Such transformations are widely used in linear algebra, see, e.g.,[17].But the set of matrices { R, ˜ P A } with R and ˜ P A given in (18) and (26), (28), (29)is completely reducible since by an accordant permutation of rows and columns theycan be transformed to direct sums of 2 × { R × , ˜ P × } where R × = (cid:18) ω − ω (cid:19) ≡ ωσ , ˜ P × = (cid:18) µµ (cid:19) ≡ µσ , µ = µ , µ , ..., µ r (30)and of 1 × R × = ± ω, ˜ P × = µ, µω = 0 (31)6ere ω and µ are arbitrary real numbers. The transformation of matrices (18) and(26), (28), (29) to the direct sum of matrices (30) and (31) can be given explicitly as R → U RU † , ˜ P → U RU † where U is a unitary matrix whose nonzero entries are: U a a = U b m + b − = U m + b b +1 = 1 ,a = 1 , m + s, m + s + 1 , m + s + 2 , · · · , n, b = 2 , , · · · , s + 1 . Thus up to unitary equivalence we have only two versions of irreducible matrices R and P which are given by equations (30) and (31).The remaining equation (22) is easily integrable, thus we can find all inequivalentirreducible superpotentials (9) in explicit form. There are six different types of solutions of equation (22), namely Q = 0 , ν = 0 , (32)and Q = − αx , ν = 0 ,Q = − λα , ν = − λ α < ,Q = − λα tanh λx, ν = − λ α < ,Q = − λα coth λx, ν = − λ α < ,Q = λα tan λx, ν = λ α > x → x + c, c is an integration constant, and α issupposed to be nonzero.The corresponding matrices ˜ P are easily calculated using equations (23) and (30)or (31).Let us note that using solutions (32) or solutions (33) for scalar P and R givenby relations (31), we simply recover the known list of shape invariant potentialswhich is presented, e.g., in [4], see the table on pages 291-292 (this list includes alsothe harmonic oscillator (10)). We will not present this list here but note that ourapproach gives a simple and straightforward way to find it.Consider the case when P and R are 2 × Q and solutions with trivial matrices P are not available since they lead to reducible7uperpotentials. However, solutions (33) are consistent. Substituting (23), (30), (33)into (9) we obtain the following list of matrix superpotentials W κ,µ = ((2 µ + 1) σ − κ −
1) 12 x + ω κ + 1 σ , µ > − , (34) W κ,µ = λ (cid:16) − κ + µ exp( − λx ) σ − ωκ σ (cid:17) , (35) W κ,µ = λ (cid:16) κ tan λx + µ sec λxσ + ωκ σ (cid:17) , (36) W κ,µ = λ (cid:16) − κ coth λx + µ csch λxσ − ωκ σ (cid:17) , µ < , ω > , (37) W κ,µ = λ (cid:16) − κ tanh λx + µ sech λxσ − ωκ σ (cid:17) , (38)where we introduce the normalized parameter κ = kα . These superpotentials are de-fined up to translations x → x + c , κ → κ + γ , and up to unitary transformations W κ,µ → U a W κ,µ U † a where U = σ , U = √ (1 ± i σ ) and U = σ . In particular thesetransformations change signs of parameters µ and ω in (35)–(38) and of µ + in (34),thus without loss of generality we can set ω > , µ > µ and κ in the way indicated in(37).Notice that the transformations k → k ′ = k + α correspond to the followingtransformations for κ : κ → κ ′ = κ + 1 . (40)If µ = 0 and ω = 1 then operator (34) coincides with the well known superpoten-tial for PS problem (4), but for µ = 0 superpotential (34) is not equivalent to (4). Theother found superpotentials are new also and make it possible to formulate consis-tent, exactly solvable problems for Schr¨odinger equation with matrix potential. Thecorresponding potentials V κ can be found starting with (34)–(37) and using definition(6). Let us rewrite equation (6) as follows: W κ,µ − W ′ κ,µ = V κ = ˆ V κ + c κ (41)where c κ is a constant and ˆ V κ does not include constant terms proportional to theunit matrix. As a result we obtainˆ V κ = (cid:0) µ ( µ + 1) + κ − κ (2 µ + 1) σ (cid:1) x − ωx σ , (42)8 V κ = λ (cid:0) µ exp( − λx ) − (2 κ − µ exp( − λx ) σ + 2 ωσ (cid:1) , (43)ˆ V κ = λ (cid:0) ( κ ( κ −
1) + µ ) sec λx + 2 ω tan λxσ + µ (2 κ −
1) sec λx tan λxσ ) , (44)ˆ V κ = λ (cid:0) ( κ ( κ −
1) + µ ) csch ( λx ) + 2 ω coth λxσ + µ (1 − κ ) coth λx csch λxσ ) , (45)ˆ V κ = λ (cid:0) ( µ − κ ( κ − λx + 2 ω tanh λxσ − µ (2 κ −
1) sech λx tanh λxσ ) . (46)Potentials (42), (43), (44) (45) and (46) are generated by superpotentials (34), (35),(36), (37) and (38) respectively. The corresponding constants c κ in (41) are c κ = ω (2 κ + 1) (47)for potential (42), c κ = λ (cid:18) κ + ω κ (cid:19) (48)for potentials (43), (45), (46) and c κ = λ (cid:18) ω κ − κ (cid:19) (49)for potential (44).All the above potentials are shape invariant and give rise to exactly solvable prob-lems for systems of two coupled Schr¨odinger equations, i.e., for systems of Schr¨odinger-Pauli type. To find potentials (42)–(45) we ask for their shape invariance w.r.t. shifts of parameter κ . The shape invariance condition together with the supposition concerning thegeneric form (9) of the corresponding superpotential make it possible to define thesepotentials up to arbitrary parameters λ, ω, κ and µ .Starting with superpotentials (34)–(37) we can find the related potentials (42)–(45) in a unique fashion. But let us consider the inverse problem: to find possiblesuperpotentials corresponding to given potentials wich in our case are given by equa-tions (42)–(45). The problems of this kind are very interesting since their solutionscan be used to generate families of isospectral hamiltonians. It happens that in thecase of matrix superpotentials everything is much more interesting since there existadditional superpotentials compatible with the shape invariance condition.9o find the mentioned additional superpotentials we use the following observation:potentials (42), (44) and (45) are invariant with respect to the simultaneous change µ → κ − , κ → µ + 12 . (50)In addition, there exist another transformations of µ and κ but they lead to the sameresults.Thus in addition to the shape invariance w.r.t. shifts of κ potentials (42), (44)and (45) should be shape invariant w.r.t. shifts of parameter µ also. In other words,superpotentials in Section 5, should be considered together with superpotentials whichcan be obtained from (34), (36) and (37) using the change (50).Thus, we also can represent potentials (34), (36) and (37) in the following form f W µ,κ − f W ′ µ,κ = ˆ V µ + c µ (51)where ˆ V µ = ˆ V κ , and f W µ,κ = κσ − µ − x + ω µ + 1) σ , c µ = ω µ + 1) (52)for ˆ V k given by equation (42), f W µ,κ = λ (cid:18) (2 µ + 1) tan λx + (2 κ −
1) sec λxσ + 4 ω µ + 1 σ (cid:19) (53)for potential (44), and f W µ,κ = λ (cid:18) − (2 µ + 1) coth λx + (2 κ −
1) csch λxσ − ω µ + 1 σ (cid:19) (54)for potential (45). The related constant constant c µ is: c µ = λ (cid:18) ±
14 (2 µ + 1) + 4 ω (2 µ + 1) (cid:19) (55)where the sign ”+” and ” − ” corresponds to the cases (53) and (54) respectively.We stress that superpartners of potentials (51) constructed using superpotentials f W µ,κ , i.e., V + µ = f W µ,κ + f W ′ µ,κ (56)satisfy the shape invariance condition since V + µ = V µ +1 + C µ C µ = c µ +1 − c µ .Thus potentials (34), (36) and (37) admit a dual supersymmetry, i.e., they areshape invariant w.r.t. shifts of two parameters, namely, κ and µ . More exactly,superpartners for potentials (42), (44) and (45) can be obtained either by shifts of κ or by shifts of µ while simultaneous shifts are forbidden. We call this phenomena dual shape invariance .Notice that the remaining potentials (43) and (46) do not posses the dual shape in-variance in the sense formulated above. In potential (43) parameter µ is not essential.It is supposed to be non-vanishing (since for µ = 0 the corresponding superpotentialis reducible) and can be normalized to the unity by shifting independent variable x .The hamiltonian with potential (46) is not invariant w.r.t. change (50). Howeverif we suppose that parameter µ be purely imaginary, i.e., set µ = i˜ µ with ˜ µ real, thecorresponding potential admits discrete symmetry (50) for parameters κ and ˜ µ andthus possesses dual supersymmetry with shape invariance. In this way we obtain aconsistent model of ”PT-symmetric quantum mechanics [15]” with the dual shapeinvariance. Discussion of this model lies out the scope of present paper. We only notethat for ω = 0 the corresponding potential is decoupled to a direct sum of potentialsdiscussed in [16]. Consider the Schr¨odinger equationsˆ H κ ψ ≡ (cid:18) − ∂ ∂x + ˆ V κ (cid:19) ψ = E κ ψ (57)where ˆ H κ = a + κ,µ a − κ,µ + c κ and ˆ V κ are matrix potentials represented in (43)–(48). Sinceall these potentials are shape invariant, equations (57) can be integrated using thestandard technique of SSQM. An algorithm for construction of exact solutions ofsupersymmetric a shape invariant Schr¨odinger equations includes the following steps(see, e.g., [4]): • To find the ground state solutions ψ ( κ, µ, x ) which are proportional to squareintegrable solutions of the first order equation a − κ,µ ψ ( κ, µ, x ) ≡ (cid:18) ∂∂x + W κ,µ (cid:19) ψ ( κ, µ, x ) = 0 . (58)In view of (41) function ψ ( κ, µ, x ) solves equation (57) with E κ = E κ, = − c κ . (59)11 To find a solution ψ ( κ, µ, x ) for the first excited state which is defined by thefollowing relation: ψ ( κ, µ, x ) = a + κ,µ ψ ( κ + 1 , µ, x ) ≡ (cid:18) − ∂∂x + W κ,µ (cid:19) ψ ( κ + 1 , µ, x ) . (60)Since a ± κ and ˆ H κ satisfy the interwining relationsˆ H κ a + κ,µ = a + κ,µ ˆ H κ +1 function (60) solves equation (57) with E κ = E κ, = − c κ +1 . • Solutions for the second excited state can be found as ψ ( κ, µ, x ) = a + κ,µ ψ ( κ +1 , µ, x ), etc. Finally, solutions which correspond to n th exited state for anyadmissible natural number n > ψ n ( κ, µ, x ) = a + κ,µ a + κ +1 ,µ · · · a + κ + n − ,µ ψ ( κ + n, µ, x ) . (61)The corresponding eigenvalue E κ,n is equal to − c κ + n . • For systems admitting the dual shape invariance it is necessary to repeat thesteps enumerated above using alternative (or additional) superpotentials.All potentials presented in the previous section generate integrable models withHamiltonian (57). However, it is desirable to analyze their consistency. In particular,it is necessary to verify that there exist square integrable solutions of equation (58)for the ground state.In the following sections we prove that such solutions exist for all superpotentialsgiven by equations (34)–(37) and (52)–(54). We will see that to obtain normalizableground state solutions it is necessary to impose certain conditions on parameters ofthese superpotentials.To finish this section we present energy spectra for models (57) with potentials(42)–(45): E = − ω (2 N + 1) (62)for potential (42), E = − λ (cid:18) N + ω N (cid:19) (63)for potentials (43), (46), (45), and E = λ (cid:18) N − ω N (cid:19) (64)12or potentials (44).In equations (62)–(62) we omit subindices labeling the energy levels. The spectralparameter N can take the following values N = n + κ, (65)and (or) N = n + µ + 12 (66)where n = 0 , , , ... are natural numbers which can take any values for potentials(42)–(44). For potentials (43), (46) and (45) with a fixed k < n are bound by the condition ( k + n ) > | ω | , see section 9.For potential (43) the spectral parameter is defined by equation (65). For poten-tials (42), (44), (46) the form of N depends on relations between parameters κ and µ , see section 9. Let us show that for some values of parameters µ and κ potentials (42)–(46) areisospectral with direct sums of known scalar potentials.Considering potential (42) and using its dual shape invariance it is possible todiscover that for half integer µ V κ can be transformed to a direct sum of scalarCoulomb potentials. Indeed, its superpartner obtained with using superpotential(51) with opposite sign, i.e., ˆ W µ,κ = − f W µ,κ looks as:ˆ V + κ,µ = ˆ W µ,κ + ˆ W ′ µ,κ + c µ = (cid:0) µ ( µ −
1) + κ − κ (2 µ − σ (cid:1) x − ωx σ . (67)Considering ˆ V + κ,µ = ˆ V κ,µ +1 as the main potential and calculating its superpartnerwith using superpotential ˆ W µ +1 ,κ we come to equation (67) with µ → µ −
2, etc. Itis easy to see that continuing this procedure we obtain on some step the followingresult:ˆ V + κ, ˜ µ = l ( l + 1) x − ωx σ , l = κ −
12 (68)where ˜ µ = µ + n, n = − µ − . Diagonalizing matrix σ → σ we reduce (68) to adirect sum of attractive and repulsive Coulomb potentials written in radial variables.It means that for negative and half integer µ our potential (42) is isospectral withthe Coulomb one. 13n analogous way we can show that potentials (44) with half integer κ or integer µ is isospectral with the potentialˆ V κ = λ (cid:0) r ( r −
1) sec λx + 2 ω tan λxσ (cid:1) , r = 12 ± µ or r = κ, (69)which is equivalent to the direct sum of two trigonometric Rosen-Morse potentials.Under the same conditions for parameters µ and κ potential (46) is isospectral withthe following potential:ˆ V κ = λ (cid:0) r ( r −
1) csch ( λx ) + 2 ω coth λxσ (cid:1) (70)which is equivalent to the direct sum of two Eckart potentials. Finally, potential (46)is isospectral withˆ V κ = λ (cid:0) r ( r −
1) sech λx + 2 ω tanh λxσ (cid:1) , r = 12 ± r µ + 12 (71)provided κ is negative half integer. Potential (71) is equivalent to the direct sum oftwo hyperbolic Rosen-Morse potentials.Thus for some special values of parameters µ and κ we can establish the isospec-trality relations of matrix potentials (42)–(46) with well known scalar potentials. Thisobservation is supported by the direct comparison of spectra (62)–(62) with the spec-tra of Schr¨odinger equation with Coulomb, Rosen-Morse and Eckart potentials whichcan be found, e.g., in [4].Let us note that setting in (43)–(46) ω = 0 we also come to the direct sums ofshape invariant potentials, namely, Morse, Scraft and generalized P¨oshl-Teller ones.However for nonzero ω = 0 and µ , κ which do not satisfy conditions imposed to obtain(68)–(71) the found potentials cannot be transformed to the mentioned directs sumsusing the consequent Darboux transformations. Let us find the ground state solutions for equations (57) with shape invariant po-tentials (42)–(45). To do this it is necessary to solve equations (58) where W κ,µ are superpotentials given in (34)–(37), and analogous equation with superpotentials(52)–(54). The corresponding solutions are square integrable two component func-tions which we denote as: ψ ( κ, µ, x ) = (cid:18) ϕξ (cid:19) . (72)In this section we find the ground state solutions considering consequently all thementioned potentials. 14 .1 Ground states for systems with potentials (42) and (43) Let us start with the superpotential defined by equation (34). Substituting (34) and(72) into (58) we obtain the following system: ∂ϕ∂x + ( µ − κ ) ϕx + ω κ + 1 ξ = 0 , (73) ∂ξ∂x − ( µ + κ + 1) ξx + ω κ + 1 ϕ = 0 . (74)Solving (74) for ϕ , substituting the solution into (73) and making the change ξ = y κ +1 ˆ ξ ( y ) , y = ωx k + 1 (75)we obtain the equation y ∂ ˆ ξ∂y + y ∂ ˆ ξ∂y − (cid:0) y + µ (cid:1) ˆ ξ = 0 . (76)Its solution is a linear combination of modified Bessel functions:ˆ ξ = C K µ ( y ) + C I µ ( y ) . (77)To obtain a square integrable solution we have to set in (77) C = 0 since I µ ( y )turns to infinity with x → ∞ . Then substituting (77) into (75) and using (74) weobtain solutions for system (73), (74) in the following form: ϕ = y κ +1 K µ +1 ( y ) , ξ = y κ +1 K | µ | ( y ) (78)where y is the variable defined in (75), ωx/ (2 κ + 1) ≥ κ is positive and satisfiesthe following relation: κ − µ > . (79)If this condition is violated, i.e., κ − µ ≤ a − µ,κ ˜ ψ ( µ, κ, x ) ˜ ψ ( µ, κ, x ) = 0 . (81)15here (and in the following)˜ a − µ,κ = ∂∂x + f W µ,κ , ˜ a + µ,κ = − ∂∂x + f W µ,κ . (82)Indeed, solving (81) we obtain a perfect ground state vector:˜ ψ ( µ, κ, x ) = (cid:18) ˜ ϕ ˜ ξ (cid:19) , ˜ ϕ = y µ + K | ν | ( y ) , ˜ ξ = y µ + K | ν − | ( y ) (83)where y = ωx µ +1) and ν = κ + 1 / . The normalizability conditions for solution (83)are: κ − µ < , if κ ≥ κ + µ > , if κ < . (85)It is important to note that conditions (79) and (84) are compatible provided κ > , < κ − µ < . (86)Conditions (79) and (85) are incompatible.Thus if parameters µ and κ satisfy (80) and (84) , equation (57) admits groundstate solutions (83). If (79) is satisfied but (84) is not true, the ground state solutionsare given by relations (78). If condition (86) is satisfied both solutions (78) and (83)are available. In the special case κ = µ + 1 / µ is positive excludes the case µ = − / ξ = y − κ ˆ ξ ( y ) , ϕ = y − κ ˆ ϕ ( y ) , y = µ exp( − λx )we find the following solutions: ϕ = y − κ K | ν | ( y ) , ξ = − y − κ K | ν − | ( y ) (87)where ν = ω/κ + 1 / ω and κ should satisfy the conditions κ < , κ > ω. (88)Since potential (43) does not admit the dual shape invariance, there are no otherground state solutions. 16 .2 Ground states for systems with potentials (44)–(46) In analogous manner we find solutions of equations (58) and (81) for the remainingsuperpotentials (35)–(37). Let us present them without calculational details.Solving equation (58) for superpotential (36) we obtain two normalizable solutions,the first of which is: ϕ = y κ − µ (1 − y ) κ + µ F ( a, b, c ; y ) ,ξ = 2 ωκ (2 µ − y κ − µ (1 − y ) κ + µ F ( a + 1 , b + 1 , c + 1; y ) . (89)Here F ( a, b, c ; y ) is the hypergeometric function, a = − i ωκ , b = i ωκ , c = 12 − µ = 0 ,y = 12 (sin λx + 1) , − π ≤ λx ≤ π , (90)and parameters µ and κ are constrained by the conditions (79) and κ + µ > . (91)The second solution is ϕ = y κ + µ (1 − y ) κ + µ F ( a, b, c ; y ) ,ξ = − (2 µ + 1) κ ω (cid:18) − yy (cid:19) ϕ − κ (2 µ + 1) + 4 ω ωκ (2 µ + 3) y κ + µ (1 − y ) κ + µ F ( a + 1 , b + 1 , c + 1; y ) (92)where variable y is the same as in (89), a = µ + 12 − i ωκ , b = µ + 12 + i ωκ , c = µ + 32 , (93)and parameters κ, µ again should satisfy conditions (79) and (91).Using the dual shape invariance of potential (44) we can find additional (or alter-native) ground state solutions using equation (81) with superpotential (53). In thisway we obtain˜ ϕ = y µ − κ +12 (1 − y ) κ + µ F ( a, b, c ; y ) , ˜ ξ = 4 ω (2 κ − µ + 1) y µ − κ (1 − y ) κ + µ F ( a + 1 , b + 1 , c + 1; y ) . (94)Here variable y and its domain are the same as given in (90), − a = b = 2i ω µ + 1 , c = 1 − κ, (95)17nd parameters µ and κ are constrained by the conditions (91) and κ − µ < . (96)The second solution is˜ ϕ = ϕ , ˜ ξ = ξ (97)where ϕ and ξ are functions defined by equation (92) where arguments a, b and c of the hypergeometric function differs from (93) and have the following form: a = κ − ω µ + 1 , b = κ − ω µ + 1 c = κ + 12 , and parameters κ, µ should satisfy conditions (91) and (96).Thus for potential (44) we have three versions of constraints for parameters µ and κ : κ − µ ≥ , (98) κ − µ ≤ < κ − µ < . (100)In addition, condition (91) should be imposed.For the cases (98) and (99) the ground state solutions are given by equations (89),(92) and (94), (97) correspondingly while in the case (100) all solutions (89), (92),(94) and (97) are available.For superpotential (37) we obtain the following solution of equation (58): ϕ = (1 − y ) ωκ − κ y κ − µ F (cid:0) a, b, c, y (cid:1) ,ξ = − yϕ + a + cc y κ − µ +1 (1 − y ) − κ + ωκ F (cid:0) a + 1 , b + 1 , c + 1; y (cid:1) (101)where a = ωκ , b = a + c, c = 12 − µ, y = tanh λx κ, µ, ω satisfy conditions (79) and (88). These conditions are com-patible iff µ < ξ = (1 − y ) ωκ − κ y κ − µ +12 F (cid:0) a, b, c ; y (cid:1) ,ϕ = (cid:18) κ (2 µ − y − ωy − y (cid:19) ϕ + κabωc (1 − y ) − κ + ωκ y κ − µ +22 F ( a + 1 , b + 1 , c + 1; y ) (103)18here y is the variable given in (102), a = 1 + ωκ , b = c − a, c = 32 − µ. (104)Solution (103) is normalizable provided conditions (79) and (88) are satisfied.Potential (45) possesses the dual shape invariance thus we also should find so-lutions of equation (81) with superpotential (54). The explicit expression of thesesolutions can be obtained from (101) and (103) using the change (50). To obtainconsistent solutions the additional conditions µ < , (2 µ + 1) > ω > ϕ = y − κ + ω κ (1 − y ) − κ − ω κ F ( a, b, c ; y ) ,ξ = − µκ ω + κ y − κ + ω κ (1 − y ) − κ − ω κ F ( a + 1 , b + 1 , c + 1; y ) , (106)where the parameters and variables are a = − i µ, b = i µ, c = 12 + ωκ = 0 , y = 12 (tanh λx + 1) (107)with κ and µ satisfying (88). The second solution looks as: ϕ = y − κ − ω κ (1 − y ) − κ − ω κ F ( a, b, c ; y ) ,ξ = 2 ω − κ κµ (cid:18) − yy (cid:19) ϕ − ( κ − ω ) + 4 µ κ µκ (3 κ − ω ) y − κ − ω κ (1 − y ) − κ − ω κ F ( a + 1 , b + 1 , c + 1; y ) (108)where y is the variable defined in (107), a = 12 − ωκ − i µ, b = 12 − ωκ + i µ, c = 32 − ωκ , (109)and parameters κ, ω should satisfy condition (88).Due to the absence of dual shape invariance of potential (46) there are no alter-native ground state solutions.Thus we find ground state solutions for equation (57) with all potentials (42)–(45).These solutions are square integrable and correspond to the eigenvalues E κ = − c κ or E µ = − c µ where c κ and c µ are given by equations (47)–(49) and (55).19 We already know ground state solutions for Schr¨odinger equations with potentialsgiven by relations (42)–(45). Solutions which correspond to n th energy level can beobtained starting, e.g., with the ground state solutions (78), (87), (89), (92), (101),(103), (106), (108) and applying equation (61). However it is necessary to make surethat such defined ground and exited states are square integrable.Let us first consider potential (42) which admits the dual shape invariance. Thisinvariance enables to make factorization of the corresponding Schr¨odinger equationand find ground state solutions using either superpotential (34) or (52), or even bothof them, depending on given initial values of parameters κ and µ . Namely if (79) issatisfied but (84) is not true , the ground state solutions are given by equations (72)and (78) and exited states are given by equation (61).The corresponding energy levels are given by equations (62) and (65).If parameters µ and κ satisfy (80) and one of conditions (84) or (84), equation(57) admits ground state solutions (83) and exited states are defined by the followingequation:˜ ψ n ( κ, µ, x ) = ˜ a + κ,µ ˜ a + κ +1 ,µ · · · ˜ a + κ + n − ,µ ˜ ψ ( κ + n, µ, x ) (110)where ˜ a + κ,µ = − ∂∂x + f W µ,κ and f W µ,κ is the alternative superpotential given by (52).The corresponding energy levels are given by formulae (62) and (66)If condition (86) is satisfied both versions of solutions and energy levels givenabove are available. In the special case κ = µ + 1 / y = 0,and decreases exponentially at infinity. Moreover, at y = 0 there is the inverse powersingularity, i.e., K ν ∼ y ν with ν = µ or ν = µ + 1 which is perfectly compensated bythe multiplier y κ +1 provided κ satisfies (79).Calculating the first exited state (60) we can use relation (58) where κ → κ + 1,thus ψ ( k, µ, y ) = ( W κ +1 ,µ + W κ,µ ) ψ ( κ + 1 , µ, y ) . (111)Again we recognize a good behavior at the singularity point y = 0 since ( W κ +1 ,µ + W κ,µ ) ∼ y + · · · and ψ ( κ + 1 , µ, y ) ∼ yψ ( κ, µ, y ).Let us suppose that the wave function corresponding to n th exited state is regularat y = 0 and has the following form ψ n ( κ, µ, y ) = W ψ ( κ + n, µ, y ) (112)where ψ ( κ + n, µ, y ) is the ground state solution given by relations (72), (78) and W
20s a matrix depending on y and k . Then ψ n +1 ( κ, µ, y ) = (cid:18) − ∂∂y + W κ (cid:19) ψ n ( κ + n + 1 , µ, y )= (cid:18) − ∂W∂y + W κ W + W W κ + n +1 (cid:19) ψ ( κ + n + 1 , µ, y ) (113)where we use the fact that ψ ( κ + n + 1 , µ, y ) solves the equation (58) where κ → κ + n + 1.Using (113) it is not difficult to show that if function ψ n ( κ + 1 , µ, y ) be squareintegrable then ψ n +1 ( κ, µ, y ) is square integrable too. Indeed, at the neighborhoodof the singularity point y = 0 the ground state functions are related as ψ ( κ + n +1 , µ, y ) ∼ yψ ( κ + n, µ, y ), and (cid:16) − ∂W∂y + W κ W + W W κ +1 (cid:17) ∼ y W . Thus ψ n +1 ( κ, µ, y )is regular at y = 0 provided ψ n ( κ, µ, y ) be regular. Since for n = 0 , ψ n ( κ, µ, y ) (61) are normalizablefor any n .By direct repeating the above speculations we can prove the square integrability ofground state vector (83) and exited state vectors given by relation (110). In fact theonly thing we need is to change ψ n ( κ, µ, y ) and W κ,µ by their counterparts ˜ ψ n ( κ, µ, y )and ˜ W µ,κ .In complete analogy with the above one can prove the normalizability of the exitedstates for the case when the superpotential is given by equation (35). However thereis an essentially new point which is generated by condition (88). The think is thatsolutions (87) being well defined for κ and µ satisfying conditions (79) and (88), canloose their square integrability after the change κ → κ + n for a sufficiently large n .Namely, in order to obtain a normalizable solution ψ ( κ + n, µ, x ) we have to ask for( κ + n ) ≥ ω >
0. Since κ is negative we have the following restriction for n : n < | κ | − √ ω. (114)It is possible to show that if ψ ( κ + n, µ, x ) is not normalizable the same is true forexited states (61).Let us consider the ground state solutions (89). This solution like all the re-maining solutions (92), (94), (97), (101), (103), (106), (108) is expressed via linearcombinations of the following elements: y A (1 − y ) B F ( a, b, c ; y ) (115)where parameters a, b and c are given by equation (90), A = κ − µ , B = κ + µ forcomponent ϕ , etc.In accordance with its definition, variable y belongs to the interval [0 ,
1] and thereare two points which are ”suspicious w.r.t. singularity”, namely, y = 0 and y = 1.In order the solution to be regular (and equal to zero) in these points it is necessary21nd sufficient to ask for A > , B > ℜ e ( B + c − a − b ) >
0. Exactly theseconditions generate restrictions (88) for parameters κ which guarantee the solutionnormalizability. The same is true for solutions (92), (106), (108), (101) and (103).To analyze solutions for exited states we rewrite superpotential (36) in terms ofvariable y : W κ,µ = λ (cid:16) κ (2 y −
1) + µ p y (1 − y ) σ + ωκ σ (cid:17) . (116)We see that W κ is nonsingular at y = 0 and y = 1, the same is true for W κ + n,µ forany natural number n . Functions (89) are still regular at these points if we change κ → κ + n , thus we can again apply relations (111)–(113) to prove the normalizabilityof the corresponding solutions (61) for arbitrary n .In a similar way we can prove the square integrability for solutions (61) cor-responding to ground state solutions (92), (94), (97), (101), (103), (106) and (108).However in these cases we again have constraint (88) which generates restriction (114)for the number n enumerating the exited states (61). Analogously, starting with con-dition (105) we come to conclusion that solutions (110) for the Schr¨odinger equationwith potential (45) are square integrable provided quantum number n satisfies thecondition n < | µ | − √ ω − . (117)Thus for any fixed κ , µ and ω equation (57) with potentials (43) (45) and (46)describes a system which has a finite number of states with discrete spectrum. Thesestates are enumerated by non-negative natural numbers n satisfying condition (114)or (117).The systems with potentials (42)–(46) can also have states with continuous spec-trum. In particular, such states should change the bound states when conditions(114) and (117) are violated. Analysis of the states with continuous spectra lies outof frames of the present paper.
11 Discussion
Generalizing the supersymmetric PS problem we find a family of matrix potentialsfor Shr¨odinger equation satisfying the shape invariance condition. In this way wefind five exactly solvable problems for systems of coupled Shr¨odinger equations. Therelated matrix potentials are given by equations (42)– (46).Let us stress that we present the completed classification of shape invariant super-potentials of the generic form (9) where P and R are hermitian matrices of arbitraryfinite dimension and Q is proportional to the unit matrix. Namely, we show that suchobjects can be reduced to direct sums of known scalar superpotentials and superpo-tentials presented in section 5. 22he found potentials include parameters λ, κ, µ and ω whose possible values arerestricted but quite arbitrary. Moreover, parameters ω in (42) and µ in (43) can bereduced to unity by scaling and shifting the independent variable x correspondingly.Taking into account all possibilities enumerated in (79), (84), (85) and (98)–(100)we conclude that in the case of potentials (42), (44) and (45) there are discretespectrum states for all real values of arbitrary parameters λ, κ and µ except the case κ = µ . Parameter ω can be constrained by equations (88) or (105).Potential (42) is a slightly generalized effective potential for the PS problem.Moreover, these potentials coincide for a particular value µ = 0 of arbitrary parameter µ . However, if µ = 0 potential (42) is not equivalent to the potential appearing inthe PS problem and corresponds to a more general interaction in the initial three-dimension problem.At the best of our knowledge the remaining potentials (43)–(46) are new. Therelated Schr¨odinger equations can be integrated using tools of the SUSY quantummechanics. The corresponding spectrum and eigenvectors are given by equations(62)–(66) and (61) or (110) while the ground state solutions are discussed in section9. solutions Notice that the ”matrix supersymmetry” has a new feature in comparisonwith the standard (i.e., scalar) one. Namely, matrix models with shape invariance canhave degenerated ground states in spite of that there exists a normalizable solutionfor equation (58). Example of models with such specific spontaneously broken SUSYis given by the Schr¨odinger equation (57) with potentials (44)–(46).Mathematically, there are natural reasons for appearance of a zero energy doubletof the ground states in systems with the matrix supersymmetry. The thing is thatequation (58) is a system of two the first order equations whose solutions are linearcombinations of two functions while in the ordinary SUSY quantum mechanics wehave a one first order equation for ground states. For potentials (42) and (43) onlyone of these functions is normalizable but for potentials (44)–(43) there are two groundstate solutions.Let us note that existence of zero energy doublets of the ground states was alreadyregistered in periodic quantum systems, see [18] and [19] for discussion of this phe-nomenon. In this connection it seems to be interesting to extend our approach to thecase of periodic systems. Formally speaking, the only new constructive elements ofpotentials (42)–(46) in comparison with the standard scalar shape invariant potentialsare matrices σ and σ which are involutions anticommuting between themselves. Infact the nature of these involutions is not essential for deducing the shape invariantpotentials, and many of the results discussed in present paper can be generalized tothe case of another involutions. For example, it is possible to change the mentionedmatrices by reflection and shift operators which also can be anticommuting involu-tions being applied to functions with an appropriate parity and periodicity. In thisway it seems to be possible to extend the list of potentials which admit supersym-metry including shifts of arguments [20]. A classification of anticommuting discretesymmetries and the corresponding supersymmetric versions of the Schr¨odinger and23auli equations can be found in [22].An interesting phenomena which appears to be typical for systems with matrixSUSY is the dual shape invariance discussed in Section 6. It enables to imposemuch less restrictive constrains on parameters of potentials then the ordinary shapeinvariance. In addition, it can be used to explain the insensibility of the spectra(62)–(62), (65) on parameter µ . Namely, hamiltonians with shifted µ should bealmost isospectral thanks to the dual shape invariance, which is incompatible with µ -dependence of energy values (excluding the exotic case when these values are periodicfunctions of µ ).For some values of parameters µ and κ the additional branch of spectrum causedby the dual shape invariance can appear. In particular it is true for potential (42)with µ = 0 and 0 < κ < /
2. Enhanced analysis of such potentials was made inpaper [12]. In the present paper we slightly refine results of [12].Let us note that the dual shape invariance can be recognized for two potentials ofthe ordinary SUSY quantum mechanics, namely, for the trigonometric Scraft 1 andgeneralized P¨oshl-Teller potentials: V = ( κ ( κ −
1) + µ ) sec x + µ (1 − κ ) sec λx tan λx,V = ( κ ( κ −