aa r X i v : . [ m a t h - ph ] O c t Exact solvability of PDM systems with extended Liesymmetries A. G. Nikitin
Institute of Mathematics, National Academy of Sciences of Ukraine,3 Tereshchenkivs’ka Street, Kyiv-4, Ukraine, 01004
Abstract
It is shown that all PDM Schr¨odinger equations admitting more than fivedimensional Lie symmetry algebras (whose completed list can be found in pa-per [
J. Math. Phys. , , 083508 (2017)] are exactly solvable. The correspondingexact solutions are presented. The supersymmetric aspects of the exactly solvablesystems are discussed. E-mail: [email protected]
Introduction.
Group classification of differential equations consists in the specification of non-equivalentclasses of such equations which possess the same symmetry groups. It is a rather attrac-tive research field which has both fundamental and application values.A perfect example of group classification of fundamental equations of mathematicalphysics was presented by Boyer [1] who had specified all inequivalent Schr¨odinger equa-tions with time independent potentials admitting symmetries with respect to Lie groups,see also papers [2, 3, 4] where particular important symmetries were discussed, and paper[5] were the Boyer results are corrected. These old results have a big impact since includea priori information about all symmetry groups which can be admitted by the fundamen-tal equation of quantum mechanics. Let us mention also that the nonlinear Schr¨odingerequation as well as the generalized Ginsburg–Landau quasilinear equations have beenclassified also [6, 7] as well as symmetries of more general systems of reaction-diffusionequations [8, 9]. For general discussion of supersymmetries of Schr¨odinger equation see,e.g., papers [10, 11, 12, 13].In contrary, the group classification of Schr¨odinger equations with position dependentmass (PDM) was waited for a very long time. There were many papers devoted to PDMSchr¨odinger equations with particular symmetries, see, e.g., [14, 15, 16, 17]. But thecomplete group classification of these equations appears only recently in papers [18] and[19, 20] for the stationary and time dependent equations correspondingly. A systematicsearch for the higher order symmetries if the PDM systems started in paper [21]. So latemaking of such important job have to cause the blame for experts in group analysis ofdifferential equations, taking into account the fundamental role played by such equationsin modern theoretical physics!Let us remind that the PDM Schr¨odinger equations are requested for description ofvarious condensed-matter systems such as semiconductors, quantum liquids, and metalclusters, quantum wells, wires and dots, super-lattice band structures, etc., etc.It happens that the number of PDM systems with different Lie sym-metries is ratherextended. Namely, in [20] seventy classes of such systems are specified. Twenty of themare defined up to arbitrary parameters, the remaining fifty systems include arbitraryfunctions.The knowledge of all Lie groups which can be admitted by the PDM Schr¨odingerequations has both fundamental and application values. In particular, when constructthe models with a priory requested symmetries we can use the complete lists of inequiv-alent PDM systems presented in [19] for d = 2 and [20] for d = 3. Moreover, in manycases a sufficiently extended symmetry induces integrability or exact solvability of thesystem, and just this aspect will be discussed in the present paper.It will be shown that all PDM systems admitting six parametric Lie groups of symme-tries or more extended symmetries are exactly solvable. Moreover, the complete sets ofsolutions of the corresponding stationary PDM Schr¨odinger equations will be presentedexplicitly. 1here exist a tight connection between the complete solvability and various typesof higher symmetries and supersymmetries. We will see that extended Lie symmetriesalso can cause the exact solvability. More-over, the systems admitting extended Liesymmetries in many cases are supersymmetric and superintegrable. In paper [20] we present the group classification of PDM Schr¨odinger equations Lψ ≡ (cid:18) i ∂∂t − H (cid:19) ψ = 0 , (2.1)where H is the PDM Hamiltonian of the following generic form H = (cid:0) m α p a m β p a m γ + m γ p a m β p a m α (cid:1) + ˆ V , p a = − i ∂∂x a . (2.2)Here m = m ( x ) and ˆ V = ˆ V ( x ) are the mass and potential depending on spatial variables x = ( x , x , x ), and summation w.r.t. the repeating indices a is imposed over the values a = 1 , ,
3. In addition, α , β and γ are the ambiguity parameters satisfying the condition α + β + γ = − H = p a f p a + V, (2.3)where V = ˆ V + ( α + γ ) f aa + αγ f a f a f (2.4)with f = m , f a = ∂f∂x a and f aa = ∆ f = ∂f a ∂x a .In the following text just representation (2.4) will be used.In accordance with [20] there is a big variety of Hamiltonians (2.4) generating non-equivalent continuous point symmetries of equation (2.2). The corresponding potentialand mass terms are defined up to arbitrary parameters or even up to arbitrary functions.In the present paper we consider the PDM systems defined up to arbitrary parame-ters. Just such systems admit the most extended Lie symmetries. Using the classificationresults presented in [18] and [20] we enumerate these systems in the following Table 1,where ϕ = arctan x x and the other Greek letters denote arbitrary constants parameters,which are supposed not to be zero simultaneously. Moreover, λ and ω are either real orimaginary, the remaining parameters are real.2able 1. PDM systems with extended Lie symmetries.No Inverse mass f Potential V Symmetries1 (cid:0) r + 1 (cid:1) − r M , M , M ,M , M , M (cid:0) r − (cid:1) − r M , M , M ,M , M , M x ν ln( x ) P , P , M , D + νt r κx + λ ˜ r P + κt, D + i t∂ t , M x λx + κx P + κt, P , D + i t∂ t x σ +23 κx σ P , P , M , D + i σt∂ t ,σ = 0 , , −
27 ˜ r σ +2 e λϕ κ ˜ r σ e λϕ M + i λt∂ t , P ,D + i σt∂ t , σ = 08 ˜ r λ ϕ + µϕ + ν ln(˜ r ) B , B ,D + νt, P r e σϕ κ e σϕ + ω e − σϕ N , N , P , D, K r ν ln( r ) + λ ln( r ) B , B , L , L , L r σ κr σ + ω r − σ N , N , L , L , L The symmetry operators presented in column 4 of the table are given by the followingformulae: P i = p i = − i ∂∂x i , D = x n p n − ,M ij = x i p j − x j p j , M i = (cid:0) K i + P i (cid:1) , M i = (cid:0) K i + P i (cid:1) ,B = λ sin( λt ) M (cid:0) λ ϕ + ν (cid:1) cos( λt ) , B = ∂∂t B ,B = sin( λt ) D − cos( λt ) (cid:0) λ ln( r ) + νλ (cid:1) , B = ∂∂t B ,N = ω cos( ωσt ) L − sin( ωσt ) (cid:0) i ∂ t − ω e − σ Θ (cid:1) , N = ∂∂t N ,N = ω cos( ωσt ) D + sin( ωσt ) (cid:0) i ∂ t − ω r − σ (cid:1) , N = ∂∂t N , (2.5)where K i = x n x n p i − x i D and indices i , j , k , n take the values 1, 2, 3. In addition, allthe presented systems admit symmetry operators P = i ∂∂t and the unit operator, thelatter is requested to obtain the closed symmetry algebras.Rather surprisingly, all systems presented in Table 1 (except ones given in items4 and 5 with κ = 0) are exactly solvable. In the following sections we present theirexact solutions. To obtain these solutions we use some nice properties of the considered3ystems like superintegrability and supersymmetry with shape invariance. Let us remindthat the quantum mechanical system is called superintegrable if it admits more integralsof motion than its number of degrees of freedom. In accordance with Table 1 we canindicate 11 inequivalent PDM systems which are defined up to arbitrary parameters andadmit Lie symmetry algebras of dimension five or higher. Notice that the systems fixedin items 4 and 5 admit five dimension symmetry algebras while the remaining systemsadmit more extended symmetries. First we consider systems whose mass and potential terms are fixed, i.e., do not includearbitrary parameters. These systems are presented in items 1, 2 of Table 1 and othersprovided the mass does not depends on parameters and parameters of the potential aretrivial. so (4). Consider Hamiltonian (2.3) with functions f and V presented in item 1 of Table 1: H = p a (cid:0) r (cid:1) p a − r . (3.1)The eigenvalue problem for this Hamiltonian can be written in the following form: Hψ = 2 Eψ, (3.2)where E are yet unknown eigenvalues.Equation (3.2) admits six integrals of motion M AB , A, B = 1 , , ,
4, presented inequation (2.5). Let us write them explicitly M ab = x a p b − x b p a , M a = (cid:0) r − (cid:1) p a − x a x b p b + x a . (3.3)Operators (3.3) form a basis of algebra so (4). Moreover, the first Casimir operatorof this algebra is proportional to Hamiltonian (3.1) up to the constant shift C = M AB M AB = ( H − , while the second Casimir operator C = ε ABCD M AB M CD appears to be zero.Thus like the Hydrogen atom system (3.2) admits six integrals of motion belongingto algebra so (4) and is maximally superintegrable.Using our knowledge of unitary representations of algebra so (4) is possible to findeigenvalues E algebraically: E = 4 n + 5 , (3.4)4here n = 0 , , , . . . are natural numbers.To find the eigenvectors of Hamiltonian (3.1) corresponding to eigenvalues (3.4) weuse the rotation invariance of (3.2) and separate variables. Introducing spherical vari-ables and expanding solutions via spherical functions ψ = 1 r X l,m φ lm ( r ) Y lm (3.5)we come to the following equations for radial functions (cid:18) − (cid:0) r + 1 (cid:1) (cid:18) ∂ ∂r − l ( l + 1) r (cid:19) − r (cid:0) r + 1 (cid:1) ∂∂r − r (cid:19) ϕ lm = (cid:0) n + 1 (cid:1) ϕ lm , where l = 0 , , , . . . are parameters numerating eigenvalues of the squared orbital mo-mentum. The square integrable solutions of these equations are ϕ lm = C nlm (cid:0) r + 1 (cid:1) − n − r l +1 F (cid:0) [ A, B ] , [ C ] − r (cid:1) , (3.6)where A = − n + l + 1 , B = − n + , C = l + . F ( · · · ) is the hypergeometric function and C nlm are integration constants. Solutions (3.6)tend to zero at infinity provided n is a natural number and l ≤ n − so (1 , The next Hamiltonian we consider corresponds to functions f and V presented in item 2of Table 1. The related eigenvalue problem includes the following equation Hψ ≡ − (cid:0) ∂ a (cid:0) − r (cid:1) ∂ a + 6 r (cid:1) ψ = Eψ. (3.7)Equation (3.7) admits six integrals of motion M µν , µ, ν = 0 , , , , given by equation(2.5), which can be written explicitly in the following form M ab = x a p b − x b p a ,M a = (cid:0) r + 1 (cid:1) p a − x a x b p b + x a , a, b = 1 , , . (3.8)These operators form a basis of algebra so (1 , C = M ab M ab − M a M a = ( H + 9) , (3.9)5hile the second one appears to be zero.Using our knowledge of irreducible unitary representations of Lorentz group we findeigenvalues of C and C in the form [22, 23]: c = 1 − j − j , c = 2i j j , where j and j are quantum numbers labeling irreducible representations. Since the sec-ond Casimir operator C is trivial, we have c = j = 0. So there are two possibilities [22]:either j is an arbitrary imaginary number, and the corresponding representation be-longs to the principal series, or j is a real number satisfying | j | ≤
1, and we come tothe subsidiary series of IRs. So j = i λ, c = 1 − j = λ + 1 , (3.10)where λ is an arbitrary real number, or, alternatively,0 ≤ j ≤ , c = 1 − j . (3.11)In accordance with (3.9) the related eigenvalues E in (3.7) are E = − − j . (3.12)In view of the rotational invariance of equation (3.7) it is convenient to representsolutions in form (3.5). As a result we obtain the following radial equations (cid:18) − (cid:0) r − (cid:1) (cid:18) ∂ ∂r − l ( l + 1) r (cid:19) − r (cid:0) r − (cid:1) ∂∂r − r (cid:19) ϕ lm = ( ˜ E + 4) ϕ lm . (3.13)The general solution of (3.13) is ϕ lm = (cid:0) − r (cid:1) − − k (cid:0) C klm r l +1 F (cid:0) [ A, B ] , [ C ] , r (cid:1) + ˜ C klm r − l F (cid:0) [ ˜ A, ˜ B ] , [ ˜ C ] , r (cid:1)(cid:1) , (3.14)where A = − k + l + 1 , B = − k + , C = l + , ˜ A = − k − l, ˜ B = − k + , ˜ C = − l, k = p − ˜ E − r = 1. However, for ˜ C klm = 0 and k = j the solutions are normalizablein some specific metric [18].Thus the system presented in item 7 of Table 1 is exactly solvable too. The corre-sponding eigenvalues and eigenvectors are given by equations (3.10), (3.11), (3.12) and(3.14) correspondingly. 6 .3 Scale invariant systems. Consider one more PDM system which is presented in item 3 of the table and includesthe following Hamiltonian: Let us note that the free fall effective potential appears alsoone more system specified in Table 1. Thus, considering the inverse mass and potentialspecified in item 3 we come to the following Hamiltonian H = − (cid:18) x ∂∂x x ∂∂x + x ∂∂x + x (cid:18) ∂ ∂x + ∂ ∂x (cid:19)(cid:19) + ν ln( x ) . (3.15)Equation (3.7) with Hamiltonian given in (3.15) can be easily solved by separation ofvariables in Cartesian coordinates. Expanding the wave function ψ via eigenfunctionsof integrals of motion P and P : ψ = exp( − i( k x + k x ))Φ( k , k , x ) (3.16)and introducing new variable y = ln( x ) we come to the following equation for Φ =Φ( k , k , x ): − ∂ Φ ∂y + (cid:0)(cid:0) k + k (cid:1) exp(2 y ) + 2 νy (cid:1) Φ = ˜ E Φ (3.17)where ˜ E = 2 E − .Here we consider the simplest version of equation (3.17) when parameter ν is trivial: − ∂ Φ ∂y + (cid:0) k + k (cid:1) exp(2 y )Φ = ˜ E Φ . (3.18)This equation is scale invariant and can be easily solved. Its square integrable solutionsare given by Bessel functionsΨ = C Ek k K i √ ˜ E (cid:16)q k + k ln( x ) (cid:17) , where C Ek k are integration constants and ˜ E are arbitrary real parameters.It is interesting to note that there are rather non-trivial relations between the resultsgiven in the present and previous sections. Equation (3.18) admits six integrals of motionwhich are nothing but the following operators P , P , K , K , M , D, (3.19)which are presented in equations (2.5).Like operators (3.8) integrals of motion (3.19) form a basis of the Lie algebra ofLorentz group, and we again can find the eigenvalues of Hamiltonian (3.18) algebraicallyby direct analogy with the above. We will not present this routine procedure since there7xist strong equivalence relations between Hamiltonians (3.18) with zero ν and (3.1).To find them we note that basis (3.19) is equivalent to the following linear combinationsof the basis elements: M , M , M , M M , M , (3.20)whose expressions via operators (3.19) are given by equation (2.5). To reduce (3.20) tothe set (3.8) it is sufficient to change subindices 4 to 3, i.e., to make the rotation in theplane 43. The infinitesimal operator for such rotation is given by the following operator M = ( K + P ) = (cid:0) r − (cid:1) p − x x b p b + x , which belongs to the equivalence group of equations. Solving the corresponding Lieequations and choosing the group parameter be equal π we easily find the requestedequivalence transformations.One more scale invariant system is presented in item 8 where all parameters ofpotential are zero. The relation Hamiltonian looks as follows: H = − ˜ r ∂∂x α ˜ r ∂∂x α − x α ∂∂x α − ˜ r ∂ ∂x , α = 1 , . (3.21)Considering the eigenvalue problem for (3.21) it is convenient to use the cylindricalvariables˜ r = q x + x , ϕ = arctan x x , x = z (3.22)and expand solutions via eigenfunctions of M and P = − i ∂∂z :Ψ = exp[i( κϕ + ωz )]Φ κω (˜ r ) , κ = 0 , ± , ± , ..., −∞ < ω < ∞ . (3.23)As a result we come to the following equations for radial functions Φ = Φ κω (˜ r ): − (cid:18) ˜ r ∂∂ ˜ r ˜ r ∂∂ ˜ r + ˜ r ∂∂ ˜ r + ω (cid:19) Φ = ( ˜ E − κ . )ΦSquare integrable (with the weight ˜ r ) solutions of this equation are:Φ κω = 1˜ r J α ( ω ˜ r ) , α = κ + 1 − ˜ E (3.24)where J α ( ω ˜ r ) is Bessel function of the first kind. Functions (3.24) are normalizable anddisappear at ˜ r = 0 provided α ≤
0. The rescaled energies ˜ E continuously take the values κ ≤ ˜ E ≤ ∞ .The last scale invariant system which we have to consider is fixed in item 10 where ν = λ = 0 We will do it later in Section 8 Systems defined up to arbitrary parameters.
In previous section we present exact solutions for systems with fixed potential and massterms. In the following we deal with the systems defined up to arbitrary parameters.
Let us consider equation (2.1) with f and V are functions fixed in item 10 of Table 1,i.e., i ∂ψ∂t = (cid:18) − ∂∂x a r ∂∂x a + ν ln( r ) + λ r ) (cid:19) ψ. These equations admit extended symmetries Lie symmetries (whose generators areindicated in the table) being invariant w.r.t. six-parametrical Lie group. Let us showthat they also admit hidden supersymmetries.In view of the rotational invariance and symmetry of the considered equations withrespect to shifts of time variable, it is reasonable to to search for their solutions inspherical variables, i.e., in the following formΨ = e − iEt R lm ( r ) Y lm ( ϕ, θ ) , (4.1)where ϕ and θ are angular variables and Y lm ( ϕ, ϕ ) are spherical functions, i.e., eigen-vectors of L = L + L + M and M . As a result we come to the following radialequations (cid:18) − r ∂R lm ∂r r ∂R lm ∂r − r ∂R lm ∂r + l ( l + 1) + ν ln( r ) + λ r ) (cid:19) R lm = 2 ER lm . (4.2)Introducing new variable y = √ r ) we can rewrite equation (4.2) in the followingform: (cid:18) − ∂ ∂y + l ( l + 1) + νy + λ y (cid:19) R lm ( y ) = ˜ ER lm ( y ) , (4.3)where ˜ E = E − .Let λ = 0 then equation (4.3) is reduced to the 1D harmonic oscillator up to theadditional term l ( l + 1) The admissible eigenvalues ˜ E are given by the following formula˜ E = n + l ( l + 1) . where n is a natural number. The corresponding eigenfunctions are well known andwe will not presented them here. The same is true for supersymmetric aspects of theconsidered system.If parameter λ is equal to zero then (4.3) reduces to equation with free fall potentialslightly modified by the term l ( l + 1). The corresponding solutions can be found intextbooks devoted to quantum mechanics. 9 .2 The systems with potentials equivalent to 3d oscillator. Consider now the system represented in item 11 of the table. The corresponding equation(2.1) takes the following form:i ∂ψ∂t = (cid:18) − ∂ a r σ +2 ∂ a + κr σ + ω r σ (cid:19) ψ. (4.4)Like in previous section we represent the wave function in the form given in (4.1) andcame to the following radial equation − r σ +2 ∂ R lm ∂r − (2 σ + 4) r σ +1 ∂R lm ∂r + (cid:0) r σ ( l ( l + 1) + κ ) + ω r − σ (cid:1) R lm = 2 ER lm . (4.5)Using the Liouville transform r → z = r − σ , R lm → ˜ R lm = z σ +32 σ R lm , we reduce (4.5) to the following form − σ ∂ ˜ R lm ∂z + (cid:18) l ( l + 1) + δz + ω z (cid:19) ˜ R lm = 2 E ˜ R lm , (4.6)where δ = ( σ + 1)( σ + 3) + 2 κ .Equation (4.6) describes a deformed 3d harmonic oscillator including two deformationparameters, namely, σ and κ .Let2 κ = − σ − σ − , (4.7)then equation (4.6) is reduced to the following form H l ˜ R lm ≡ (cid:18) − σ ∂ ∂z + (2 l + 1) − σ z + ω z (cid:19) ˜ R lm = 2 E ˜ R lm . (4.8)Equation (4.8) is shape invariant. Hamiltonian H r can be factorized H l = a + l a l − C l , (4.9)where a = − σ ∂∂z + W, a + = σ ∂∂z + W,W = 2 l + 1 + σ z + ωz, C l = ω (2 l + 2 σ + 1) . H l of Hamiltonian (4.9) has the following propertyˆ H l ≡ a l a + l + C l = H l + σ + C l . Thus our Hamiltonian is shape invariant.Thus to solve equation (4.8) we can use the standard tools of SUSY quantum me-chanics and find the admissible eigenvalues in the following form E n = ω (cid:0) nσ + l + σ + (cid:1) = ω (cid:0) n + l + (cid:1) + δω (2 n + 1) , (4.10)where δ = σ − δ .For equation (4.6) we obtain in the analogous way E n = ω (cid:0) σ (2 n + 1) + p (2 l + 1) + ˜ κ (cid:1) , (4.11)where ˜ κ = 8( κ + 1) + σ ( σ + 3) . The related eigenvectors are expressed via the confluenthypergeometric functions F : R n = e − ωrσ σ r σn − Enω F (cid:18) − n, E n σω − n, ωσ r − σ (cid:19) , where n is integer and E n is eigenvalue (4.11). The next system which we consider is specified by the inverse mass and potential pre-sented in item 8 of the table. The corresponding Hamiltonian is: H = p a r p a + λ ϕ + σϕ + ν ln(˜ r ) . The corresponding eigenvalue equation is separable in cylindrical variables, thus it isreasonable to represent the wave function as follows ψ = Ψ(˜ r )Φ( ϕ ) exp( − i kx ) . (4.12)As a result we obtain the following equations for radial and angular variables (cid:0) − ˜ r∂ ˜ r ˜ r∂ ˜ r − ˜ r∂ ˜ r + ν ln(˜ r ) + k ˜ r − µ (cid:1) Ψ(˜ r ) = 0 (4.13)and (cid:18) − ∂ ∂ϕ + λ ϕ + σϕ − µ (cid:19) Φ( ϕ ) = 0 , (4.14)11here µ is a separation constant.For λ nonzero equation (4.14) is equivalent to the Harmonic oscillator. The specificityof this system is that, in contrast with (4.3), it includes angular variable ϕ whose originis 0 ≤ ϕ ≤ π. (4.15)For trivial λ our equation (4.14) is reduced to equation with free fall potential, butagain for the angular variable satisfying (4.15).The radial equation (4.13) is simple solvable too. In the case k = 0 we again cometo the free fall potential. The next system we consider is specified by the inverse mass and potentials representedin item 9 of Table 1. The corresponding Hamiltonian is H = − ∂∂x a ˜ r e σϕ ∂∂x a + κ e σϕ + ω − σϕ . Introducing again the cylindric variables and representing the wave function in theform (4.12) we come to the following equations for the radial and angular variables (cid:18) − (cid:18) ∂ ∂y + ∂∂y (cid:19) + µ + k e y (cid:19) Ψ(˜ r ) = µ Ψ(˜ r )and (cid:18) − e σϕ (cid:18) ∂ ∂ϕ + κ − µ (cid:19) + ω − σϕ (cid:19) Φ( ϕ ) = ˜ E Φ( ϕ ) . (4.16)Dividing all terms in (4.16) by exp( σϕ ) we obtain the following equation: (cid:18) − (cid:18) ∂ ∂ϕ + κ − µ (cid:19) + ω − σϕ (cid:19) Φ( ϕ ) = e − σϕ ˜ E Φ( ϕ ) . or (cid:18) − (cid:18) ∂ ∂ϕ (cid:19) + ω − σϕ − ˜ E e − σϕ (cid:19) Φ( ϕ ) = ˆ E Φ( ϕ ) , (4.17)where we denote ˆ E = µ − κ .Formula (4.17) represents the Schr¨odinger equation with Morse potential. This equa-tion is shape invariant and also can be solved using tools of SUSY quantum mechanics.We demonstrate this procedure using another system.12onsidering the mass and potential presented in item 6 of Table 1 we come to thefollowing Hamiltonian H = 12 p a x σ +23 p a + κx σ . Equation (3.7) with Hamiltonian (3.15) can be solved by separation of variables inCartesian coordinates. Expanding the wave function ψ via eigenfunctions of integrals ofmotion P and P in the form (3.16) and introducing new variable y = ln( x ) we reducethe problem to the following equation for Φ( k , k , x ): (cid:18) − ∂∂x x σ +23 ∂∂x + x σ +23 k + 2 κx σ (cid:19) Φ = 2 E Φ (4.18)were k = k + k .Dividing all terms in (4.18) by x σ we can rewrite it in the following form: (cid:18) − ∂ ∂y − ( σ + 1) ∂∂y − E exp( − σy ) + k exp(2 y ) + 2 κ (cid:19) Φ = 0In the particular case σ = 2 we again come to the equation with Morse effectivepotential.One more system which can be related to Morse potential is represented in item 7and include the following Hamiltonian: H = 12 p a exp( λϕ )˜ r σ +2 p a + ν exp( λϕ )˜ r σ . The corresponding equation (3.7) is separable in the cylindrical variables (3.22) provided σ · λ = 0 and again includes the Morse effective potential.Let us return to equation (4.5) and solve it using approach analogous to the presentedabove. In other words, we will change the roles of eigenvalues and coupling constants.First we divide all terms in by r σ and obtain − r ∂ R lm ∂r − (2 σ + 4) r ∂R lm ∂r + (cid:0) ω r − σ + µr − σ (cid:1) R lm = εR lm , (4.19)where ε = − l ( l + 1) − κ, µ = − E. (4.20)Applying the Liouville transform r → ρ = ln( r ) , R lm → ˜ R lm = e − σ +32 R lm
13e reduce (4.19) to a more compact form H ν ˜ R lm ≡ (cid:18) − ∂ ∂ρ + ω e − σρ + (2 ων + ωσ )e − σρ (cid:19) ˜ R lm = ˆ ε ˜ R lm , (4.21)whereˆ ε = ε − (cid:18) σ + 32 (cid:19) , ν = µ ω − σ . (4.22)Like (4.17) equation (4.21) includes the familiar Morse potential and so is shapeinvariant. Indeed, denoting µ = 2 ω ( ν + σ ) we can factorize hamiltonian H ν like it wasdone in (4.9) where index l should be changed to ν and W = ν − ω e − aρ , C ν = ν and the shape invariance is easy recognized.To find the admissible eigenvalues ε and the corresponding eigenvectors we can di-rectly use the results presented in paper [24], see item 4 of Table 4.1 thereˆ ε = ˆ ε n = − ( ν − nσ ) , (cid:0) ˜ R lm (cid:1) n = y νσ − n e − y L νσ − n ) n ( y ) , where y = ωσ r − σ .Thus we find the admissible values of ˆ ε n . Using definitions (4.20) and (4.22) we canfind the corresponding values of E which are in perfect accordance with (4.11). The results presented above in Section 2 include the complete list of continuous symme-tries which can be admitted by PDM Schr¨odinger equations, provided these equationsare defined up to arbitrary parameters.It is important to note that the list of symmetries presented in the fourth columnof the table is valid only for the case of nonzero parameters defining the potential andmass terms. If some (or all) of these parameters are trivial, the corresponding PDMSchr¨odinger equation can have more extended set of symmetries. For example, it isthe case for the potential and PDM presented in item 3 of the table, compare the listof symmetries presented in column 4 with (24). The completed list of non-equivalentsymmetries can be found in [13] which generalizes the Boyer results [3] to the case ofPDM Schr¨odinger equations. As other extensions of results of [3] we can mention thegroup classification of the nonlinear Schr¨odinger equations [15] and the analysis of itsconditional symmetries [6].Thanks to their extended symmetries the majority of the presented systems is exactlysolvable. In Sections 3 and 4 we present the corresponding solutions explicitly anddiscuss supersymmetric aspects of some of them. However, two of the presented systems14whose mass and potential are presented in items 4 and 5 of Table 1) are not separable,if arbitrary parameter κ is nonzero. And just these systems have ”small” symmetry,admitting five parametrical Lie groups. For κ equal to zero these systems are reducedto particular cases presented in items 6 and 11.On the other hand, all systems admitting six- or higher-dimensional Lie symmetryalgebras are separable and exactly solvable. In addition to the symmetry under thesix parameter Lie group, equation (32) (which we call deformed 3d isotropic harmonicoscillator) possesses a hidden dynamical symmetry w.r.t. group SO(1, 2). The effectiveradial Hamiltonian is shape invariant, and its eigenvalues can be found algebraically.In spite on the qualitative difference of its spectra (37) and (38) of the standard 3doscillator, it keeps the main supersymmetric properties of the latter. We note that theshape invariance of PDM problems usually attends their extended symmetries. References [1] C. P. Boyer, ”The maximal kinematical invariance group for an arbitrary potential”,Helv. Phys. Acta, , 450–605 (1974).[2] C. R. Hagen, ”Scale and conformal transformations in Galilean-invariant conformalfield theory”, Phys. Rev. D , 377–388 (1972).[3] U. Niederer, ”The maximal kinematical invariance group of the free Schr¨odingerequations”, Helv. Phys. Acta, , 802–810 (1972).[4] R. L. Anderson, S. Kumei, C. E. 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