Singular Isotonic Oscillator, Supersymmetry and Superintegrability

In the case of a one-dimensional nonsingular Hamiltonian H and a singular supersymmetric partner Hα, the Darboux and factorization relations of supersymmetric quantum mechanics can be only formal relations. It was shown how we can construct an adequate partner by using infinite barriers placed where...

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Дата:	2012
Автор:	Marquette, I.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2012
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/148465
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Цитувати:	Singular Isotonic Oscillator, Supersymmetry and Superintegrability / I. Marquette // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 47 назв. — англ.

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spelling	irk-123456789-1484652019-02-19T01:25:46Z Singular Isotonic Oscillator, Supersymmetry and Superintegrability Marquette, I. In the case of a one-dimensional nonsingular Hamiltonian H and a singular supersymmetric partner Hα, the Darboux and factorization relations of supersymmetric quantum mechanics can be only formal relations. It was shown how we can construct an adequate partner by using infinite barriers placed where are located the singularities on the real axis and recover isospectrality. This method was applied to superpartners of the harmonic oscillator with one singularity. In this paper, we apply this method to the singular isotonic oscillator with two singularities on the real axis. We also applied these results to four 2D superintegrable systems with second and third-order integrals of motion obtained by Gravel for which polynomial algebras approach does not allow to obtain the energy spectrum of square integrable wavefunctions. We obtain solutions involving parabolic cylinder functions. 2012 Article Singular Isotonic Oscillator, Supersymmetry and Superintegrability / I. Marquette // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 47 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81R15; 81R12; 81R50 DOI: http://dx.doi.org/10.3842/SIGMA.2012.063 http://dspace.nbuv.gov.ua/handle/123456789/148465 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In the case of a one-dimensional nonsingular Hamiltonian H and a singular supersymmetric partner Hα, the Darboux and factorization relations of supersymmetric quantum mechanics can be only formal relations. It was shown how we can construct an adequate partner by using infinite barriers placed where are located the singularities on the real axis and recover isospectrality. This method was applied to superpartners of the harmonic oscillator with one singularity. In this paper, we apply this method to the singular isotonic oscillator with two singularities on the real axis. We also applied these results to four 2D superintegrable systems with second and third-order integrals of motion obtained by Gravel for which polynomial algebras approach does not allow to obtain the energy spectrum of square integrable wavefunctions. We obtain solutions involving parabolic cylinder functions.
format	Article
author	Marquette, I.
spellingShingle	Marquette, I. Singular Isotonic Oscillator, Supersymmetry and Superintegrability Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Marquette, I.
author_sort	Marquette, I.
title	Singular Isotonic Oscillator, Supersymmetry and Superintegrability
title_short	Singular Isotonic Oscillator, Supersymmetry and Superintegrability
title_full	Singular Isotonic Oscillator, Supersymmetry and Superintegrability
title_fullStr	Singular Isotonic Oscillator, Supersymmetry and Superintegrability
title_full_unstemmed	Singular Isotonic Oscillator, Supersymmetry and Superintegrability
title_sort	singular isotonic oscillator, supersymmetry and superintegrability
publisher	Інститут математики НАН України
publishDate	2012
url	http://dspace.nbuv.gov.ua/handle/123456789/148465
citation_txt	Singular Isotonic Oscillator, Supersymmetry and Superintegrability / I. Marquette // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 47 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT marquettei singularisotonicoscillatorsupersymmetryandsuperintegrability
first_indexed	2025-07-12T19:32:30Z
last_indexed	2025-07-12T19:32:30Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 063, 14 pages Singular Isotonic Oscillator, Supersymmetry and Superintegrability? Ian MARQUETTE School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia E-mail: i.marquette@uq.edu.au Received July 20, 2012, in final form September 14, 2012; Published online September 19, 2012 http://dx.doi.org/10.3842/SIGMA.2012.063 Abstract. In the case of a one-dimensional nonsingular Hamiltonian H and a singular supersymmetric partner Ha, the Darboux and factorization relations of supersymmetric quantum mechanics can be only formal relations. It was shown how we can construct an adequate partner by using infinite barriers placed where are located the singularities on the real axis and recover isospectrality. This method was applied to superpartners of the harmonic oscillator with one singularity. In this paper, we apply this method to the singular isotonic oscillator with two singularities on the real axis. We also applied these results to four 2D superintegrable systems with second and third-order integrals of motion obtained by Gravel for which polynomial algebras approach does not allow to obtain the energy spectrum of square integrable wavefunctions. We obtain solutions involving parabolic cylinder functions. Key words: supersymmetric quantum mechanics; superintegrability; isotonic oscillator; poly- nomial algebra; special functions 2010 Mathematics Subject Classification: 81R15; 81R12; 81R50 1 Introduction In recent years, many papers [2, 4, 5, 8, 13, 17, 19, 23, 38] were devoted to a nonsingular isotonic oscillator and its various generalizations. It is also referred in literature as CPRS system and also often written using the following parameter ω = ~ 2a2 or taking ~ = 1. This quantum system is exactly solvable and given by the following equation (i.e. with the parameter given by a = ia0, a0 ∈ R) H = P 2 x 2 + ~2 ( x2 8a4 + 1 (x− a)2 + 1 (x+ a)2 ) . (1) Let us also define the singular isotonic oscillator that correspond to the Hamiltonian given by the equation (1) with alternatively the parameter defined as a ∈ R and the singularities are on the real axis. Let us mention that in the nonsingular case the solutions in terms of series was obtained [8] that involve polynomials that are linear combinations of Hermite polynomials. Moreover, it was shown by J. Sesma [38] that the corresponding Schrödinger equation can be transformed into the confluent Heun equation [39, 40] and studied quasi-exactly solvable analogs. In addition, the CPRS system was related to position-dependent effective mass Schrödinger equations (PDMS) [23] and it was shown recently, using Bethe ansatz method [2], that this system possesses also a hidden sl(2) algebraic structure. Let us mention that the nonsingular isotonic oscillator was obtained and studied in earlier works of Spiridonov and Veselov in the context of the dressing chain method [41, 43] and that ?This paper is a contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”. The full collection is available at http://www.emis.de/journals/SIGMA/SESSF2012.html mailto:i.marquette@uq.edu.au http://dx.doi.org/10.3842/SIGMA.2012.063 http://www.emis.de/journals/SIGMA/SESSF2012.html 2 I. Marquette some results presented in [13] were obtained in the papers [37, 42]. This system is also a parti- cular case of a system [3, 6, 7, 26, 28] involving the fourth Painlevé transcendent [20] and related to third-order supersymmetric quantum mechanics (SUSYQM). Recently, using a modified fac- torization a family that include this system was also obtained [13]. The nonsingular isotonic oscillator also appears in context of 2D quantum superintegrable systems with a second and a third-order integrals of motion with separation of variables in Carte- sian coordinates classified by S. Gravel [18]. Four of these systems that possess more integrals than degrees of freedom, contain the nonsingular isotonic oscillator. These four superintegrable systems were studied [27, 30] from the point of view of SUSYQM [16, 22, 45] and polynomial algebras. The square integrable wavefunctions and the energy spectrum were obtained. Fur- thermore, the superintegrability property does not depend on the nature of the singularity and thus the 2D superintegrable systems remain superintegrable when we take a ∈ R and thus these 2D superintegrable systems also admit two types, i.e. one singular and one regular type. The singular version of these 2D superintegrable Hamiltonian remain to be solved. Let us mention, that the singular isotonic oscillator (i.e. equation (1) with a ∈ R) belongs to the class of 1D Hamiltonian constructed by Robnik [36] from SUSYQM with higher excited states to construct the superpotential. It was recognized that the factorisation and interwining (or Darboux) relations of SUSYQM are only formal. It was observed by many authors that for systems with singularities on the real axis related to a regular superpartner, SUSYQM does not allow [9, 10, 21, 31, 46, 47] to relate the wave- functions and the energy spectrum of the superpartners using supercharges. Supersymmetry and it algebraic equations (factorization, Darboux or intertwining) are formal and do not take into account boundaries or singularities. A method to reobtain the isospectrality property, i.e. relate the wavefunctions and the energy spectrum for a singular Hamiltonian and its regular superpartner, consists to use infinite barriers to modify the regular Hamiltonian [31]. However, this idea was only applied to systems with only one singularity on the real axis. The study of Hamiltonians with many singularities using this approach is an unexplored subject. The purpose of this paper is to apply this approach to the singular isotonic oscillator which has 2 singularities of the real axis and also use these results to obtain the energy spectrum and wavefunctions for four 2D singular superintegrable systems with a second and third-order integrals that remains to be solved. Let us present the organization of the paper. In Section 2, we recall results obtained for the constrained harmonic oscillator. In Section 3, present definition and results concerning super- symmetric quantum mechanics and discussed how we can construct the adequate superpartner for singular and nonsingular superpartners. In Section 4, we apply the results of Section 2 and the method of Marquez, Negro and Nieto [31] to the singular isotonic oscillator. We obtain the wavefunctions and square integrable wavefunctions. In Section 5, we discuss how the polynomial algebra obtained for four 2D singular superintegrable systems with a second and a third integrals of motion that contain the singular isotonic oscillator is only formal and related to SUSYQM. We apply the results of Section 4 to solve these systems and obtain the corresponding energy spectrum and wavefunctions in terms of the parabolic cylinder functions. 2 Constrained harmonic oscillator Let us recall known results concerning the 1D unconstrained and constrained harmonic oscillator (i.e. with one or two symmetric infinite barriers). The unconstrained harmonic oscillator that we denote by VI is given by the following well known Hamiltonian Hψ = ( −~2 2 d2 dx2 + VI(x) ) ψ(z) = Eψ(x), VI(x) = ω2x2 2 , −∞ < x <∞. (2) Singular Isotonic Oscillator, Supersymmetry and Superintegrability 3 The energy spectrum and the wavefunctions are also well known En = ( n+ 1 2 ) ~ω, ψn = (√ πn! )− 1 2 hn (( 2ω ~ ) 1 2 x ) , where hn are the Hermite polynomials. We can defined on [B,∞) and [−B,B] respectively the constrained harmonic oscillators VII and VIII [11, 32]: VII(x) =  ω2x2 2 , x ≥ B, B > 0, ∞, x < B, B ∈ R, (3) VIII(x) =  ω2x2 2 , −B ≤ x ≤ B, B > 0, ∞, x < −B, x > B, B ∈ R. (4) We consider also the following transformation z = √ 2ω ~ x, ε = − E ~ω , b = √ 2ω ~ B, ψ(x) = y(z), and the Schrödinger equation corresponding to the non vanishing part of these potentials VII and VIII thus become the Weber differential equation d2y(z) dz2 − ( z2 4 + ε ) y(z) = 0. (5) 2.1 Constrained harmonic oscillator VII For the quantum system VII given by equation (3) we have the following constraints on the solution y(z) of the equation (5) [11, 31, 32]: (i) y(z) must be square integrable, (ii) y(z) must be continuous on R, i.e. y(z)→ 0 when x→ b+. The equation (5) can be solved in terms of the parabolic cylinder functions [1, 44] y1(ε, z) = e− 1 4 z2 1F1 ( 1 2 ε+ 1 4 , 1 2 , 1 2 z2 ) , y2(ε, z) = e− 1 4 z2 1F1 ( 1 2 ε+ 3 4 , 3 2 , 1 2 z2 ) , where 1F1 is the confluent hypergeometric function also related to the Whittaker function. We can also introduce the following and more appropriate functions in the case of this constrained harmonic oscillator [1, 44] U(ε, z) and V (ε, z) that are combinations of y1(ε, z) and y2(ε, z) and given by U(ε, z) = Y1 cos ( π ( 1 4 + 1 2 ε )) − Y2 sin ( π ( 1 4 + 1 2 ε )) , V (ε, z) = 1 Γ ( 1 2 − ε ) (Y1 sin ( π ( 1 4 + 1 2 ε )) + Y2 cos ( π ( 1 4 + 1 2 ε ))) , where Y1(ε, z) = π 1 2 sec ( π ( 1 4 + 1 2ε )) 2 1 2 ε+ 1 4 Γ ( 3 4 + 1 2ε ) y1(ε, z), Y2(ε, z) = π 1 2 cosec ( π ( 1 4 + 1 2ε )) 2 1 2 ε− 1 4 Γ ( 1 4 + 1 2ε ) y2(ε, z). 4 I. Marquette The asymptotic behavior of the functions U(ε, z) and V (ε, z) is given by the following expressions U(ε, z) ∼ e− z2 4 zε+ 1 2 ( 1− ( ε+ 1 2 ) ( ε+ 3 2 ) 2z2 + · · · ) , V (ε, z) ∼ √ 2 πe z2 4 z−ε+ 1 2 ( 1 + ( ε− 1 2 ) ( ε− 3 2 ) 2z2 + · · · ) . V (ε, x) diverge for z →∞ but U(ε, x) is an appropriate solution and we thus take yII(ε, z) = { U(ε, z), z ≥ b, 0, z < b. (6) Let us present some results [11] concerning the discrete ensemble of solutions. It can be shown (from U(εIIn (b), b) = 0) that the energy levels depend of the position denoted b of the infinite barrier ( with b assumed to be positive ) in the following way εIIn (b) = −1 2 + ε0(n) + ε1(n)b+ ε2(n)b2 + · · · , n = 0, 1, 2, . . . , and the first coefficients are given by the following recurrence relations ε0(n) = −(2n+ 1), n = 0, 1, 2, . . . , ε1(n+ 1) = 2n+ 3 2n+ 2 ε1(n), ε1(0) = −2 1 2π− 1 2 , ε2(n+ 1) = ( 2n+ 3 2n+ 2 )2 ε2(n) + (2n+ 3)!(2n+ 2)! 16π((n+ 1)!)4(n+ 1)24n , ε2(0) = −2π−1(1− Log[2]). These recurrence relations can be solved to obtain explicitly the value of the coefficients. The coefficients of higher-order terms can be calculated and this calculation could also be imple- mented numerically. Therefore, for every fixed value of b where the infinite barrier is located, the zeros of this equation provide a discrete ensemble of solutions [11, 31, 32] εIIn (b), n = 0, 1, 2, . . . and for the Hamiltonian with the potential given as in equation (3) we have EII n (b) = −~ωεIIn (b). The corresponding eigenfunctions are then U(εIIn (b), z). The Fig. 1 presents the energy for the ground state and the 10 first excited states as a function of the position of the infinite barrier. The Fig. 3 presents the ground state for an infinite barrier located at b = 1. The only exactly solvable case [11, 31, 32] is when b = 0. In this particular case we obtain from the condition U(εIIn , 0) = 0 EII n = −~ωεn = ~ω ( 2n+ 1 + 1 2 ) , n = 0, 1, 2, . . . . The wavefunctions correspond to the odd levels of the harmonic oscillator. The limiting case where b→∞ allow to recover the spectrum of the harmonic oscillator given by equation (2). 2.2 Constrained harmonic oscillator VIII For the system VIII given by equation (4), we have the following constraints [11, 31] (i) y(z) must be square integrable, (ii) y(z) must be continuous on R, i.e. y(z)→ 0 when z → b+ and z → −b−. Singular Isotonic Oscillator, Supersymmetry and Superintegrability 5 -2 0 2 4 6 0 10 20 30 40 50 b Figure 1. EII n (b) for the ground state and the 10 first excited states, ~ = 1 and ω = 1 for the Case VII. -2 -1 0 1 2 3 4 5 0 10 20 30 40 50 b Figure 2. EIII n (b) for the ground state and the 10 first excited states, ~ = 1 and ω = 1 for the Case VIII. The associated eigenfunctions are the even and odd functions y1(ε, z) and y2(ε, z). yIII(ε, z) =  y1(ε, z), n odd, −b ≤ z ≤ b, y2(ε, z), n even, −b ≤ z ≤ b, 0, z < −b, z > b. The expansion of the eigenvalues as a function of the positions of the barriers takes the form in this case [11] εIIIn (b) = ε−2(n) b2 + ε0(n) + ε2(n)b2 + ε4(n)b4 + ε6(n)b6 + · · · , n = 0, 1, 2, . . . , with the first coefficients given by ε−2(n) = − ( n+ 1 2 )2 π2, n = 0, 1, 2, . . . , ε0(n) = ε4 = 0, ε2(n) = − 1 12 − 1 8ε−2(n) , ε6(n) = 1 720ε−2(n) + 5 192ε−2(n)2 + 7 128ε−2(n)3 . These relations thus provide for a fixed value of b a discrete ensemble of solutions εIIIn (b), n = 0, 1, 2, . . . . For the system given by the equation (4) the energy spectrum is thus EIII n (b) = −~ωεIIIn (b). The corresponding eigenfunctions are then y1(ε III n (b), z) for the even levels and y1(ε III n (b), z) for the odd levels. The Fig. 2 presents the energy for the ground state and the 10 first excited states as a function of the position of the infinite barrier. The Fig. 4 presents the ground state for an infinite barrier located at b = 1. 3 Factorization method and singular Hamiltonian In supersymmetric quantum mechanics [16, 22, 45], we can define two first-order operators A and A† (supercharges) in the following way A = ~√ 2 d dx +W (x), A† = − ~√ 2 d dx +W (x). where W (x) is called the superpotential. 6 I. Marquette -2 0 2 4 6 8 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 z Figure 3. The wavefunction for the ground state U(εII0 , z) (with ~ = 1 and b = 1) for the Case VII. -2 -1 0 1 2 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 z Figure 4. The wavefunction for the ground state y1(εIII0 , z) (with ~ = 1 and b = 1) for the Case VIII. We consider the following two Hamiltonians which are called “superpartners” H1 = A†A = −~2 2 d2 dx2 +W 2 − ~√ 2 W ′, H2 = AA† = −~2 2 d2 dx2 +W 2 + ~√ 2 W ′. (7) It can be shown that these algebraic relations allow to relate eigenfunctions and energy spectrum of the two superpartners H1 and H2. There are two cases that can be considered. The first one corresponds to broken supersymmetry and happen when Aψ (1) 0 6= 0, E (1) 0 6= 0, A†ψ (2) 0 6= 0 and E (2) 0 6= 0. The wavefunctions and the energy spectra are thus related in the following way E(2) n = E(1) n > 0, ψ(2) n = ( E(1) n )− 1 2Aψ(1) n , ψ(1) n = ( E(2) n )− 1 2A†ψ(2) n , (8) and the two Hamiltonians are strictly isospectral. For the second case the supersymmetry is unbroken and one of the Hamiltonians allows a zero mode, i.e. a states is annihilated by one of the supercharges. Without lost of generality we take H1 as having a zero mode (also referred as zero energy ground state) and thus we have Aψ (1) 0 = 0, E (1) 0 = 0, A†ψ (2) 0 6= 0 and E (2) 0 6= 0. We obtain the following relations between the wavefunctions and the energy spectra E(2) n = E (1) n+1, E (1) 0 = 0, ψ(2) n = ( E (1) n+1 )− 1 2Aψ (1) n+1, ψ (1) n+1 = ( E(2) n )− 1 2A†ψ(2) n . (9) In this case, the superpotential W (x) can be written in term ψ (1) 0 as W (x) = ψ (1)′ 0 ψ (1) 0 . These relations (8) and (9) allow, if one of these systems (H1 or H2) is exactly solvable, to obtain the energy spectrum and the wavefunctions of its superpartner. A such construction can be extended more generally to families of superpartners and superpartners of kth-order. It can be shown, that in the case of Hermitian operators, the factorization relations (7) are equivalent to the following intertwinning (also called Darboux) relations H1A = AH2, A†H1 = H2A †. (10) The factorization and interwining relations can be combined and written in terms of matrices that generate a superalgebra. These algebraic relations also allow to relate the creation and annihilation operators of the Hamiltonians H1 and H2. The ladder operators c† and c of H1 and the ladder operators of H2 that we denote M † and M are related by M = A†cA, M † = A†c†A, (11) Singular Isotonic Oscillator, Supersymmetry and Superintegrability 7 However, as mentioned earlier the algebraic relations given by equations (7), (10) and (11) can be formal and do not take into account boundaries and singularities. In the case of a 1D nonsingular Hamiltonian H1 and a singular supersymmetric partner H2 the wavefunctions ψ(2) obtained from those of H1 (i.e. ψ(1)) and by acting with the supercharges (i.e. using (8) or (9)) on are not guaranteed to belong in the Hilbert space of H2. If the supercharges admit singularity the resulting wavefunctions could also admit singularity. It is known that a singularity of the form 1 (x−a)2 makes that a particle is confined in one of the regions (i.e. left or right side of the singularity) because the probability of going through the barrier is zero. We thus need to compute independently the eigenfunctions for each region divided by a such singularity [15, 24]. It was shown by Marquez, Nieto and Negro [31] that for Hamiltonians with such singular terms we can construct an adequate superpartner by using infinite barriers placed where are located the singularities on the real axis and recover isospec- trality between the superpartners. The wavefunctions and the energy spectrum of the systems with singularities can now be obtained from those of the regular Hamiltonian with infinite bar- riers and the equations (8) and (9). Using this approach, families of quantum systems that are superpartners of the harmonic oscillator with one singularity were studied [31]. 4 Singular nonlinear oscillator Let us consider the following supercharges that admit singularities at a and −a A† = ~√ 2 ( − d dx − 1 2a2 x− ( −1 x− a + −1 x+ a )) , A = ~√ 2 ( d dx − 1 2a2 x− ( −1 x− a + −1 x+ a )) . We can form from these operators, the following Hamiltonian H1 and H2 H1 = A†A = P 2 x 2 + V1 = P 2 x 2 + ~2x2 8a4 + ~2 (x− a)2 + ~2 (x+ a)2 − 3~2 4a2 , (12) H2 = AA† = P 2 x 2 + V2 = P 2 x 2 + ~2x2 8a4 − 5~2 4a2 . (13) These algebraic relations can be used in the case a = ia0, a0 ∈ R to solve the nonsingular isotonic oscillator, but in the singular case a ∈ R they can not be used. The Hamiltonian H2 is the well known harmonic oscillator with an additive constant and in principle the SUSYQM relations would provide the corresponding wavefunctions and energy spectrum of the Hamil- tonian H1. However, the operators A† contains singularities and using SUSYQM relations, we obtain eigenstates of the Hamiltonian H1 but non square integrable states and thus non physical states. However, as mentioned this is possible to recover the isospectrality by using an appro- priate superpartner for each region we would like to compute the wavefunctions and defined by the singularities, i.e. R1 = (−∞,−a], R2 = [a,∞) and R3 = [−a, a]. Due to symmetry, the calculation for the first and the second regions are similar. Let us only present, the calculation in the case of the regions R2 and R3. In the case of the singular Hamiltonian H1 in the region R2 we consider a modified super- partner that is the potential given by equation (3) up to an additive constant that does not affect the results obtained in Section 2 is V2II(x) =  ~2x2 8a4 − 5~2 4a2 , x ≥ a, a > 0, ∞, x < a, a ∈ R. 8 I. Marquette In the case of the singular Hamiltonian H1 in the region R3 we take a modified superpartner as given by equation (3) up to an additive constant V2III(x) =  ~2x2 8a4 − 5~2 4a2 , −a ≤ x ≤ a, a > 0, ∞, x < −a, x > a, a ∈ R. The case with two singularities differs from the case with one singularity because of the possibility of bounded intervals as R3 and as we saw from the Fig. 1–4 the behavior of the two types of constrained harmonic oscillator differ and thus their superpartners will also admit different spectrum and wavefunctions. We can now apply results of Sections 2 and 4 and we obtain for the Hamiltonian given by equation (12). Case R2 = [a,∞): ψ ( εIIn (a), x ) = A†yII(x) =  [ ~√ 2 ( − d dx − 1 2a2 x− ( −1 x− a + −1 x+ a ))] U ( εIIn (a), x ) , x ≥ a, 0, x ≤ a, EII n = ~2 2a2 ( εIIn (a)− 5 2 ) . Case R3 = [−a, a]: ψ ( εIIIn (a), x ) = A†yIII(x) =  [ ~√ 2 ( − d dx − 1 2a2 x− ( −1 x− a + −1 x+ a ))] y1 ( εIIIn (a), x ) , n even, x ≤ \|a\|,[ ~√ 2 ( − d dx − 1 2a2 x− ( −1 x− a + −1 x+ a ))] y2 ( εIIIn (a), x ) , n odd, x ≤ \|a\|, 0, x ≥ \|a\|, EIII n = ~2 2a2 ( εIIIn (a)− 5 2 ) . The wavefunctions are square integrable and also continuous at the infinite barrier by construc- tion. This can be obtained by considering the local behavior in terms of a Taylor series about the position of the infinite barrier. The completeness of the set is also provided by the fact that the state annihilated by the operator A has singularities at the infinite barrier and is not an admissible wavefunction. In the next section we will show how these results can also be used in context of superintegrable systems that also involve such singular systems. 5 SUSYQM and superintegrability Let us now consider the following 2D superintegrable Hamiltonian Hs1 = Hx1 +Hx2 = 1 2 P 2 x1 + 1 2 P 2 x2 + ~2 ( x21 + x22 8a4 + 1 (x1 − a)2 + 1 (x1 + a)2 ) . (14) This system is one of the four irreducible superintegrable systems with a second and a third- order integrals of motion found by Gravel [18] that remained to be solved. An important property of superintegrable systems with higher-order integrals of motion is that the quantum systems and their classical analog do not necessarily coincide [18]. In this case the classical limit is the free particle. It involves the nonsingular or the singular isotonic oscillator as we Singular Isotonic Oscillator, Supersymmetry and Superintegrability 9 choose a = ia0, a0 ∈ R or a ∈ R. The existence of the integrals of motion does not depend on the nature of the singularities and thus of this choice. This system was studied [27, 30] using polynomial algebras for both cases. However, only in the nonsingular case the finite-dimensional unitary representations correspond to physical states. In the singular case, the integrals (i.e. they commute with the Hamiltonian Hs1) I1 and I2 take the form I1 = P 2 x1 − P 2 x2 + 2~2 ( x21 − x22 8a4 + 1 (x1 − a)2 + 1 (x1 + a)2 ) , I2 = 1 2 { L,P 2 x1 } + 1 2 ~2 { x2 ( 4a2 − x21 4a4 − 6(x21 + a2) (x21 − a2)2 ) , Px1 } + 1 2 ~2 { x1 ( (x21 − 4a2) 4a4 − 2 x21 − a2 + 4(x21 + a2) (x21 − a2)2 ) , Px2 } , where { , } is the anticommutator. They are respectively polynomials in the momenta of order 2 and 3. These integrals generate the following commutation relations [27, 30] [I1, I2] = I3, [I1, I3] = 4h4 a4 I2, [I2, I3] = −2~2I31 − 6~2I21Hs1 + 8~2H3 s1 + 6 ~4 a2 I21 + 8 ~4 a2 Hs1I1 − 8 ~4 a2 H2 s1 + 2 ~6 a4 I1 − 2 ~6 a4 Hs1 − 6 ~8 a6 . We can calculate the Casimir operator, the realizations in terms of deformed oscillator al- gebras, the finite-dimensional unitary representation and the corresponding energy spectrum. For a ∈ R the following solutions is obtained E = ~2(p+ 3) 2a2 , Φ(N) = ~8 a4 N(p+ 1−N)(N + 1)(N + 3), where Φ(N) is the structure functions of the deformed oscillator algebra. However, the polyno- mial algebra does not take into account boundaries and singularities and is thus only a formal algebraic construction in this case and this finite-dimensional unitary representations do not correspond to physical states. Furthermore, this is the same energy spectrum that is obtained by using the formal factorization and Darboux relations with the harmonic oscillator and given by equation (12) and (13). However, as mentioned we can show they are not square integrable. The fact that the polynomial algebra and the SUSYQM relation are formal is also related. This can be understood from the fact that the integrals and the polynomial algebras can be derived using SUSYQM [26]. In the x1-axis, the ladder operators are constructed from super- symmetry with the supercharges using equation (11) [26, 27]. The annihilation operators are given by Mx1 = ~2 4a2 ( − d dx1 − 1 2a2 x1 + ( 1 x1 − a + 1 x1 + a )) × ( x1 + 2a2 d dx1 )( d dx1 − 1 2a2 x1 + ( 1 x1 − a + 1 x1 + a )) , Mx2 = ~ 2a2 ( x2 + 2a2 d dx2 ) , with the creation operators obtained by considering (Mx1)† and (Mx2)†. The integrals can be written as I2 = −2a2i ~ ( M †x1Mx2 −Mx1M † x2 ) , I3 = −2a2i ~ ( M †x1Mx2 +Mx1M † x2 ) . 10 I. Marquette 5.1 Modified SUSYQM and singular superintegrable systems Let us consider again the Hamiltonian Hs1 given by the equation (14) and use the results of Section 4 on the 1D Hamiltonian Hx1 and takes a modified superpartner that is isospectral using infinite barrier at the singularities. The 1D Hamiltonian Hx2 is only the 1D unconstrained harmonic oscillator. The wavefunctions and the energy spectrum would be thus for a particle in the region [a,∞) for the axis x1 Φ(x1, x2) = χ(x2)A †y (a) II (x1) =  [ χ(x2) ~√ 2 ( − d dx1 − 1 2a2 x1 − ( −1 x1 − a + −1 x1 + a ))] U ( εn(a), x1 ) , x1 ≥ a, 0, x1 ≤ a, E = ~2 2a2 ( εIIn (a) + k − 2 ) . For a particle in the region [−a, a] for the axis x1 Φ(x1, x2) = χ(x2)A †y (a) III (x1) =  [ χ(x2) ~√ 2 ( − d dx1 − 1 2a2 x1 − ( −1 x1 − a + −1 x1 + a ))] y1(εn(a), x1), n even, x1 ≤ \|a\|,[ χ(x2) ~√ 2 ( − d dx1 − 1 2a2 x1 − ( −1 x1 − a + −1 x1+a ))] y2(εn(a), x1), n odd, x1 ≤ \|a\|, 0, x1 ≥ \|a\|, E = ~2 2a2 ( εIIIn (a) + k − 2 ) , with χ(x2) = Ce −1 4a2 x22Hk (√ 1 2a2 x2 ) . The eigenfunctions and energy spectrum for the three other irreducible 2D superintegrable systems with a second and third-order integrals of motion can be calculated using the same approach. Hamiltonian Hs2 Hs2 = 1 2 P 2 x1 + 1 2 P 2 x2 + ~2 ( x21 + 9x22 8a4 + 1 (x1 − a)2 + 1 (x1 + a)2 ) . For a particle in the region [a,∞) for the axis x1 Φ(x1, x2) = χ(x2)A †y (a) II (x1) =  [ χ(x2) ~√ 2 ( − d dx1 − 1 2a2 x1 − ( −1 x1 − a + −1 x1 + a ))] U(εn(a), x1), x1 ≥ a, 0, x1 ≤ a, E = ~2 2a2 ( εIIn (a) + k − 1 ) . For a particle in the region [−a, a] for the axis x1 Φ(x1, x2) = χ(x2)A †y (a) III (x1) Singular Isotonic Oscillator, Supersymmetry and Superintegrability 11 =  [ χ(x2) ~√ 2 ( − d dx1 − 1 2a2 x1 − ( −1 x1 − a + −1 x1 + a ))] y1(εn(a), x1), n even, x1 ≤ \|a\|,[ χ(x2) ~√ 2 ( − d dx1 − 1 2a2 x1 − ( −1 x1 − a + −1 x1 + a ))] y2(εn(a), x1), n odd, x1 ≤ \|a\|, 0, x1 ≥ \|a\|, E = ~2 2a2 ( εIIIn (a) + k − 1 ) , with χ(x2) = Ce −3 4a2 x22Hk (√ 3 2a2 x2 ) . Hamiltonian Hs3 Hs3 = 1 2 P 2 x1 + 1 2 P 2 x2 + ~2 ( x21 + x22 8a4 + 1 x22 + 1 (x1 − a)2 + 1 (x1 + a)2 ) . For a particle in the region [a,∞) for the axis x1 Φ(x1, x2) = χ(x2)A †y (a) II (x1) =  [ χ(x2) ~√ 2 ( − d dx1 − 1 2a2 x1 − ( −1 x1 − a + −1 x1 + a ))] U(εn(a), x1), x1 ≥ a, 0, x1 ≤ a, E = ~2 2a2 ( εIIn (a) + k ) . For a particle in the region [−a, a] for the axis x1 Φ(x1, x2) = χ(x2)A †y (a) III (x1) =  [ χ(x2) ~√ 2 ( − d dx1 − 1 2a2 x1 − ( −1 x1 − a + −1 x1 + a ))] y1(εn(a), x1), n even, x1 ≤ \|a\|,[ χ(x2) ~√ 2 ( − d dx1 − 1 2a2 x1 − ( −1 x1 − a + −1 x1 + a ))] y2(εn(a), x1), n odd, x1 ≤ \|a\|, 0, x1 ≥ \|a\|, E = ~2 2a2 ( εIIIn (a) + k ) , with χ(x2) = e− 1 4a2 x22x22L 3 2 k ( 1 2a2 x22 ) , where L 3 2 k are Laguerre polynomials. Hamiltonian Hs4 Hs4 = 1 2 P 2 x1 + 1 2 P 2 x2 + ~2 ( x21 + x22 8a4 + 1 (x1 − a)2 + 1 (x1 + a)2 ) + 1 (x2 − a)2 + 1 (x2 + a)2 ) . For a particle in the region [a,∞) for the axis x1 and x2 Φ(x1, x2) = 2∏ i A†iy (a) II (xi), 12 I. Marquette with A†iy (a) II (xi) =  [ ~√ 2 ( − d dxi − 1 2a2 xi − ( −1 xi − a + −1 xi + a ))] U(εn(a), xi), xi ≥ a, 0, xi ≤ a, E = ~2 2a2 ( εIIn (a) + k − 5 ) . For a particle in the region [−a, a] for the axes x1 and x2 Φ(x1, x2) = 2∏ i A†iy (a) III (xi), with A†iy (a) III (xi) =  [ ~√ 2 ( − d dxi − 1 2a2 xi − ( −1 xi − a + −1 xi+a ))] y1(εn(a), xi), n even, xi ≤ \|a\|,[ ~√ 2 ( − d dxi − 1 2a2 xi − ( −1 xi − a + −1 xi + a ))] y2(εn(a), xi), n odd, xi ≤ \|a\|, 0, x1 ≥ \|a\|, E = ~2 2a2 ( εIIIn (a) + k − 5 ) , 6 Conclusion In this paper, we studied the singular isotonic oscillator using the approach introduced in [31]. We obtained the energy spectrum and the wavefunctions from supersymmetry by identifying the appropriate superpartner. We also studied four singular 2D superintegrable Hamiltonians obtained by Gravel. The energy spectrum in the 1D cases is no longer equidistant and in the 2D superintegrable versions the energy spectra do not display accidental degeneracy explained by a polynomial algebra that is only formal. However, this point out how supersymmetric quantum mechanics can be used to solve such systems. The wavefunctions are also interesting from the point of view of special functions as they are written in terms of expressions involving parabolic cylinder functions. These results could also be interesting in regard of possible applications. The constrained harmonic oscillators has found applications in the context of condensed matter [14]. Moreover, the factorization is not unique [33] and it was shown that the nonsingular isotonic oscillator has more general families of superpartners [29]. Many families of nonsingular super- partners of the radial oscillator were obtained and the relations with exceptional orthogonal polynomial studied [35]. The study of the singular version of these systems and the relations with special functions is also important. 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Singular Isotonic Oscillator, Supersymmetry and Superintegrability

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