New Exactly Solvable Two-Dimensional Quantum Model Not Amenable to Separation of Variables
aa r X i v : . [ h e p - t h ] O c t NEW EXACTLY SOLVABLE TWO-DIMENSIONALQUANTUM MODEL NOT AMENABLE TO SEPARATION OFVARIABLESM.V. Iof fe , D.N. Nishnianidze , , P.A. Valinevich Saint-Petersburg State University, 198504 St.-Petersburg, Russia Akaki Tsereteli State University, 4600 Kutaisi, Republic of Georgia
The supersymmetric intertwining relations with second order superchargesallow to investigate new two-dimensional model which is not amenable tostandard separation of variables. The corresponding potential being thetwo-dimensional generalization of well known one-dimensional P¨oschl-Tellermodel is proven to be exactly solvable for arbitrary integer value of parameter p : all its bound state energy eigenvalues are found analytically, and thealgorithm for analytical calculation of all wave functions is given. The shapeinvariance of the model and its integrability are of essential importance toobtain these results.PACS numbers: 03.65.-w, 03.65.Fd, 11.30.Pb The beautiful idea of supersymmetry (SUSY) was first introduced [1] and developed inQuantum Field Theory and Elementary Particle Theory at the seventies of the last century.During these years supersymmetry became one of the most popular and promising branchesof modern High Energy Physics [2].Supersymmetry was also studied in the simplest toy model of (0+1) Quantum Field The-ory (i.e. in nonrelativistic Quantum Mechanics) in order to clarify some delicate problemsof spontaneous supersymmetry breaking [3]. Very soon, this by-product of supersymmet-rical Quantum Field Theory became a new independent tool to study many problems inQuantum Mechanics itself [4]. In particular, the notions of SUSY intertwining relations [4],[5] and of shape invariance [6] provided both new methods to derive some old results andto obtain new interesting results. As an example, all previously known one-dimensionalexactly solvable potentials were reproduced as potentials obeying the shape invariance [7].In its turn, SUSY intertwining relations were successfully used [8], [9], [10], [11] in two-dimensional Quantum Mechanics to obtain a variety of partially (quasi-exactly) solvable a a e-mail: m.ioff[email protected] b e-mail: [email protected] c e-mail: [email protected] a By definition, partial (quasi-exact) solvability of the model means that a part of its energy spectrum V ( ~x ; A, B, p ) depending on parameters
A, B, p threesteps will be done. First, to find such exclusive value of parameter (actually, p = 1 , ) thatinitial Hamiltonian H ( p = 1) does allow conventional separation of variables. Second, usingSUSY intertwining relations and shape invariance, to build eigenfunctions for Hamiltonians H ( p ) , p = 2 , , ..., which are not already amenable to separation of variables. And finally,to prove that all constructed wave functions are normalizable and that no extra levels exist.The structure of the paper is the following. In Section 2, the model and its main propertiesare formulated, and the scheme of investigation is reviewed. The separation of variablesfor first Hamiltonian H ( p = 1) is performed and some delicate properties of potential arediscussed in Section 3. The zero modes of the supercharge are built in Section 4, and theyare used for construction of wave functions in Section 5. Normalizability of wave functionsis studied in Section 6, where the absence of any other bound states was proven. A fewexamples of wave functions for low values of parameter p are given in Section 7. Conclusionsincludes the comparison of obtained results with the limiting case which is explicitly solvable.Rather cumbersome calculation of coefficients necessary for wave functions and spectrum arepresented in Appendix. and corresponding wave functions are known. Such models take up an intermediate place between exactlysolvable ones and models with unknown spectra [12]. Formulation of the Model and the General Scheme.
We consider the intertwining relations of the form Q − H = e HQ − ; HQ + = Q + e H, (1)where H and e H are two-dimensional Hamiltonians of the Schr¨odinger type H = − ( ∂ + ∂ ) + V ( x , x ); e H = − ( ∂ + ∂ ) + e V ( x , x ) , (2)and intertwining operators Q ± are second-order differential operators. A number of modelsof this kind were investigated in the series of papers [8], [9], [10], [11]. In [14], it was proventhat one of them - the generalized two-dimensional Morse - possesses exact solvability. Herewe shall show that one more model [11] involved in intertwining relations (1) is also exactlysolvable. It reads: H ( p ) = − ∂ − ∂ − p ( p − (cid:0) cosh − ( x + ) + cosh − ( x − ) (cid:1) + k (cid:0) sinh − ( x ) − cosh − ( x ) (cid:1) + k (cid:0) cosh − ( x ) − sinh − ( x ) (cid:1) , (3) e H ( p ) = − ∂ − ∂ − p ( p + 1) (cid:0) cosh − ( x + ) + cosh − ( x − ) (cid:1) + k (cid:0) sinh − ( x ) − cosh − ( x ) (cid:1) + k (cid:0) cosh − ( x ) − sinh − ( x ) (cid:1) , (4) Q ± = ∂ − ∂ ± p (tanh( x + ) + tanh( x − )) ∂ ± p (tanh( x − ) − tanh( x + )) ∂ + 4 p tanh( x + ) tanh( x − )+ k (cid:0) sinh − ( x ) + cosh − ( x ) (cid:1) + k (cid:0) cosh − ( x ) + sinh − ( x ) (cid:1) , (5)where p and k , are real parameters, so far arbitrary, and x ± ≡ x ± x . The Hamiltoniansin (3), (4) can be represented in the form: H P − T ( x ) + H P − T ( x ) + f ( x , x ) , (6)where H P − T ( x ) are well known one-dimensional P¨oschl-Teller Hamiltonians, and f ( x , x )- specific term mixing x and x variables in potentials. Due to this expansion, potentials V ( x , x ) , e V ( x , x ) may be considered [11] as a two-dimensional generalization of P¨oschl-Teller potential. These models are shape-invariant [6] with respect to the parameter p : H ( p + 1) = e H ( p ) . (7)Properties (1) and (7) will be essential for the proof of exact solvability of the model forpositive integer values of p . 3e remark that, by construction, these models are integrable since from intertwiningrelations (1) it follows that[ H, R ] = 0 , [ e H, e R ] = 0; R = Q + Q − , e R = Q − Q + , (8)with symmetry operators of fourth order in momenta.The general scheme to determine the spectrum of the model (3) could be adopted fromthe paper [14], where the full spectrum of two-dimensional generalization of Morse poten-tial [9], [10] was found. In the present context, the plan of construction could be the fol-lowing: we start with H ( p = 1) and find all normalizable solutions Ψ( ~x ; p = 1) for thecorresponding Schr¨odinger equation (Section 3) as far as it is amenable to separation of vari-ables. Then, by means of intertwining relations (1), we find eigenfunctions e Ψ( ~x ; p = 1)of e H (1) . In general, they might be of two types [9]: some of them are inherited from H (1) as: e Ψ( ~x ; 1) = Q − (1)Ψ( ~x ; 1) , and others are zero modes of the intertwining opera-tor: Q + (1) e Ψ( ~x ; 1) = 0 . In such a way we obtain all eigenfunctions of H (2) . Due to theshape-invariance (7) of the model, Ψ( ~x ; p + 1) = e Ψ( ~x ; p ) , and therefore, we have calcu-lated already Ψ( ~x ; 2) . Following this strategy step by step, we expect to find the eigenfunc-tions and eigenvalues for the Hamiltonians e H ( p ) = H ( p + 1) with arbitrary integer values p = 1 , , .... At each step, the full variety of eigenfunctions of H ( p + 1) will belong to one oftwo classes: 1) each normalizable wave function Ψ( ~x ; p ) leads to normalizable wave functionΨ( ~x ; p + 1) ≡ e Ψ( ~x ; p ) = Q − ( p )Ψ( ~x ; p ); 2) the same Hamiltonian H ( p + 1) has also somenumber of extra normalizable functions which are specific linear combinations of zero modesΩ( ~x ; p ) of the operator Q + ( p ) . We shall see below that this plan has to be modified suitablyfor the case of our present model, but the main ideas will be analogous to that of [9], [10],[14]. H ( p = 1) . For the Hamiltonian H (1) the standard procedure of separation of variables in Carte-sian coordinates can be applied. Looking for the solutions of the Schr¨odinger equation H (1)Ψ( ~x ; 1) = E Ψ( ~x ; 1) in the form Ψ( ~x ; 1) = η ( x ) ρ ( x ) , one obtains two one-dimensionalequations for unknown functions ρ , η − η ′′ ( x ) − (cid:18) k cosh x + k sinh x (cid:19) η ( x ) = εη ( x ); (9) − ρ ′′ ( x ) + (cid:18) k cosh x + k sinh x (cid:19) ρ ( x ) = ˜ ερ ( x ) , (10)where prime denotes the derivative of the function with respect to its argument, ε + ˜ ε = E is the energy value for H (1) , and both ε and ˜ ε must be negative for the discrete part of4he spectrum. Thus, we have to consider solutions of one-dimensional Scr¨odinger equations(9)-(10) with P¨oschl-Teller potentials V P − T ( x ). It is convenient to replace parameters k , k by A, B according to k ≡ B ( B − k ≡ − A ( A − . Avoiding the case of fall onto center[15], we shall restrict ourselves with reasonably attracting singularity in V P − T ( x ) , V P − T ( x )with coefficients k ∈ ( − / , , k ∈ (0 , / , i.e. it is sufficient to take A, B ∈ (0 , / . The substitution η ( x ) = sinh A ( x ) cosh B ( x ) F ( x ) , and the subsequent change of variable x to z ≡ − sinh ( x ) , turns (9) into the hypergeometric equation for the function F ( z ) : z (1 − z ) d F ( z ) dz + (cid:18) A + 12 − ( A + B + 1) z (cid:19) dF ( z ) dz + (cid:18) −
14 ( A + B ) − ε (cid:19) F ( z ) = 0 . The pair of independent solutions for the given value of ε reads (see 2.3.1(1) in [16]): η (1) ε ( x ) = sinh A ( x ) cosh B ( x ) F (cid:18) A + B + √− ε , A + B − √− ε A + 12 ; − sinh ( x ) (cid:19) ; (11) η (2) ε ( x ) = sinh − A ( x ) cosh B ( x ) F (cid:18) − A + B + √− ε , − A + B − √− ε − A ; − sinh ( x ) (cid:19) . (12)The similarity of expressions (11) and (12) reflects the obvious symmetry of potential in(9) under A → (1 − A ) . The potential under consideration obeys also the similar symmetryunder B → (1 − B ) . But the corresponding independent solutions are related to solutions(11), (12) according to relations between hypergeometric functions (see 2.1.4(23) in [16]).The formulae analogous to (11) and (12) hold also for ρ (1) , (2)˜ ε ( x ) , but with the necessarychanges A → B, B → A and ε → ˜ ε. To provide the normalizability of Ψ( ~x ; 1) , both ρ and η must be normalizable. To ana-lyze the possible bound states of H ( p = 1) it will be sufficient to consider the asymptoticbehaviour for large | x | , | x | . The general solution η ε ( x ) is a linear combination: η ε ( x ) = α η (1) ε ( x ) + α η (2) ε ( x ) (13)with arbitrary constants α , α . The asymptotic behaviour of the analytic continuation ofhypergeometric functions for large z = − sinh x (see 2.10(2), 2.10(5) in [16]) reads: F ( a, b ; c ; z ) = B ( a, b, c ) (cid:20) ( − z ) − a + O (( − z ) − a − ) (cid:21) + B ( a, b, c ) (cid:20) ( − z ) − b + O (( − z ) − b − ) (cid:21) , (14)where constants B , are expressed in terms of Gamma functions: B ( a, b, c ) = Γ( c )Γ( b − a )Γ( b )Γ( c − a ) ; B ( a, b, c ) = Γ( c )Γ( a − b )Γ( a )Γ( c − b ) . η ε ( x ) : proportional to ( − z ) −√− ε/ (1 + O ( z − )) and proportional to ( − z ) + √− ε/ (1 + O ( z − )) , correspondingly. In order to forbid the growing term in wave function, one has to requirethe coefficient to vanish: α B (cid:18) A + B + √− ε , A + B − √− ε , A + 12 (cid:19) ++ α B (cid:18) − A + B + √− ε , − A + B − √− ε , − A (cid:19) = 0 . In general, there are two options to fulfil this requirement: α = 0; B (cid:18) A + B + √− ε , A + B − √− ε , A + 12 (cid:19) = 0; α = 0; B (cid:18) − A + B + √− ε , − A + B − √− ε , − A (cid:19) = 0;In the case of arbitrary A, B, these conditions can be achieved by means of suitable choicesof energy values ε, due to Gamma functions in denominators of coefficient B : arguments a or ( c − b ) of these Gamma functions must be equal − n with n = 0 , .... Just this conditionmight give the energies of bound states. But one can check easily, that in our present case ofparameters
A, B ∈ (0 , / , these conditions can not be fulfilled for positive values of √− ε .Thus, we are not able to kill the growing terms in asymptotic of η ε and ρ ˜ ε . Therefore, theHamiltonian H ( p = 1) has no bound states because of asymptotic behaviour at large | x | , andthe first source for construction of eigenfunctions of e H (1) : e Ψ( ~x ; 1) = Q − (1)Ψ( ~x ; 1) - doesnot work. We stress that this statement depends crucially on a chosen region for values of A, B, which in its turn was dictated by conditions on both potentials V , V , simultaneously.Taken separately, these Hamiltonians would have bound states, but for different values of A, B.
We notice also that the conclusion above does not depend on the behaviour of solutionsat the singular point x = 0 . Nevertheless, we will discuss the x → η (1) , (2) ε here, since it will be necessary for the analysis in subsequent Sections. The point is that(in contrast to standard situation of nonsingular potentials) both solutions (11), (12) havezero limit at the origin x → A ∈ (0 , / . Namely, their behaviour is η (1) ∼ x A and η (2) ∼ x − A . This is the typical situation of the so called ”limit circle” kind (see [17],Appendix to Section 10.1), which was widely discussed in the literature in the context ofone-dimensional potential (the so-called Calogero potential) g/x on the semi-axis or onthe whole axis (e.g., see [18], [19], [20]). In such a case, there is continuous freedom inchoosing (among many opportunities) some ”good kind of behaviour” for wave functions.The resulting spectrum of the model depends on this choice, thereby defining the kind ofits quantization. The preferences are usually motivated by physical arguments [19], [18],621]. In the case of Hamiltonian H ( p = 1) this problem is of little importance due toasymptotic behaviour of solutions at infinity discussed above. One more remark concernsthe extension of solutions to negative semiaxis: it is reasonable to choose an odd way, taking η ( −| x | ) = − η ( | x | ) . This choice provides continuity of the derivative η ′ ( x ) at the origin.We have to remark that from mathematical point of view, both one-dimensionalSchr¨odinger operator with P¨oschl-Teller potential in (9), (10) and two-dimensional oper-ator in (3), (4) produce rather nontrivial problem. It is possible to check that both of them(in two-dimensional case, due to Green’s identity of vector calculus), are symmetric opera-tors, but in strictly mathematical approach, they are unbounded and not self-adjoint for theconventional choice of smooth functions with a compact support (dense in L ) as a domain D ( H P − T ( x )) . Similarly to the analysis given in [19] for the case V = α/x , the self-adjointextension of H P − T ( x ) includes also the functions from L with specific asymptotic at thesingular point x = 0 . For details, we refer readers to the papers [19], [18], [20] and referencestherein, where one will find also the description of some paradoxes induced by too naiveapproach to singular potentials of α/x type. Q + . As it was mentioned above, the second possible source of eigenfunctions for e H ( p ) are thezero modes of operator Q + ( p ). It is well known [9], [10], that the subspace of zero modes of Q + ( p ) is invariant under the action of e H ( p ) . This means that if e Ω k ( ~x ; p ) is the zero-mode of Q + ( p ) , i.e. Q + ( p ) e Ω k ( ~x ; p ) = 0 , then due to intertwining relations (1) e H ( p ) e Ω k ( ~x ; p ) = N X i =0 C ki e Ω i ( ~x ; p ) (15)( C ki is the ( N + 1) × ( N + 1) matrix with complex elements). If the matrix C ki can bediagonalized by some matrix B : BC = Λ B ; Λ = diag ( E , E , ..., E N ) , (16)the functions e Ψ i ( ~x ; p ) = N X k =0 B ik e Ω k ( ~x ; p ) (17)are the eigenfunctions of e H ( p ) : e H ( p ) e Ψ i ( ~x ; p ) = E i e Ψ i ( ~x ; p ) . At first, one needs to calculate e Ω i ( ~x ; p ) . For this purpose, it is useful to perform thesimilarity transformation, which will help to separate variables: q + ( p ) = e pχ ( ~x ) Q + ( p ) e − pχ ( ~x ) ; e Ω i = e − pχ ( ~x ) e ω i , (18)7here χ = ln (cosh( x + ) cosh( x − )) . After that, the problem Q + ( p ) e Ω i ( ~x ; p ) = 0 becomes q + ( p ) e ω i ( ~x ; p ) = 0 , (19)where q + reads q + = ∂ − ∂ + k (cid:0) sinh − ( x ) + cosh − ( x ) (cid:1) + k (cid:0) cosh − ( x ) + sinh − ( x ) (cid:1) . (20)The choice of the function χ provides that (19) is amenable to separation of variablesin Cartesian coordinates. The two-dimensional equation (19) is equivalent to the pair ofone-dimensional ones if one takes e ω i = η ( x ) ρ ( x ), and it appears that they are exactly theequations (9)-(10), but with ε = ˜ ε. The solutions can be written as linear combinations of e ω ε = η ε ( x ) ρ ε ( x ) , (21)where η ε (and analogously, ρ ε ) must be built from solutions (11), (12). Of course, we areinterested only in normalizable zero modes Ω( ~x ; p ) of the two-dimensional operator Q + ( p ) , but the normalizability condition, in comparison with Section 3, is essentially less restrictivenow: Z | e Ω( ~x ; p ) | d x = Z e − pχ ( ~x ) | η ( x ) | | ρ ( x ) | d x == Z ( cosh ( x + ) cosh ( x − )) − p | η ( x ) | | ρ ( x ) | d x < ∞ . (22)The factor exp ( − pχ ( ~x )) in (22) is exponentially decreasing at infinity in all directionson the plane, and it is able to compensate even growing functions e ω ε . Due to asymptoticequivalence cosh x ∼ sinh x at infinity, asymptotical behaviour of the integrand of (22) canbe represented as | e Ω( ~x ; p ) | ∼ (cosh x + cosh x − ) − p (cosh x + − cosh x − ) √− ε . Therefore, the functions e Ω are normalizable for arbitrary values of p and ε, satisfying: ε > − p . This fact has to be taken into account in calculation of the spectrum of H ( p + 1) (seeSection 5).But at first, we must define the variety of functions e Ω ε , which may be used for constructionof actual wave functions. In this context, functions η ε ( x ) and ρ ε ( x ) are the auxiliary objectsfor construction of zero modes e Ω according to (18), (21). Therefore, all four possible combi-nations can be used, in general. The first of them e Ω (1) ε ( ~x ) = exp (cid:18) − pχ ( ~x ) (cid:19) η (1) ε ( x ) ρ (1) ε ( x )8s: e Ω (1) ε ( ~x ) == ± (cosh( x + ) cosh( x − )) − p sinh A ( | x | ) cosh B ( | x | ) ·· sinh B ( | x | ) cosh A ( | x | ) F (cid:18) a ε , b ε ; A + 12 ; z (cid:19) · F (cid:18) a ε , b ε ; B + 12 ; z (cid:19) (23)where a ε ≡ A + B − √− ε b ε ≡ A + B + √− ε z ≡ − sinh x ; z ≡ − sinh x , (24)and the sign ± depends on a quarter on a plane ( x , x ) , according to the choice at the endof Section 3. Other zero modes e Ω (2) ε ( ~x ) , e Ω (3) ε ( ~x ) , e Ω (4) ε ( ~x ) are obtained from e Ω (1) ε ( ~x ) by meansof substitutions of pairs of parameters (cid:18) − A, B (cid:19) for e Ω (2) ε ( ~x ) , of (cid:18) A, − B (cid:19) for e Ω (3) ε ( ~x ) , and (cid:18) − A, − B (cid:19) for e Ω (4) ε ( ~x ) , instead of (cid:18) A, B (cid:19) in e Ω (1) ε ( ~x ) . H . In this Section we shall look for linear combinations of zero modes of Q + ( p ) , which are simul-taneously the eigenfunctions of the Hamiltonian e H ( p ) = H ( p + 1) in (4). Being interested inthe discrete energy spectrum E n , we suppose that the corresponding wave functions are builtfrom the finite number of zero modes e Ω ( γ ) ε k ( ~x ; p ); γ = 1 , , , , with parameters a k ≡ a ε k in(24), numbering by discrete values k = 0 , , , ..., N ( γ ) , so that the constants a k are orderedas: a > a > ... > a N ( γ ) . We suppose also that four kinds of such wave functions exist: eachis built from the corresponding zero modes e Ω ( γ ) with fixed value of γ (the value of γ definesbehaviour at the origin). According to (15),˜ H ( p ) e Ω ( γ ) ε k ( ~x ) = N X i =0 C ( γ ) ki e Ω ( γ ) ε i ( ~x ) , (25)where C ( γ ) ki are constants, and N also depend on γ = 1 , , , . Performing with e H ( p ) the similarity transformation analogous to that with Q + ( p ) in (18),one obtains:˜ h ( p ) ≡ e pχ ( ~x ) e H ( p ) e − pχ ( ~x ) = − ∂ − ∂ + ˆ D − k cosh x − k sinh x + k cosh x + k sinh x − p , where the mixing operator ˆ D is defined as:ˆ D ≡ p cosh x + cosh x − (sinh(2 x ) ∂ + sinh(2 x ) ∂ ) . (26)9hen, exploring (9), (10), the action of ˜ h ( p ) on e ω (1) ε k ( p ) (see its definition in (18) and (21))can be expressed as:˜ h ( p ) e ω (1) ε k ( p ) = 2 (cid:18) p ( A + B ) + ε k − p (cid:19)e ω (1) ε k + sinh A ( x ) cosh B ( x ) sinh B ( x ) ·· cosh A ( x ) ˆ D (cid:18) F ( a k , b k ; A + 1 / − sinh x ) F ( a k , b k ; B + 1 / − sinh x ) (cid:19) , and (25) takes the form:2 (cid:18) p ( A + B ) + ε k − p (cid:19) F ( a k , b k ; A + 1 / z ) F ( a k , b k ; B + 1 / z ) ++ ˆ D (cid:18) F ( a k , b k ; A + 1 / z ) F ( a k , b k ; B + 1 / z ) (cid:19) == N X i =0 C (1) ki F ( a i , b i ; A + 1 / z ) F ( a i , b i ; B + 1 / z ) . After straightforward calculations, (25) can be rewritten as:2 (cid:18) p ( A + B ) + ε k − p (cid:19) F ( a k , b k ; c ; z ) · F ( a k , b k ; c ; z ) ++ a k b k p − z − z (cid:18) z (1 − z ) c F ( a k + 1 , b k + 1; c + 1; z ) · F ( a k , b k ; c ; z ) ++ z (1 − z ) c · F ( a k + 1 , b k + 1; c + 1; z ) · F ( a k , b k ; c ; z ) (cid:19) == P Ni =0 C (1) ki · F ( a i , b i ; c ; z ) · F ( a i , b i ; c ; z ) , (27)where c = A + 12 , c = B + 12 , b k ≡ b ε k = A + B + √− ε k . (28)For z = 0 (27) reads:2 (cid:18) p ( A + B ) − p + ε k (cid:19) F ( a k , b k ; c ; z ) + a k b k pc z · F ( a k + 1 , b k + 1; c + 1; z ) == P Ni =0 C (1) ki · F ( a i , b i ; c ; z ) . (29)In the z → −∞ limit, the largest power in the l.h.s. of (29) is ( − z ) − a k . Therefore, C (1) k,i = 0 f or i > k, (30)10.e. Eq.(29) takes the form:2 (cid:18) p ( A + B ) − p + ε k (cid:19) · F ( a k , b k ; c ; z ) + a k b k pc z · F ( a k + 1 , b k + 1; c + 1; z ) == P ki =0 C (1) ki · F ( a i , b i ; c ; z ) . (31)Further, for particular values z = 0 and k = 0 , it gives: C (1)00 = 2 (cid:18) p ( A + B ) − p + ε (cid:19) . (32)Substitution of (32) back into (31) with arbitrary z and k = 0 leads to a · b = 0 . Sinceall b k are positive, the only opportunity is a = 0 . Comparing next powers in Eq.(31) for z → −∞ , we obtain: a k + 1 = a k − . Together with a = 0 , this relation uniquely defines all values of a k : a k = − k ; k = 0 , , ..., N (1) . (33)In turn, comparison of coefficients of ( − z ) − a k in Eq.(31), gives values of elements C (1) kk . Due to (30), matrix C (1) ki is triangular. Its diagonal elements C (1) kk coincide with elementsof diagonal matrix Λ in (16), and therefore, C (1) kk gives a part of the eigenvalues of discreteenergy spectrum of the Hamiltonians e H ( p ) = H ( p + 1) : e E (1) k ( p ) = E (1) k ( p + 1) = − (cid:18) ( A + B + 2 k − p ) + p (cid:19) ; k = 0 , , ..., N (1) . (34)It is clear from (34) that the lowest energy state corresponds to the maximal k, i.e. to k = N (1) , which can be defined from conditions of normalizability of e Ω (1) e k ( ~x ) . These conditionswere formulated in Section 4, and they can be rewritten now as: k (1) <
12 ( p − A − B ); N (1) = (cid:20)
12 ( p − A − B ) (cid:21) , (35)where [ c ] means the integer part of c. Analogously one can construct three other kinds of energy levels e E ( γ ) k ( p ) of e H ( p ) = H ( p + 1) by replacing everywhere above ( A, B ) by (1 − A, B ) , or by ( A, − B ) , or by(1 − A, − B ) , correspondingly. The result is the following: e E (2) k ( p ) = E (2) k ( p + 1) = − (cid:18) (1 − A + B + 2 k − p ) + p (cid:19) ; k (2) = 0 , , ..., N (2) ; (36) e E (3) k ( p ) = E (3) k ( p + 1) = − (cid:18) (1 + A − B + 2 k − p ) + p (cid:19) ; k (3) = 0 , , ..., N (3) ; (37) e E (4) k ( p ) = E (4) k ( p + 1) = − (cid:18) (2 − A − B + 2 k − p ) + p (cid:19) ; k (4) = 0 , , ..., N (4) , (38)11here: k (2) <
12 ( p − A − B ); (39) k (3) <
12 ( p − − A + B ); (40) k (4) <
12 ( p − A + B ) . (41)The energy spectra and the corresponding wave functions for several lowest values of p willbe given in Section 7.According to (17), the eigenfunctions of H ( p + 1):Ψ ( γ ) k ( ~x ; p + 1) = e Ψ ( γ ) k ( ~x ; p ) = N X i =0 B ( γ ) ki e Ω ( γ ) i ( ~x ; p ); γ = 1 , , , C ( γ ) ki in (25) and, after that, of B ( γ ) ki from (16). The first step - calculation of C ( γ ) ki with general k, i - is given in Appendix,while the calculation of B ( γ ) ki seems to be rather complicated in a general form. Instead, thiswill be done explicitly for small values of p in Section 7.Thus, each Hamiltonian H ( p + 1) (for N ( γ ) ≥
0) possesses N (1) + N (2) + N (3) + N (4) + 4bound states Ψ ( γ ) k ( ~x ; p + 1) with energy levels E ( γ ) k ( p + 1) , k = 0 , , ..., N ( γ ) , γ = 1 , , , , which are absent in the spectrum of H ( p ) . As we know, due to SUSY intertwining relations(1) (see also Section 2), each of these wave functions produce the tower of extra eigenfunctionsfor higher Hamiltonians H ( p + n + 1) , n = 1 , , ... with the same energy values E ( γ ) k ( p + 1).These wave functions Ψ ( γ ) kn ( ~x ; p + n + 1) are built by the action of n operators Q − ( p + m ) , m =1 , , ..., n :Ψ ( γ ) kn ( ~x ; p + n + 1) = e Ψ ( γ ) kn ( ~x ; p + n ) = Q − ( p + n ) Q − ( p + n − ...Q − ( p + 1)Ψ ( γ ) k ( ~x ; p + 1) , (43)and their indices indicate the number k among N ( γ ) bound states of original Hamiltonian H ( p + 1) , and the number n of one after another acting operators Q − . According to results of previous Sections, the Hamiltonian H ( p + 1) = e H ( p ) has two classesof bound state wave functions. The second one (in terminology of Section 2) Ψ ( γ ) k ( ~x ; p +1) , k = 0 , , ..., N ( γ ) is produced by normalizable zero modes of Q + via their suitable linearcombinations. The first class is obtained from the eigenfunctions of lower Hamiltonians bymeans of operators Q − . In notations introduced above, they are:Ψ ( γ ) m, ( p − n ) ( ~x ; p + 1) = Q − ( p ) Q − ( p − ...Q − ( n + 1)Ψ ( γ ) m ( ~x ; n + 1) , (44) n = 1 , , ..., p − , E ( γ ) m ( n + 1) (see (34)). The restrictions for values of m depend on γ : m < ( n − A − B ) / , γ = 1; m < ( n − A − B ) / , γ = 2; m < ( n − − A + B ) / , γ = 3; m < ( n − A + B ) , γ = 4 . This is an appropriate point to remark that the situation of general position corresponds tothe simple spectrum of H ( p + 1) , which consists of levels E ( γ ) k ( p + 1) (see (34) - (38)) forΨ ( γ ) k ( p + 1) and levels E ( γ ) m ( n + 1) for the states (44). But nothing prohibits from the possibleoccasional degeneracy of the spectrum for some specific values of parameters. Indeed, thissituation is nongeneric: an occasional degeneracy of levels may occur only for some singlevalues of parameters A, B.
In such a case, the degeneracy can be removed easily by anarbitrary small variations of
A, B.
The normalizability of functions of the second class is obvious by construction, but thisproperty for the wave functions (44) will be proven now. The Hamiltonian e H ( p ) has thesymmetry operator e R ( p ) = Q − ( p ) Q + ( p ) (see (8)), and in turn, H ( p + 1) − its own symmetryoperator R ( p + 1) = Q + ( p + 1) Q − ( p + 1) . As far as these Hamiltonians coincide (shapeinvariance) H ( p + 1) = e H ( p ) , the corresponding symmetry operators must coincide as well,but up to the function of the Hamiltonian. Indeed, by straightforward calculation one obtainsthe relation: R ( p + 1) − e R ( p ) = 8(2 p + 1) (cid:18) e H ( p ) + 2(2 p + 2 p + 1) (cid:19) , (45)which will help to analyze the normalizability.The norm of the arbitrary wave function Ψ ( γ ) m, ( p − n ) ( ~x ; p + 1) (44) can be written as: k Ψ ( γ ) m, ( p − n ) ( ~x ; p + 1) k = h Ψ ( γ ) m ( ~x ; n + 1) | Q + ( n + 1) Q + ( n + 2) ...Q + ( p ) ·· Q − ( p ) Q − ( p − ...Q − ( n + 2) Q − ( n + 1)Ψ ( γ ) m ( ~x ; n + 1) i . To simplify it, one may explore Eq.(45) and its consequence: (cid:18) Q + ( n + 1) Q − ( n + 1) − Q − ( n ) Q + ( n ) (cid:19) Q − ( n ) Q − ( n − . . . Q − ( m ) == Q − ( n ) Q − ( n − . . . Q − ( m )Γ mn , where Γ mn is the function of the Hamiltonian:Γ mn = 8(2 n + 1) (cid:18) H ( m ) + 2(2 n + 2 n + 1) (cid:19) . Q + ( n + 1) Q + ( n + 2) ...Q + ( p ) Q − ( p ) Q − ( p − ...Q − ( n + 1) == R ( n + 1) (cid:18) R ( n + 1) + Γ n +1 ,n +1 (cid:19) · (cid:18) R ( n + 1) + Γ n +1 ,n +1 + Γ n +1 ,n +2 (cid:19) ...... (cid:18) R ( n + 1) + Γ n +1 ,n +1 + Γ n +1 ,n +2 + ... + Γ n +1 ,p − (cid:19) , and finally, one obtains that norms of the wave functions Ψ ( γ ) m, ( p − n ) ( ~x ; p + 1) of second classfor H ( p + 1) are proportional to the norms of wave functions Ψ ( γ ) m ( ~x ; n + 1) of the first classfor the Hamiltonian H ( n + 1) : k Ψ ( γ ) m, ( p − n ) ( ~x ; p + 1) k = 64(2 n + 1) (cid:18) E ( γ ) m ( n + 1) + 2 n + 2 n + 1 (cid:19) ·· p − Y q = n +1 (cid:18) ( q + 1) − n (cid:19)(cid:18) E ( γ ) m ( n + 1) + 2(( q + 1) + n ) (cid:19) k Ψ ( γ ) m ( ~x ; n + 1) k . It is easy to check explicitly that the coefficient of proportionality is positive. Hence, asfar as the initial state Ψ ( γ ) m ( n + 1) is normalizable by the construction, any wave functionΨ ( γ ) m, ( p − n ) ( p + 1) is normalizable too.One more statement is necessary to prove in order to be sure that the full variety ofeigenfunctions for H ( p + 1) was constructed above. Namely, we must prove that no addi-tional normalizable wave functions exist besides those in (42), (43). Starting from the lowestHamiltonians, let us suppose that H (2) has such additional eigenfunction Φ(2), which differsfrom the linear combination of zero modes of Q + (1) . Then, it follows from the intertwin-ing relations, that Q + (1)Φ(2) must satisfy the Schr¨odinger equation with Hamiltonian H (1) . One may check that supercharges Q ± ( p ) do not change the normalizability neither at infinity,nor at coordinate axes x , x : the detail analysis is presented below in the next paragraphsof this Section. As we already know from Section 3, the Hamiltonian H (1) has no boundstates at all, and therefore, our supposition was wrong. Let us suppose now that the firstHamiltonian possessing such additional state Φ( p + 1) is H ( p + 1) , while all previous Hamil-tonians H ( p ) , H ( p − , ...H (2) have bound states of the forms (42), (43), only. Then, dueto intertwining relations, Q + ( p )Φ( p + 1) is the eigenfunction Ψ( p ) of H ( p ) , and therefore,coincides either with Ψ k ( p ) or with Ψ l, ( p ) , by our assumption. For simplicity, we do notconsider here the case of possible degeneracy of levels of H ( p ) (the conclusion will be thesame in this case). Acting by Q + ( p ) onto Φ( p + 1) and using the relation (45), one obtainsby straightforward calculations that for both options, Ψ( p ) is proportional to Q + Ψ n, ( p + 1)with some suitable n. Therefore, the wave function Φ( p + 1) coincides with Ψ n, ( p + 1) upto zero modes of Q + ( p ) :Φ( ~x ; p + 1) = c ( p + 1)Ψ n, ( ~x ; p + 1) + c ( p + 1)Ψ k ( ~x ; p + 1) , c , ( p + 1) are constants.Thus, the problem is reduced to the question: whether the operators Q ± ( p ) are ableto change the normalizability of functions. If they are not, no additional normalizableeigenfunctions of H ( p + 1) exist. It is evident from the explicit expressions (5) of Q ± ( p ) andfrom taking into account the exponential decreasing of Ψ at infinity, that Q ± can not violateintegrability of | Ψ | at ±∞ . More difficult problem arises in the neighborhood of x → x → . The part of Q ± linear in derivatives coincides with the operator ˆ D, defined in (26). In the limit x → x = 0 , it is: ˆ D ∼ p (tanh x ) ∂ − p (1 − tanh x ) x ∂ , i.e. it does not change the asymptotic behaviour of the function. Analogous conclusion istrue in the limit x → x = 0 . To analyze the limit when both x , → , it is convenientto use the polar coordinates x + = R cos ϕ, x − = R sin ϕ. Asymptotically, ˆ D for R → D ∼ p (sin(2 ϕ ) R∂ R + cos(2 ϕ ) ∂ ϕ ) , i.e. in this limit ˆ D can not change the behaviour of function as well.Coming back to the operators Q ± , the only terms which could in principle change thebehaviour of function at x → x → Q ± ( p ) ∼ − (cid:18) − ∂ − k sinh − ( x ) (cid:19) + (cid:18) − ∂ + k sinh − ( x ) (cid:19) . Comparing these terms with (3), (4), we observe the same parts (although with differentsigns) in asymptotical expressions: H ( p ) ∼ (cid:18) − ∂ − k sinh − ( x ) (cid:19) + (cid:18) − ∂ + k sinh − ( x ) (cid:19) . Since Ψ( ~x ; p ) are eigenfunctions of H ( p ) , Q ± ( p ) are not able to change the behaviour of Ψ , and the absence of any additional wave functions besides that of (42), (43) types was thusproven. The explicit expressions for matrix elements B ik from (16), which are necessary to build theeigenfunctions (42) of H ( p +1), seem to be difficult to present in a general form. Nevertheless,the problem can be solved straightforwardly for low values of p. By means of separation ofvariables, we demonstrated in Section 3, that the Hamiltonian H (1) has no bound states.The next Hamiltonian H (2) (it corresponds to p = 1 in formulas above) has two boundstates: one bound state with k (1) = 0, due to inequality (35) with p = 1 , and the second15ound state with k (2) = 0 or k (3) = 0 , due to inequalities (39), (40), depending on thepositivity of ( A − B ) or ( B − A ) . Of course, these bound states are of the second class, i.e.are built from the zero modes:Ψ (1)0 ( ~x ; 2) ∼ e Ω (1)0 ( ~x ; 1) = ± (cosh( x + ) cosh( x − )) − ·· sinh A ( | x | ) cosh B ( | x | ) sinh B ( | x | ) cosh A ( | x | ) (46)with energy E (1)0 (2) = − (cid:18) ( A + B − + 1 (cid:19) , and (for A > B )Ψ (2)0 ( ~x ; 2) ∼ e Ω (2)0 ( ~x ; 1) = ± (cosh( x + ) cosh( x − )) − ·· sinh − A ( | x | ) cosh B ( | x | ) sinh B ( | x | ) cosh − A ( | x | ) (47)with energy E (2)0 (2) = − (cid:18) ( B − A ) + 1 (cid:19) . No bound states of the first class exist in thiscase.For the next value p = 2 , i.e. for the Hamiltonian H (3) , four bound states are of thesecond class being built by the zero modesΨ ( γ )0 ( ~x ; 3) ∼ e Ω ( γ )0 ( ~x ; 2); γ = 1 , , , E (1)0 (3) = − (cid:18) ( A + B − + 1 (cid:19) ; E (2)0 (3) = − (cid:18) ( B − A − + 1 (cid:19) ; E (3)0 (3) = − (cid:18) ( A − B − + 1 (cid:19) ; E (4)0 (3) = − (cid:18) ( A + B ) + 1 (cid:19) . But in this case, two wave functions of the first class also can be built from (46) and (47) bythe procedure (43): Ψ ( γ )01 ( ~x ; 3) ∼ Q − (2)Ψ ( γ )0 ( ~x ; 2); γ = 1 , A > B. (48)Their energies E ( γ )01 (3) coincide with E ( γ )0 (2); γ = 1 , H ( p + 1) with p = 3 for A > B has six states which are builtfrom bound states of H (3) by means of operator Q − (4) . They have the same energies E (1)0 (3) , E (2)0 (3) , E (3)0 (3) , E (4)0 (3) and E (1)0 (2) , E (2)0 (2) . As for the second class bound states, sixsuch bound states exist: k (1) = 0 , k (2) = 0 , k (3) = k (4) = 0 . This set includes two wavefunctions coinciding with e Ω (3)0 ( ~x ; 4); e Ω (4)0 ( ~x ; 4) , and four other wave functions have to be builtas linear combinations of pairs of zero modes e Ω (1)0 ( ~x ; 4) , e Ω (1)1 ( ~x ; 4) and e Ω (2)0 ( ~x ; 4) , e Ω (2)1 ( ~x ; 4) , N (1) = N (2) = 1 for p = 4 . The matrix elements of 2 × C (1) ki aredefined by (55), (56): C (1)00 = E (1)0 (4) = − (cid:20) ( A + B − +9 (cid:21) ; C (1)11 = E (1)1 (4) = − (cid:20) ( A + B − +9 (cid:21) ; C (1)10 = − . These elements are necessary to determine coefficients B (1) ki for (42). From Eq.(16), one canfind that for C (1)00 = C (1)11 , as in the present case, the matrix B (1) ki is also triangular B (1)01 = 0.More of that, while B (1)10 = C (1)10 C (1)11 − C (1)00 B (1)11 , the last coefficient B (1)00 is arbitrary. This fact is not discouraging, since the linear combination(42) for Ψ (1)0 (4) includes only one term (with arbitrary B (1)00 ). Thus, the value of B (1)00 is fixedby unity norm of Ψ (1)0 (4) . The second linear combination (42) for Ψ (1)1 (4) includes both B (1)10 and B (1)11 , which are proportional to each other. The absolute values of these coefficients willalso be fixed by normalization of the wave function. Analogous calculations can be easilyrepeated for γ = 2 . An exhaustive procedure of analytical solution of two-dimensional generalization of P¨oschl-Teller model with integer values of parameter p was presented above. Being based on SUSYintertwining relations and shape invariance of the model, the procedure replaces the standardmethod of separation of variables which is not applicable here, and it can be considered asa special - SUSY - separation of variables.In order to confirm obtained results for H ( p + 1), it is useful to compare them with thelimiting case which possesses the direct solution by means of separation of variables. Indeed,if the parameters A, B, which originally belong to the interval (0 , / , are chosen on thelimit of range A, B → , the procedure above (starting from Sect. 3) does not work. Butdue to conventional separation of variables, the Hamiltonian H ( p + 1) from (3) is reduced(up to a trivial multiplier 2) to a sum of two one-dimensional Hamiltonians with well knownreflectionless potentials in variables x ± : h ( p + 1)( x ) = − ∂ − p ( p + 1)cosh x . The spectra of these Hamiltonians are well known: for p ∈ [ L, L + 1) they have exactly L bound states. To compare the properties of spectra with that in Section 7, one has to explorethe original inequalities (35), (39) - (41) and (34), (36) - (38) where A and B are written17xplicitly. The point is that some of bound states described in Section 7 disappear in thelimit A, B → . Thus, H ( p + 1) with p = 1 has one bound state with energy E (1)0 (2) = − k (1) = 0 . Taking into account the multiplier 2mentioned above, this value of energy coincides with double value of eigenvalue of h (2) . For p = 2 , three bound states of the second class for H ( p + 1) exist in the limiting case,with k (1) = k (2) = k (3) = 0 and energies E (1)0 (3) = − E (2)0 (3) = E (3)0 (3) = − , and onebound state of the first class with energy E (1)0 (2) = − . The one-dimensional Hamiltonian h (3) has two bound states, leading just to four possible bilinear combinations in the standardseparation of variables.For p = 3 with A > B the limiting Hamiltonian H (4) has nine bound states: five ofsecond class k (1) = 0 , k (2) = k (3) = k (4) = 0 , and additionally four bound states of firstclass inherited from four bound states of H (3) . As it should be, this number coincides with3 × Acknowledgements.
The work was partially supported by the RFFI grant 09-01-00145-a (M.V.I. and P.A.V.).P.A.V. is indebted to the International Centre of Fundamental Physics in Moscow and thenon-profit foundation ”Dynasty” for financial support.
Appendix: Calculation of the Coefficients C ki . We will calculate the coefficients C (1) ki , but C ( γ ) ki with γ = 2 , , A → (1 − A ) etc. The calculation will be started fromEq.(27), where parameters are defined according to (24), (33), (28). It is useful to express18he hypergeometric functions (with a k = − k ) in terms of Jacobi polynomials: F ( a k , b k ; c ; z ) = k !( α + 1) k P ( α,β ) k ( y ) , F ( a k + 1 , b k + 1; c + 1; z ) = ( k − α + 2) k P ( α +1 ,β +1) k − ( y ) , F ( a k , b k ; c ; z ) = k !( β + 1) k P ( β,α ) k ( y ) , F ( a k + 1 , b k + 1; c + 1; z ) = ( k − β + 2) k P ( β +1 ,α +1) k − ( y ) , where Pokhgammer symbols (Γ) k are defined as (Γ) k ≡ Γ(Γ + 1) · ... (Γ + k − , and newvariables and new parameters will be more suitable: y , ≡ − z , , α ≡ A − , β ≡ B − , and now: b k = 1 + α + β + k, c = 1 + α, c = 1 + β, √− e k = 1 + α + β + 2 k. Taking into account, that ( α + 1)( α + 2) k − = ( α + 1) k , the l.h.s. of (27) becomes: M ≡ ( k !) ( α + 1) k ( β + 1) k (cid:26) (cid:18) p (1 + α + β ) − (1 + α + β + 2 k ) − p (cid:19) P ( α,β ) k ( y ) P ( β,α ) k ( y ) −− p (1 + α + β + k ) y + y (cid:18) (1 − y ) P ( α +1 ,β +1) k − ( y ) P ( β,α ) k ( y ) + (1 − y ) P ( β +1 ,α +1) k − ( y ) P ( α,β ) k ( y ) (cid:19)(cid:27) . (49)Let us use the relation for Jacobi polynomials 22.17.15 from [22] with n replaced by n + 1and β by β − . Multiplying it by (1+ x ) and using the relation 22.7.16 from [22], one obtains: (cid:18) n + α + β + 12 (cid:19) (1 − x ) P ( α +1 ,β +1) n − ( x ) = 2( n + α )( n + β )2 n + α + β P ( α,β ) n − ( x ) −− n ( n + 1)2( n + 1) + α + β P ( α,β ) n +1 ( x ) + 2 (cid:18) n ( n + α )2 n + α + β − n ( n + 1 + β )2( n + 1) + α + β (cid:19) P ( α,β ) n ( x ) . (50)Let us write the same relation but for x = ⇒ y and α ⇐⇒ β, and add it to the initial Eq.(50).Then, rewriting P n +1 as a combination of P n − and P n (according to 22.7.1 from [22]), weobtain: (1 − x ) P ( α +1 ,β +1) n − ( x ) P ( β,α ) n ( y ) + (1 − y ) P ( β +1 ,α +1) n − ( y ) P ( α,β ) n ( x ) == 2 n + 1 + α + β (cid:26) n + α )( n + β )2 n + α + β (cid:18) P ( α,β ) n − ( x ) P ( β,α ) n ( y ) + P ( β,α ) n − ( y ) P ( α,β ) n ( x ) (cid:19) −− n ( x + y ) P ( α,β ) n ( x ) P ( β,α ) n ( y ) . (cid:27) . M = ( k !) ( α + 1) k ( β + 1) k (cid:26) − (cid:18) (1 + α + β + 2 k − p ) + p (cid:19) P ( α,β ) k ( y ) P ( β,α ) k ( y ) −− p ( k + α )( k + β )2 k + α + β y + y (cid:18) P ( α,β ) k − ( y ) P ( β,α ) k ( y ) + P ( β,α ) k − ( y ) P ( α,β ) k ( y ) (cid:19)(cid:27) . (51)The r.h.s. of (51) includes the combination:Φ k − ,k ≡ P ( α,β ) k − ( y ) P ( β,α ) k ( y ) + P ( β,α ) k − ( y ) P ( α,β ) k ( y ) , which (by means of recurrent formula 22.7.1 from [22]) satisfy:Φ k − ,k = ( y + y ) a k − a k − P ( α,β ) k − ( y ) P ( β,α ) k − ( y ) − a k − a k − Φ k − ,k − , (52)where the following definitions were introduced: a n = 2( n + 1)( n + α + β + 1)(2 n + α + β ); a n = (2 n + α + β + 1)( α − β ); a n = (2 n + α + β )(2 n + α + β + 1)(2 n + α + β + 2); a n = 2( n + α )( n + β )(2 n + α + β + 2) . Since Φ , = ( y + y ) a a P ( α,β )0 ( y ) P ( β,α )0 ( y ) , (53)Eqs.(52), (53) give:Φ k − ,k = ( y + y ) (cid:18) a k − a k − P ( α,β ) k − ( y ) P ( β,α ) k − ( y ) − a k − a k − a k − a k − P ( α,β ) k − ( y ) P ( β,α ) k − ( y ) ++ a k − a k − a k − a k − a k − a k − P ( α,β ) k − ( y ) P ( β,α ) k − ( y ) + ... ( − k − a a a k − ...a a k − ...a P ( α,β )0 ( y ) P ( β,α )0 ( y ) (cid:19) . 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