- Email: [email protected]
Existence of different intermediate Hamiltonians in type A N -fold supersymmetry
Annals of Physics 324 (2009) 2438–2451
Contents lists available at ScienceDirect
Annals of Physics journal homepage: www.elsevier.com/locate/aop
Existence of different intermediate Hamiltonians in type A N -fold supersymmetry Bijan Bagchi a, Toshiaki Tanaka b,c,* a b c
Department of Applied Mathematics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700 009, India Department of Physics, National Cheng-Kung University, Tainan 701, Taiwan, ROC National Center for Theoretical Sciences, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 27 May 2009 Accepted 6 August 2009 Available online 13 August 2009 Keywords: N -fold supersymmetry Parasupersymmetry Factorization method Intertwining operators Pöschl–Teller potentials
a b s t r a c t Type A N -fold supercharge admits a one-parameter family of factorizations into product of N first-order linear differential operators due to an underlying GLð2; CÞ symmetry. As a consequence, a type A N -fold supersymmetric system can have different intermediate Hamiltonians corresponding to different factorizations. We derive the necessary and sufficient conditions for the latter system to possess intermediate Hamiltonians for the N ¼ 2 case. We then show that whenever it has (at least) one intermediate Hamiltonian, it can admit second-order parasupersymmetry and a generalized 2-fold superalgebra. As an illustration, we construct a set of generalized Pöschl–Teller potentials of this kind. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Supersymmetric quantum mechanics (SUSY QM) has constituted an active research field in theoretical sciences over two decades. Although the original motivation for studying SUSY QM was to unveil the mechanisms of its dynamical breaking in quantum field theories [1], it turned out that SUSY QM, as the minimum building block of SUSY, contains various relevant concepts which provide convenient platforms to uncover many useful properties of quantum mechanics [2–4]. In particular, it is consistent with factorization schemes [5] and intertwining relationships [6] thereby providing a powerful tool to construct solvable Schrödinger equations. Furthermore, interesting extensions to higher-order SUSY schemes were carried out by taking recourse to higher-derivative versions of the factorization operators [7–9].
* Corresponding author. Address: Department of Physics, National Cheng-Kung University, Tainan 701, Taiwan, ROC. E-mail addresses: [email protected] (B. Bagchi), [email protected] (T. Tanaka). 0003-4916/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2009.08.002
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
2439
The original extension in Ref. [7] was considered by using higher-order intertwining operators which are expressed as products of first-order linear differential operators and then applying the ordinary SUSY results. Later in Refs. [10,11], through the analysis of the general second-order case, the concept of reducibility was introduced. In this regard, a higher-order intertwining operator is said to be reducible if it is factorized into a product of first-order differential operators such that with respect to each factor there exists an intermediate real Hamiltonian satisfying a (shifted) SUSY relation. Otherwise it is called irreducible. This concept, however, seems less useful in view of the current status where non-Hermitian quantum theories have been investigated intensively since the discovery of PT symmetry [12]. In fact, the SUSY method turned to be useful also in constructing a complex potential with real spectrum [13–15]. In addition to the usefulness of the reality constraint, there arises a natural question about the well-definiteness of the concept if we take into account the fact that in general factorization of higher-order linear differential operators is not unique. The latter fact indeed has been reported in the context of the factorization method and ordinary SUSY QM, that is, there exist several Schrödinger operators that admit different factorizations [16–22], see also Ref. [23] for a recent approach. Hence, a reducible higher-order intertwining operator may admit simultaneously another factorization for which there are no intermediate real Hamiltonians. On the other hand, the non-uniqueness in factorizing intertwining operators of arbitrary finite orders was, to the best of our knowledge, first reported in Ref. [24] for the well-known quasi-solvable sextic anharmonic oscillator potentials in the framework of type A N -fold supersymmetry (see, Eq. (47) in the latter reference). Later it was shown that the non-uniqueness of factorizations in type A N -fold SUSY is a consequence of the underlying GLð2; CÞ symmetry [25]. In a recent communication [26], the following two-parameter family of second-order supersymð1Þ ð2Þ b b BÞ, characterized by the two parameters A and B, satisfying metric (SSUSY) system ðh ; h ; A ð1Þ ð2Þ b b b b A Bh ¼ h A B was constructed: 2
ð1Þ
h
ð2Þ
h
c b þ c ¼ d þ V A;B ðxÞ e by B ¼B Eþ ; 2 2 dx2 2 bA b y c ¼ d þ V A;B;ext ðxÞ E c ; ¼A dx2 2 2
ð1Þ ð2Þ
where 2
V A;B ðxÞ ¼ ½B2 þ AðA þ 1Þcsch x Bð2A þ 1Þcsch x coth x;
ð3Þ
2ð2A þ 1Þ 2½4B2 ð2A þ 1Þ2 V A;B;ext ðxÞ ¼ V A;B ðxÞ þ ; 2B cosh x 2A 1 ð2B cosh x 2A 1Þ2
ð4Þ
and the constants E, e E, and c are given by
2 1 E¼ B ; 2
2 1 e E ¼ B ; 2
c ¼ 2B:
ð5Þ
b and B b are respectively given by The intertwining operators A
2B sinh x b ¼ d B 1 coth x A þ 1 cschx A ; dx 2 2 2B cosh x 2A 1 b ¼ d B 1 coth x A þ 1 cschx: B dx 2 2
ð6Þ ð7Þ
It was further shown in Ref. [26] that the system admits the intermediate Hamiltonians h given by 2 2 c b c ¼ d þ V A;B1 ðxÞ E c ¼ B by A bB b y þ c ¼ d þ V A;B1 ðxÞ e h¼A Eþ ; 2 2 2 2 dx2 dx2
ð8Þ
with 2
V A;B1 ðxÞ ¼ ½ðB 1Þ2 þ AðA þ 1Þcsch x ðB 1Þð2A þ 1Þcschx coth x;
ð9Þ
2440
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
b and Bh b ¼ hð2Þ A b ð1Þ ¼ h B. b We shall hereafter call the system which satisfy the ordinary SUSY relations Ah (1)–(9) the BQR SSUSY model. Taking into account the fact that type A 2-fold SUSY is the necessary and sufficient condition for the existence of (at least) two linearly independent analytic (local) solutions to Schrödinger equation of one degree of freedom [27], we immediately know that the above BQR SSUSY model, which is exactly solvable, also belongs to type A 2-fold SUSY. Therefore, it would be natural that it has (at least) two different factorizations of the second-order intertwining operator in view of the aforementioned GLð2; CÞ symmetry and has two different intermediate Hamiltonians correspondingly. Regarding the existence of intermediate Hamiltonians, we note the following fact shown in Ref. [24] that the most general form of type A N -fold SUSY quantum systems constructed directly from the N th-order intertwining operators of type A, namely, type A N -fold supercharge P N ¼ þ P N 1 . . . P NN (cf., Eq. (11)) by solving P N H ¼ H P N is more general than that of the systems constructed from the N repeated applications of the first-order intertwining operators P N k by solving ðk1Þ ð0Þ ¼ HðkÞ P and Hþ ¼ HðN Þ . It is apparent that P N kH N k ðk ¼ 1; . . . ; N Þ with the identification H ¼ H in the latter construction we automatically obtain a series of the intermediate Hamiltonians Hð1Þ ; . . . ; HðN 1Þ in addition to the N -fold SUSY pair Hamiltonians H at the cost of the generality. In the former construction, on the other hand, the existence of intermediate Hamiltonians is not guaranteed in general. Hence, the fact that the type A N -fold supercharge has the factorized form by definition does not necessarily imply the existence of intermediate Hamiltonians. In particular, the fact that type A N -fold supercharge admits different factorizations does not automatically mean that the most general type A N -fold SUSY system has different sets of intermediate Hamiltonians accordingly. Motivated by the backgrounds described above, we investigate in this article under what conditions type A N -fold SUSY systems admit intermediate Hamiltonians in the case of N ¼ 2. Furthermore, we also examine under the satisfaction of the conditions how many sets of such Hamiltonians are admissible for the type A 2-fold SUSY systems. In addition, we show that any such a system has another type of non-linear supersymmetries, namely, parasupersymmetry of order 2 [28]. We organize the article as follows. In the next section, we review the framework of type A N -fold SUSY by putting emphasis on the GLð2; CÞ symmetry. In Section 3, we investigate in details under what conditions a type A 2-fold SUSY quantum system has one or more intermediate Hamiltonians. In particular, we show that the maximum number of different intermediate Hamiltonians in type A 2-fold SUSY is two. In Section 4, we further show that when a type A 2-fold SUSY system admits (at least) one intermediate Hamiltonian, the system can have second-order parasupersymmetry. A novel generalization of 2-fold superalgebra is discussed briefly. As an application of the results, we construct in Section 5 a type A 2-fold SUSY system with two different intermediate Hamiltonians of generalized Pöschl–Teller type which includes the BQR SSUSY model as a particular case. Then, we close the article with discussion and perspectives of further developments in the last section. 2. Type A N -fold supersymmetry and GLð2; CÞ covariance Roughly speaking, type A N -fold SUSY quantum systems are composed of a pair of scalar Hamilto1 nians H and an N th-order linear differential operator P N of the following forms: 2
2
1 d 1 N 1 N þ WðxÞ2 ð2E0 ðxÞ EðxÞ2 Þ W 0 ðxÞ R; 2 dx2 2 24 2 N 1 Y d N 1 2k ¼ þ WðxÞ þ EðxÞ ; dx 2 k¼0
H ¼
ð10Þ
PN
ð11Þ
where R is a constant while EðxÞ and WðxÞ are analytic functions satisfying
1 We keep the original notations as far as possible. Thus, note that the function EðxÞ is different from the constant E in the BQR SSUSY model (1)–(9). Similarly, the function AðzÞ introduced later in (19) is different from the parameter A in the latter model.
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
d d d EðxÞ þ EðxÞ WðxÞ ¼ 0 for N P 2; dx dx dx d d d d 2EðxÞ EðxÞ þ EðxÞ EðxÞ ¼ 0 for N P 3: dx dx dx dx
2441
ð12Þ ð13Þ
The product of operators appeared in (11) is defined by n Y
Ak A n . . . A 1 A0 :
ð14Þ
k¼0
The operators H and P N satisfy an intertwining relation
PN H ¼ Hþ PN :
ð15Þ
One of the most important features of type A N -fold SUSY quantum systems is that the gauged Hame and Hþ introduced by iltonians H
e ¼ eW N H eW N ; H
W N ðxÞ ¼
N 1 2
Z
dx EðxÞ
Z
dx WðxÞ;
ð16Þ
e ðAÞ : preserve the so-called type A monomial space V N
eV e ðAÞ V e ðAÞ ; H N N
e ðAÞ ¼ h1; zðxÞ; . . . ; zðxÞN 1 i; V N
ð17Þ
where the new variable zðxÞ satisfies
z00 ðxÞ ¼ EðxÞz0 ðxÞ:
ð18Þ
Explicitly, they are given by
2 e ¼ AðzÞ d þ N 2 A0 ðzÞ Q ðzÞ d ðN 1ÞðN 2Þ A00 ðzÞ N 1 Q 0 ðzÞ þ R ; H 2 dz 12 2 dz2
ð19Þ
where the new functions AðzÞ and Q ðzÞ are defined by
2AðzÞ ¼ z0 ðxÞ2 ;
Q ðzÞ ¼ z0 ðxÞWðxÞ:
ð20Þ
The conditions (12) and (13) for type A N -fold SUSY are reduced to the following simple forms in terms of z: 3
d Q ðzÞ ¼ 0 for N P 2; dz3 5 d AðzÞ ¼ 0 for N P 3: dz5
ð21Þ ð22Þ
In particular, the condition (21) indicates that Q ðzÞ is a polynomial of at most second-degree in z for all N P 2:
Q ðzÞ ¼ b2 z2 þ b1 z þ b0 :
ð23Þ
In terms of zðxÞ, the potential terms V ðxÞ of type A N -fold SUSY Hamiltonians in (10) are expressed as
V ðxÞ ¼
1 3 2 ðN 1Þ AðzÞA00 ðzÞ A0 ðzÞ2 3Q ðzÞ2 3N ðA0 ðzÞQ ðzÞ 2AðzÞQ 0 ðzÞÞ 12AðzÞ 4
Rjz¼zðxÞ : ð24Þ The type A N -fold SUSY systems (10) and (11) have an underlying symmetry which, as we shall show, plays a central role in investigating the existence of intermediate Hamiltonians. It is GLð2; CÞ linear fractional transformations on the variable z introduced as
2442
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
z¼
aw þ b ða; b; c; d 2 C; D ad bc–0Þ: cw þ d
ð25Þ
Then, the type A monomial space is invariant under the GLð2; CÞ transformations induced by (25):
b e ðAÞ ½w ¼ ðcw þ dÞN 1 V e ðAÞ ½z # V e ðAÞ ½z V N N N
wþb z¼acwþd
e ðAÞ ½w: ¼V N
ð26Þ
e and Hþ are both such linear differential operators that preserve the type The gauged Hamiltonians H A monomial space, as was shown in (17). As a consequence, they are covariant under the following GLð2; CÞ transformations:
b e ½z # H e ½w ¼ ðcw þ dÞN 1 H e ½zðcw þ dÞðN 1Þ H
wþb z¼acwþd
ð27Þ
;
b b þ both have the same forms as given in (19) with z ree and H that is, the transformed operators H b b ðwÞ given by placed by w and with AðzÞ and Q ðzÞ replaced by the transformed functions AðwÞ and Q
b AðzÞ # AðwÞ ¼ D2 ðcw þ dÞ4 AðzÞ
wþb z¼acwþd
b ðwÞ ¼ D1 ðcw þ dÞ2 QðzÞ Q ðzÞ # Q
ð28Þ
;
z¼acwþb wþd
ð29Þ
:
b ðwÞ for arbitrary N P 2 is given by In particular, the explicit form of Q
^1 w þ b ^0 ; ^2 w2 þ b b ðwÞ ¼ b Q
ð30Þ
with
0
1 0 2 10 1 ^2 b2 a ac c2 b B^ C CB C 1 B @ b1 A ¼ D @ 2ab ad þ bc 2cd A@ b1 A: ^0 b0 bd d2 b2 b
ð31Þ
Utilizing the transformation (29) and the formulas
wðxÞ ¼
dzðxÞ b
czðxÞ a
;
w0 ðxÞ ¼
Dz0 ðxÞ ðczðxÞ aÞ2
¼ D1 ðcwðxÞ þ dÞ2 z0 ðxÞ;
ð32Þ
we obtain the transformations of WðxÞ and EðxÞ as
b c ðxÞ ¼ Q ðwÞ ¼ Q ðzÞ ¼ WðxÞ; WðxÞ # W z0 ðxÞ w0 ðxÞ 00 00 w ðxÞ z ðxÞ 2cz0 ðxÞ 2cz0 ðxÞ ¼ EðxÞ : ¼ 0 EðxÞ # b EðxÞ ¼ 0 w ðxÞ z ðxÞ czðxÞ a czðxÞ a
ð33Þ ð34Þ
The invariance of the pair of type A N -fold SUSY Hamiltonians H under the GLð2; CÞ transformations also follows from a direct application of (33) and (34):
c; b H ½W; E ¼ H ½ W E;
ð35Þ
c ðxÞ ¼ WðxÞ and from (18) and (34) we have since W
2b E 0 ðxÞ b EðxÞ2 ¼ 2E0 ðxÞ EðxÞ2 :
ð36Þ P N
On the other hand, the invariance of the type A N -fold supercharge is not manifest in the factorized form (11) and due to the fact that b EðxÞ – EðxÞ the factorized form is in appearance not invariant:
d c ðxÞ þ N 1 2k b EðxÞ þW dx 2 k¼0 N 1 Y d N 1 2k 2cz0 ðxÞ ¼ þ WðxÞ þ EðxÞ : dx 2 czðxÞ a k¼0
c; b PN ½ W E ¼
N 1 Y
ð37Þ
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
2443
The fact that P N is also invariant under the GLð2; CÞ transformations
c; b PN ½ W E ¼ PN ½W; E;
ð38Þ
proved in Ref. [25], despite the non-invariance in appearance for c – 0, indicates that the type A N -fold supercharge admits a one-parameter family of different factorizations characterized by the parameter a=c. 3. Intermediate Hamiltonians for N ¼ 2 From now on, we shall restrict ourselves to the case of N ¼ 2. The type A N -fold SUSY systems (10) and (11) for N ¼ 2 read 2
d E0 ðxÞ EðxÞ2 þ WðxÞ2 þ 2R 2W 0 ðxÞ; 2 2 dx 4 2 d d E0 ðxÞ EðxÞ2 P2 ¼ P 21 P22 ¼ 2 þ 2WðxÞ þ W 0 ðxÞ þ WðxÞ2 þ ; dx 2 dx 4 2H ¼
ð39Þ ð40Þ
where
P21 ¼
d EðxÞ þ WðxÞ ; dx 2
P22 ¼
d EðxÞ þ WðxÞ þ : dx 2
ð41Þ
The superHamiltonian H 2 and the type A 2-fold supercharges Q 2 introduced with the ordinary fermionic variables w as
H 2 ¼ H w wþ þ Hþ wþ w ;
Q 2 ¼ P2 w ;
ð42Þ
satisfy the type A 2-fold superalgebra [25]:
½Q 2 ; H 2 ¼ fQ 2 ; Q 2 g ¼ 0;
2
fQ 2 ; Q þ2 g ¼ 4ðH 2 þ RÞ2 þ 4b0 b2 b1 :
ð43Þ
In the expanded form of the type A 2-fold supercharge components (40), its invariance under the GLð2; CÞ transformations is now manifest by applying (33) and (36):
c; b P2 ½ W E ¼ P2 ½W; E:
ð44Þ
However, each factor of the type A N -fold supercharge in the factorized form is not invariant since b EðxÞ–EðxÞ as shown in (37), and thus we generally have
b P ½ W c; b P E–P21 ½W; E; 21 21
b P ½ W c; b P E–P22 ½W; E: 22 22
ð45Þ
i1
Next, we introduce another Hamiltonian H , which we shall call an intermediate Hamiltonian, as
P22 H ¼ Hi1 P22 ;
P21 Hi1 ¼ Hþ P21 ;
ð46Þ i1
which are compatible with (15). It is evident that H is in general not invariant under the GLð2; CÞ i1 transformation in contrast with H due to the fact that both of P 21 and P 22 which intertwine H with H have no invariance (45). Hence, we can expect a family of intermediate Hamiltonians for each given type A 2-fold SUSY system. Needless to say, however, we do not always have such an intermediate Hamiltonian for a given system. The necessary and sufficient conditions for its existence are that there exist two constants C 22 and C 21 such that H and Hi1 are expressed as (see, e.g., Refs. [2–4])
2H ¼ Pþ22 P22 þ 2C 22 ;
2Hþ ¼ P21 Pþ21 þ 2C 21 ;
2Hi1 ¼ P22 Pþ22 þ 2C 22 ¼ Pþ21 P21 þ 2C 21 ; 2 where Pþ ij are the transpositions of P ij , that is,
2
Note that we do not assume the reality of the functions WðxÞ and EðxÞ.
ð47Þ
2444
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
Pþ21 ¼
d EðxÞ þ WðxÞ ; dx 2
P þ22 ¼
d EðxÞ þ WðxÞ þ : dx 2
ð48Þ
More explicitly, the conditions (47) read 2
d EðxÞ2 E0 ðxÞ þ WðxÞ2 þ EðxÞWðxÞ þ W 0 ðxÞ þ 2C 22 ; 2 2 dx 4 2 d EðxÞ2 E0 ðxÞ 2Hi1 ¼ 2 þ WðxÞ2 þ EðxÞWðxÞ þ þ W 0 ðxÞ þ 2C 22 þ 2 dx 4 2 d EðxÞ2 E0 ðxÞ ¼ 2 þ WðxÞ2 EðxÞWðxÞ þ W 0 ðxÞ þ 2C 21 ; þ 2 dx 4 2 d EðxÞ2 E0 ðxÞ þ W 0 ðxÞ þ 2C 21 : 2Hþ ¼ 2 þ WðxÞ2 EðxÞWðxÞ þ 2 dx 4 2H ¼
ð49aÞ
ð49bÞ ð49cÞ
From Eqs. (39) and (49), the necessary and sufficient conditions reduce to
W 0 ðxÞ þ EðxÞWðxÞ ¼ 2R 2C 22 ¼ C 21 C 22 ¼ 2C 21 þ 2R:
ð50Þ
Noting the relation
W 0 ðxÞ þ EðxÞWðxÞ ¼ Q 0 ðzÞ;
ð51Þ
which easily follows from (18) and (20), we find that the latter conditions (50) are equivalent to
Q ðzÞ ¼ ðC 22 C 21 Þz þ b0 ;
2R ¼ C 22 þ C 21 ;
ð52Þ
with b0 being another constant. We recall that for the most general type A N -fold SUSY systems for all N P 2, Q ðzÞ is given by a polynomial of at most second-degree (23). Hence, a given type A 2-fold SUSY system (39) admits an intermediate Hamiltonian Hi1 satisfying Eq. (46) if and only if
b2 ¼ 0;
b1 ¼ C 22 C 21 ;
2R ¼ C 22 þ C 21 :
ð53Þ
The last two conditions in (53) just determine these constants for the given values of b1 and R. Hence, only the first condition in (53) is essential for the existence of an intermediate Hamiltonian. As was discussed previously, the type A 2-fold supercharge P 2 is invariant under the GLð2; CÞ trans formation (44) while its factors P 22 and P 21 are not (45) for c – 0. As a consequence, the necessary and sufficient conditions (53) for the existence of another intermediate Hamiltonian Hi2 after a GLð2; CÞ transformation are accordingly changed as
^2 ¼ 0; b
^1 ¼ C b 22 C b 21 ; b
b 22 þ C b 21 ; 2R ¼ C
ð54Þ
b 22 and C b 21 are another set of constants. Again, only the first condition in (54) is essential for where C the existence of another intermediate Hamiltonian Hi2 after the transformation. Under the fulfillment of the conditions (54), the original type A 2-fold SUSY Hamiltonians H and the new intermediate Hamiltonian Hi2 are expressed in terms of the transformed supercharges as
b b bþ P b P b bþ 2H ¼ P 2Hþ ¼ P 21 21 þ 2 C 21 ; 22 22 þ 2 C 22 ; i2 b b bþ b P bþ b 2H ¼ P 22 22 þ 2 C 22 ¼ P 21 P 21 þ 2 C 21 ;
ð55Þ
b þ are the transpositions of P b which were defined in (45), that is, where P ij ij
b b ¼ d þ W c ðxÞ EðxÞ ; P 21 2 dx
b b ¼ d þ W c ðxÞ þ EðxÞ : P 22 2 dx
ð56Þ
It is now clear that a type A 2-fold SUSY system which satisfies the conditions (53) and thus admits an intermediate Hamiltonian Hi1 also admits another different intermediate Hamiltonian Hi2 after a GLð2; CÞ transformation with c–0 if and only if the conditions (54) are simultaneously fulfilled in ^2 ¼ 0. From the transformation formula addition to (53). It essentially means the satisfaction of b2 ¼ b (31) we immediately see that it is only possible for the GLð2; CÞ transformation with c – 0 which satisfies
2445
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
ab1 þ cb0 ¼ 0:
ð57Þ
Let us first consider the case when b1 ¼ 0. In this case we can assume that b0 – 0; otherwise Q ðzÞ ¼ 0 since b2 ¼ 0 is already assumed in order to meet the condition (53), and from (24) V ðxÞ ¼ V þ ðxÞ which means that the system is trivial as 2-fold SUSY. But for b1 ¼ 0 and b0 – 0 there is no solution to Eq. (57) except for c ¼ 0. But for any GLð2; CÞ transformation with c ¼ 0, the function EðxÞ is invariant by Eq. (34) and so is the intermediate Hamiltonian Hi1 . Hence in the case of b1 ¼ 0 the factorization of type A 2-fold supercharge admitting intermediate Hamiltonians is unique. On the other hand, in the case of b1 – 0 the solution to Eq. (57) is given by
a=c ¼ b0 =b1 :
ð58Þ
As was shown previously, any type A 2-fold supercharge admits one-parameter family of factoriza tions P 2 ¼ P 21 P 22 characterized by the parameter a=c. In addition, any type A 2-fold SUSY system with b2 ¼ 0 has at least one intermediate Hamiltonian Hi1 . Then, the result (58) tells us that if the system further satisfies b1 – 0, it can admit a one and only one additional and different intermediate Hamiltonian Hi2 at the one point (58) in the parameter space of a=c 2 C. In Table 1, we summarize the results. 4. Second-order parasupersymmetry In the previous section, we have just verified that a type A 2-fold SUSY system ðH ; P 2 ¼ P 21 P 22 Þ admits (at least) one intermediate Hamiltonian Hi1 if and only if the condition b2 ¼ 0 holds. In this section, we shall further show that any such a system can possess an additional symmetry, namely, parasupersymmetry of order 2 introduced in Ref. [28]. Indeed, for a given such type A 2-fold SUSY system we can define a triple of operators ðH P ; Q P Þ by
H P ¼ H ðwP Þ2 ðwþP Þ2 þ Hi1 ðwþP wP ðwþP Þ2 ðwP Þ2 Þ þ Hþ ðwþP Þ2 ðwP Þ2 ;
ð59aÞ
Q P ¼ Pþ22 ðwP Þ2 wþP þ Pþ21 wþP ðwP Þ2 ;
ð59bÞ
Q þP ¼ P22 wP ðwþP Þ2 þ P21 ðwþP Þ2 wP ;
where w P are parafermions of order 2 satisfying [29]
ðwP Þ2 –0;
ðwP Þ3 ¼ 0;
fwP ; wþP g þ fðwP Þ2 ; ðwþP Þ2 g ¼ 2I:
ð60Þ
Then, using Eq. (47) and the parafermionic algebra of order 2 in Ref. [29] we can show that the triple ðH P ; Q P Þ defined as (59) satisfies the second-order paraSUSY relations in Ref. [28]:
ðQ P Þ2 –0;
ðQ P Þ3 ¼ 0;
ðQ P Þ2 Q P
Q P Q P Q P
þ
þ
½Q P ; H P ¼ 0; Q P ðQ P Þ2
¼
ð61Þ
4Q P H P ;
ð62Þ
if and only if the constants C ij in (47) satisfy
C 22 ¼ C 21 ¼ b1 =2;
ð63Þ
and thus in particular R ¼ 0 by (53). Hence, we conclude that any type A 2-fold SUSY quantum system with (at least) one intermediate Hamiltonian also has paraSUSY of order 2 when R ¼ 0. The additional restriction R ¼ 0 arises since one of the paraSUSY conditions (62) is not invariant under any constant shift of H P . Furthermore, as was shown in Ref. [29] this type of realization of second-order paraSUSY
Table 1 The admissible numbers of different intermediate Hamiltonians in type A 2-fold supersymmetry. Conditions
Number of Hi
b2 – 0 b2 ¼ b1 ¼ 0 b2 ¼ 0, b1 – 0
0 1 2
2446
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
systems admits an additional novel non-linear relation as the following (cf., Eq. (6.66) in the latter reference): 2
ðQ P Þ2 ðQ þP Þ2 þ Q P ðQ P Þ2 Q P þ ðQ þP Þ2 ðQ P Þ2 ¼ 4ðH P Þ2 b1 ;
ð64Þ
which can be regarded as a generalized (type A) 2-fold superalgebra. In fact, on the one hand we immediately have from the paraSUSY relations in (61)
fðQ P Þ2 ; ðQ P Þ2 g ¼ 0;
½ðQ P Þ2 ; H P ¼ 0;
ð65Þ
while on the other hand the non-linear relation (64) reduces, in the subsector with the parafermion number zero and two, to 2
fðQ P Þ2 ; ðQ þP Þ2 g ¼ 4ðH P Þ2 b1 :
ð66Þ
Then, the commutation and anti-commutation relations (65) and (66) are, under the assumed condition b2 ¼ 0 and R ¼ 0, entirely identical with the type A 2-fold superalgebra (43) with the trivial 2 identification of the type A 2-fold supercharges Q 2 with ðQ P Þ and with the observation that in the subsector H P is essentially identical with H 2 . The relation between type A 2-fold SUSY and secondorder paraSUSY was briefly referred to in Ref. [29]. Here we have firstly shown the necessary and sufficient conditions for a type A 2-fold SUSY system to admit simultaneously second-order paraSUSY, namely, Eqs. (53) and (63). Finally, it is evident that we can construct two sets of second-order paraSUSY systems whenever a type A 2-fold SUSY system has two different intermediate Hamiltonians.
5. An application to the generalized Pöschl–Teller potential As an application of the general framework discussed in the previous two sections, we shall reconstruct the BQR SSUSY model (1)–(9) and its generalization which preserves all the SUSY and SSUSY structure therein. To this end, let us first choose the change of variable z ¼ zðxÞ which determines the relation between physical Hamiltonians and gauged ones as
x 2Aþ1 x 2A2Bþ1 x ð2Aþ2Bþ1Þ zðxÞ ¼ ðsinh xÞ2B tanh ¼ 22B sinh cosh ; 2 2 2
ð67Þ
where A and B are both constants. The function AðzÞ ¼ z0 ðxÞ2 =2 defined in (20) and its derivatives with respect to z are in general transcendental functions of z. Explicitly, they read
2B cosh x 2A 1 ; sinh x zðxÞ2 ð2B cosh x 2A 1Þ2
z0 ðxÞ ¼ zðxÞ
ð68Þ
AðzÞ ¼
ð69Þ
; 2 x 2sinh aþ cosh x aþ2 A0 ðzÞ ¼ zðxÞ 4B2 1 ; 2 sinh x b cosh x b2 2A þ 1 ; A00 ðzÞ ¼ 4B2 1 2 2B cosh x 2A 1 sinh x
ð70Þ ð71Þ
where aþ i and bi are all constants given by
a1 ¼ ð2A þ 1Þð4B 1Þ; a2 ¼ ð2A þ 1Þ2 þ 2Bð2B 1Þ; b1 ¼ ð2A þ 1Þð4B þ 3Þ;
b2 ¼ ð2A þ 1Þ2 þ 2ðB þ 1Þð2B þ 1Þ:
ð72Þ ð73Þ
Substituting them into the most general form of a pair of type A 2-fold SUSY potentials, Eq. (24) with N ¼ 2, we obtain
2447
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451 2
V ðxÞ ¼
Q ðzðxÞÞ2 sinh x 2zðxÞ2 ð2B cosh x 2A 1Þ
4ð2A þ 1ÞB cosh x þ ð2A þ 1Þ2 12B2
þ 2
8ð2B cosh x 2A 1Þ2 2
ð2A þ 1ÞB cosh x AðA þ 1Þ B 2
2sinh x 2
B2 R 2
þ
2
Q ðzðxÞÞð4B sinh x aþ1 cosh x þ aþ2 Þ zðxÞð2B cosh x 2A 1Þ2
Q 0 ðzðxÞÞ;
ð74Þ
where Q ðzÞ is a polynomial of at most second-degree given as in (23). The function WðxÞ characterizing the type A system in this case reads
WðxÞ ¼
Q ðzðxÞÞ Q ðzðxÞÞ sinh x : ¼ z0 ðxÞ zðxÞð2B cosh x 2A 1Þ
ð75Þ
The other function EðxÞ defined through the relation (18), which also characterizes the type A system, is calculated as
EðxÞ ¼
z00 ðxÞ ð2B þ 1Þ cosh x 2A 1 2B sinh x þ : ¼ z0 ðxÞ sinh x 2B cosh x 2A 1
ð76Þ
Next, let us consider the case when the type A 2-fold SUSY system admits an intermediate Hamiltonian, namely, b2 ¼ 0. From (75) and (76) we have
W 0 ðxÞ ¼
Q ðzðxÞÞ ð2A þ 1Þ cosh x 2B Q ðzðxÞÞ; Q 0 ðzðxÞÞ zðxÞ zðxÞð2B cosh x 2A 1Þ2
E0 ðxÞ ¼
2ð2A þ 1ÞB cosh x 4B2 ð2B cosh x 2A 1Þ2
EðxÞ2 ¼ 4B2 þ
ð2A þ 1Þ2 4B2 ð2B cosh x 2A 1Þ
2
ð2A þ 1Þ cosh x 2B 1 2
sinh x
ð77Þ ð78Þ
;
2ð2A þ 1Þð2B þ 1Þ cosh x ð2A þ 1Þ2 ð2B þ 1Þ2 2
sinh x
:
ð79Þ
Substituting (75)–(79) into (49b), using the relations (53) among the constants, and noting that Q 0 ðzÞ ¼ b1 when b2 ¼ 0, we obtain the intermediate potential V i1 ðxÞ as 2
V i1 ðxÞ ¼
QðzðxÞÞ2 sinh x 2
2zðxÞ ð2B cosh x 2A 1Þ
2
4ð2A þ 1ÞB cosh x ð2A þ 1Þ2 4B2 8ð2B cosh x 2A 1Þ2
ð2A þ 1ÞðB þ 1Þ cosh x AðA þ 1Þ ðB þ 1Þ2 2
2sinh x
þ
B2 R: 2
ð80Þ
Next, we shall consider the case when the system admits another different intermediate Hamiltonian Hi2 , namely, b1 – 0. The GLð2; CÞ transformation which takes the type A 2-fold supercharge to another factorization for which Hi2 exists must satisfy the condition (58). The parameter d does not play an important role in our context, so we set d ¼ 0 without any loss of generality. But in this case b cannot be 0 otherwise D ¼ 0. Thus, we fix the parameters as
a=c ¼ b0 =b1 z0 ; b=c ¼ 1; d ¼ 0:
ð81Þ
In other words, we choose the following GLð2; CÞ transformation on the variable zðxÞ:
wðxÞ ¼
1 1 ¼ : zðxÞ þ z0 ðsinh xÞ2B ðtanh x=2Þ2Aþ1 z0
ð82Þ
The function WðxÞ is invariant under the transformation (see, Eq. (33)) while EðxÞ is transformed according to (34) as
2448
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
2z0 ðxÞ 2zðxÞ 2B cosh x 2A 1 b EðxÞ ¼ EðxÞ ¼ EðxÞ þ zðxÞ þ z0 zðxÞ þ z0 sinh x ¼
½ð2B 1ÞzðxÞ ð2B þ 1Þz0 cosh x ð2A þ 1ÞðzðxÞ z0 Þ 2B sinh x : þ ðzðxÞ þ z0 Þ sinh x 2B cosh x 2A 1
ð83Þ
Noting the relation Q ðzÞ ¼ b1 ðz þ z0 Þ when b2 ¼ 0, we have
b c ðxÞ ¼ EðxÞWðxÞ þ 2b1 : EðxÞ W
ð84Þ
From the second expression of b EðxÞ in (83), we obtain the following formulas:
2zðxÞ ð2A þ 1Þ cosh x 2B 2z0 zðxÞ ð2B cosh x 2A 1Þ2 b E 0 ðxÞ ¼ E0 ðxÞ þ ; 2 2 zðxÞ þ z0 ðzðxÞ þ z0 Þ2 sinh x sinh x
ð85Þ
8BzðxÞ 4zðxÞ cosh xð2B cosh x 2A 1Þ 4z0 zðxÞð2B cosh x 2A 1Þ2 b : ð86Þ EðxÞ2 ¼ EðxÞ2 þ 2 2 zðxÞ þ z0 ðzðxÞ þ z0 Þ2 sinh x ðzðxÞ þ z0 Þsinh x ^1 ¼ b1 by the transformation formula (31) in Substituting (83)–(86) into Eq. (55) and noting that b our choice of the parameters (81), we obtain for H the same potentials as the ones in (74), as they should be, while for the other intermediate Hamiltonian Hi2 the following form of the potential:
zðxÞ2 ½ð2A þ 1Þ cosh x 2B
V i2 ðxÞ ¼ V i1 ðxÞ þ
2
2
ðzðxÞ þ z0 Þ sinh x
z0 zðxÞðaþ1 cosh x aþ2 Þ
þ
2
2
ðzðxÞ þ z0 Þ sinh x
4B2 z0 zðxÞ ðz þ z0 Þ2
:
ð87Þ
If we substitute (80) for V i1 ðxÞ into the above, we finally obtain the full expression of V i2 ðxÞ as 2
V i2 ðxÞ ¼
Q ðzðxÞÞ2 sinh x 2
2
2zðxÞ ð2B cosh x 2A 1Þ
4ð2A þ 1ÞB cosh x ð2A þ 1Þ2 4B2 8ð2B cosh x 2A 1Þ2
ð2A þ 1ÞðB 1Þ cosh x AðA þ 1Þ ðB 1Þ2 2
2sinh x z20 ½ð2A
þ 1Þ cosh x 2B 2
ðzðxÞ þ z0 Þ
4B2 z0 zðxÞ ðzðxÞ þ z0 Þ2
þ
B2 z0 zðxÞða1 cosh x a2 Þ Rþ 2 2 ðzðxÞ þ z0 Þ2 sinh x ð88Þ
;
where a i are defined in (72). We are now in a position to show that the type A 2-fold system with the two intermediate Hamiltonians (74), (80), and (88) contains as a special case the BQR SSUSY model (1)–(9). For the purpose, let put b0 ¼ 0 and b1 ¼ bB. In this case, Q ðzÞ ¼ bBz and z0 ¼ b0 =b1 ¼ 0. Then, the 2-fold SUSY pair of the potentials (74) and the two intermediate potentials (80) and (88) reduce to, respectively, 2
V ðxÞ ¼
2
2
4ðb þ 1Þð2A þ 1ÞB cosh x ðb 1Þð2A þ 1Þ2 4ðb þ 3ÞB2 8ð2B cosh x 2A 1Þ2
ð2A þ 1ÞB cosh x AðA þ 1Þ B2 2
2sinh x
2
þ
b B2 ð2A þ 1Þ cosh x 2B ; þ R bB 8 2 ð2B cosh x 2A 1Þ2
ð89Þ
and 2
V i1 ðxÞ ¼ ðb 1Þ and
4ð2A þ 1ÞB cosh x ð2A þ 1Þ2 4B2 8ð2B cosh x 2A 1Þ2
ð2A þ 1ÞðB þ 1Þ cosh x AðA þ 1Þ ðB þ 1Þ2 2
sinh x
2
þ
b B2 þ R; 8 2
ð90Þ
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
2
V i2 ðxÞ ¼ ðb 1Þ
2449
4ð2A þ 1ÞB cosh x ð2A þ 1Þ2 4B2 8ð2B cosh x 2A 1Þ2
ð2A þ 1ÞðB 1Þ cosh x AðA þ 1Þ ðB 1Þ2 2
sinh x
2
þ
b B2 þ R: 8 2
ð91Þ
b The components of supercharges P ij and P ij given by (41), (48) and (56) in this case read
d ð2B þ 1Þ cosh x 2A 1 ðb 1ÞB sinh x þ þ ; dx 2 sinh x 2B cosh x 2A 1 d ð2B þ 1Þ cosh x 2A 1 ðb þ 1ÞB sinh x þ ; P22 ¼ dx 2 sinh x 2B cosh x 2A 1 b ¼ d ð2B 1Þ cosh x 2A 1 þ ðb 1ÞB sinh x ; P 21 dx 2 sinh x 2B cosh x 2A 1 d ð2B 1Þ cosh x 2A 1 ðb þ 1ÞB sinh x b þ : P 22 ¼ þ dx 2 sinh x 2B cosh x 2A 1
P21 ¼
ð92Þ ð93Þ ð94Þ ð95Þ
It is now easy to see that the BQR SSUSY model is realized when b ¼ 1. Indeed, we have the following correspondences when b ¼ 1:
1 1 þ B2 2R; 2V þ ðxÞ ¼ V A;B; ext ðxÞ þ þ B2 2R; 4 4 1 1 2V i1 ðxÞ ¼ V A;Bþ1 ðxÞ þ þ B2 2R; 2V i2 ðxÞ ¼ V A;B1 ðxÞ þ þ B2 2R; 4 4 b Pþ or P b y ; P or P b ¼ A; bþ ¼ A b ¼ B; b P þ or P bþ ¼ B by; P21 or P 21 21 22 22 21 22 22 2V ðxÞ ¼ V A;B ðxÞ þ
and in particular
P 2
¼
P 21 P 22
¼
b P b P 21 22
ð96Þ ð97Þ ð98Þ
b B. b The relations among the constants are given by ¼A
b 22 ¼ 4 C b 21 ¼ 2B; c ¼ 4C 22 ¼ 4C 21 ¼ 2B or c ¼ 4 C 1 c c e E þ ¼ E ¼ B2 þ ; R ¼ 0: 4 2 2
ð99Þ ð100Þ
The last equality R ¼ 0 means from the results in Section 4 that the BQR SSUSY model also has secondorder paraSUSY. We note that the reason why the BQR SSUSY model is realized as the particular case b2 ¼ b0 ¼ 0 of the most general type A 2-fold SUSY is the same as the one discussed in Ref. [27], Section 5. 6. Discussion and summary In this article, we have investigated in detail under what conditions type A N -fold SUSY systems can have intermediate Hamiltonians in the case of N ¼ 2. It turns out that although type A 2-fold supercharge admits a one-parameter family of factorization into product of two first-order linear differential operators due to the underlying GLð2; CÞ symmetry, at most two different intermediate Hamiltonians are admissible. As a by product of the studies, we have also obtained the necessary and sufficient conditions for a type A 2-fold SUSY system to possess paraSUSY of order 2 as well. When it is the case, the type A 2-fold superalgebra together with the second-order parasuperalgebra constitute a generalized 2-fold superalgebra. As a demonstration of the general arguments, we have constructed the generalized Pöschl–Teller potentials which are components of type A 2-fold SUSY with two intermediate Hamiltonians and reduce to the BQR SSUSY model in a particular case. As for the concept like the reducibility in Ref. [10], the present investigations indicate that it would be more natural and useful to classify higher-order intertwining operators according to the existence and the number of intermediate Hamiltonians as has been done in Table 1. After employing the latter classification scheme, we can further classify them according to the properties of the intermediate Hamiltonians such as Hermiticity, PT symmetry, and so on. Regarding the generalized Pöschl–Teller potentials constructed in Section 5, it is worth noticing that the framework of N -fold SUSY works well even when the function AðzÞ, which controls the
2450
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
change of variable z ¼ zðxÞ from the physical coordinate x to the variable z in the gauged space, is a transcendental function of z without destroying quasi-solvability. For all N P 3 cases type A N -fold SUSY requires the additional condition (22) so that AðzÞ is allowed to be at most a polynomial of fourth order in z, which results in the admissible change of variable to be at most an elliptic function, see, e.g., Ref. [25]. For the N ¼ 2 case, on the other hand, there are no such restrictions and, to the best of our knowledge, our generalized Pöschl–Teller potentials are the first quasi-solvable examples where AðzÞ is given by a transcendental function of z as Eq. (69). The analyses for N ¼ 2 carried out in this article are easily generalized to the cases N P 3, but we anticipate that richer structure could emerge for the higher N cases. In the case of N ¼ 3, for instance, according to the factorization of type A 3-fold supercharge P 3 ¼ P 31 P 32 P 33 we can consider not only the and P and between P case where intermediate Hamiltonians between P 31 32 32 and P 33 both exist, but also the cases where they exist only between the former place or only between the latter place exclusively. It is also interesting to study whether or not type A N -fold SUSY systems for higher N , when they have intermediate Hamiltonians, can admit another symmetry. The fact that in the case of N ¼ 2 they have second-order paraSUSY indicates that they could have higher-order paraSUSY [30,31] for N P 3. Indeed, it was shown in Ref. [32] that a certain realization of paraSUSY of order 3 also admits a generalized 3-fold superalgebra. Hence, at least in the case of N ¼ 3 we have a reasonable basis to expect paraSUSY as an additional symmetry. Other candidates might be quasi-paraSUSY introduced in Ref. [29] and N -fold paraSUSY in Ref. [33]. Acknowledgments This work (T.Tanaka) was partially supported by the National Cheng Kung University under the grant No. OUA:95-3-2-071. References [1] E. Witten, Dynamical breaking of supersymmetry, Nucl. Phys. B 188 (1981) 513–554. [2] F. Cooper, A. Khare, U. Sukhatme, Supersymmetry and quantum mechanics, Phys. Rep. 251 (1995) 267–385. arXiv:hep-th/ 9405029. [3] G. Junker, Supersymmetric Methods in Quantum and Statistical Physics, Springer, Berlin, 1996. [4] B.K. Bagchi, Supersymmetry in Quantum and Classical Mechanics, Chapman and Hall/CRC press, Florida, 2000. [5] E. Schrödinger, A method of determining quantum-mechanical eigenvalues and eigenfunctions, Proc. R. Irish Acad. A 46 (1940) 9–16. [6] G. Darboux, Comput. Rend. Acad. Sci. 94 (1882) 1456. [7] A.A. Andrianov, M.V. Ioffe, V.P. Spiridonov, Higher-derivative supersymmetry and the Witten index, Phys. Lett. A 174 (1993) 273–279. arXiv:hep-th/9303005. [8] H. Aoyama, M. Sato, T. Tanaka, N -fold supersymmetry in quantum mechanics: general formalism, Nucl. Phys. B 619 (2001) 105–127. arXiv:quant-ph/0106037. [9] A.A. Andrianov, A.V. Sokolov, Nonlinear supersymmetry in quantum mechanics: algebraic properties and differential representations, Nucl. Phys. B 660 (2003) 25–50. arXiv:hep-th/0301062. [10] A.A. Andrianov, M.V. Ioffe, F. Cannata, J.P. Dedonder, Second order derivative supersymmetry, q deformations and the scattering problem, Int. J. Mod. Phys. A 10 (1995) 2683–2702. arXiv:hep-th/9404061. [11] A.A. Andrianov, M.V. Ioffe, D.N. Nishnianidze, Polynomial supersymmetry and dynamical symmetries in quantum mechanics, Theoret. Math. Phys. 104 (1995) 1129–1140. [12] C.M. Bender, S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998) 5243–5246. arXiv:physics/9712001. [13] F. Cannata, G. Junker, J. Trost, Schrödinger operators with complex potential but real spectrum, Phys. Lett. A 246 (1998) 219–226. arXiv:quant-ph/9805085. [14] B. Bagchi, R. Roychoudhury, A new PT -symmetric complex hamiltonian with a real spectrum, J. Phys. A: Math. Gen. 33 (2000) L1–L3. arXiv:quant-ph/9911104. [15] M. Znojil, F. Cannata, B. Bagchi, R. Roychoudhury, Supersymmetry without hermiticity within PT symmetric quantum mechanics, Phys. Lett. B 483 (2000) 284–289. arXiv:hep-th/0003277. [16] B. Mielnik, Factorization method and new potentials with the oscillator spectrum, J. Math. Phys. 25 (1984) 3387–3389. [17] D.J. Fernández C, New hydrogen-like potentials, Lett. Math. Phys. 8 (1984) 337–343. [18] D. Zhu, A new potential with the spectrum of an isotonic oscillator, J. Phys. A: Math. Gen. 20 (1987) 4331–4336. [19] C.N. Kumar, Isospectral Hamiltonians: generation of the soliton profile, J. Phys. A: Math. Gen. 20 (1987) 5397–5401. [20] N.A. Alves, E.D. Filho, The factorization method and supersymmetry, J. Phys. A: Math. Gen. 21 (1988) 3215–3225. [21] E.D. Filho, The morse oscillator generalised from supersymmetry, J. Phys. A: Math. Gen. 21 (1988) L1025–L1028. [22] A. Mitra, P.K. Roy, A. Lahiri, B. Bagchi, Nonuniqueness of the factorization scheme in quantum mechanics, Int. J. Theoret. Phys. 28 (1989) 911–916.
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
2451
[23] C. Quesne, Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry, J. Phys. A: Math. Theor. 41 (2008) 392001. arXiv:0807.4087 [quant-ph]. [24] H. Aoyama, M. Sato, T. Tanaka, General forms of a N -fold supersymmetric family, Phys. Lett. B 503 (2001) 423–429. arXiv:quant-ph/0012065. [25] T. Tanaka, Type A N -fold supersymmetry and generalized Bender–Dunne polynomials, Nucl. Phys. B 662 (2003) 413–446. arXiv:hep-th/0212276. [26] B. Bagchi, C. Quesne, R. Roychoudhury, Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of PT symmetry, Pramana J. Phys. 73 (2009) 337–347. arXiv:0812.1488 [quant-ph].. [27] A. González-López, T. Tanaka, Nonlinear pseudo supersymmetry in the framework of N -fold supersymmetry, J. Phys. A: Math. Gen. 39 (2006) 3715–3723. arXiv:quant-ph/0602177. [28] V.A. Rubakov, V.P. Spiridonov, Parasupersymmetric quantum mechanics, Mod. Phys. Lett. A 3 (1988) 1337–1347. [29] T. Tanaka, Parasupersymmetry and N -fold supersymmetry in quantum many-body systems I. General formalism and second order, Ann. Phys. 322 (2007) 2350–2373. arXiv:hep-th/0610311. [30] M. Tomiya, Comment on generalized parasupersymmetric quantum mechanics, J. Phys. A: Math. Gen. 25 (1992) 4699– 4704. [31] A. Khare, Parasupersymmetric quantum mechanics of arbitrary order, J. Phys. A: Math. Gen. 25 (1992) L749–L754. [32] T. Tanaka, Parasupersymmetry and N -fold supersymmetry in quantum many-body systems II. Third order, Ann. Phys. 322 (2007) 2682–2702. arXiv:hep-th/0612263. [33] T. Tanaka, N -fold parasupersymmetry, Mod. Phys. Lett. A 22 (2007) 2191–2200. arXiv:hep-th/0611008.
Contents lists available at ScienceDirect
Annals of Physics journal homepage: www.elsevier.com/locate/aop
Existence of different intermediate Hamiltonians in type A N -fold supersymmetry Bijan Bagchi a, Toshiaki Tanaka b,c,* a b c
Department of Applied Mathematics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700 009, India Department of Physics, National Cheng-Kung University, Tainan 701, Taiwan, ROC National Center for Theoretical Sciences, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 27 May 2009 Accepted 6 August 2009 Available online 13 August 2009 Keywords: N -fold supersymmetry Parasupersymmetry Factorization method Intertwining operators Pöschl–Teller potentials
a b s t r a c t Type A N -fold supercharge admits a one-parameter family of factorizations into product of N first-order linear differential operators due to an underlying GLð2; CÞ symmetry. As a consequence, a type A N -fold supersymmetric system can have different intermediate Hamiltonians corresponding to different factorizations. We derive the necessary and sufficient conditions for the latter system to possess intermediate Hamiltonians for the N ¼ 2 case. We then show that whenever it has (at least) one intermediate Hamiltonian, it can admit second-order parasupersymmetry and a generalized 2-fold superalgebra. As an illustration, we construct a set of generalized Pöschl–Teller potentials of this kind. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Supersymmetric quantum mechanics (SUSY QM) has constituted an active research field in theoretical sciences over two decades. Although the original motivation for studying SUSY QM was to unveil the mechanisms of its dynamical breaking in quantum field theories [1], it turned out that SUSY QM, as the minimum building block of SUSY, contains various relevant concepts which provide convenient platforms to uncover many useful properties of quantum mechanics [2–4]. In particular, it is consistent with factorization schemes [5] and intertwining relationships [6] thereby providing a powerful tool to construct solvable Schrödinger equations. Furthermore, interesting extensions to higher-order SUSY schemes were carried out by taking recourse to higher-derivative versions of the factorization operators [7–9].
* Corresponding author. Address: Department of Physics, National Cheng-Kung University, Tainan 701, Taiwan, ROC. E-mail addresses: [email protected] (B. Bagchi), [email protected] (T. Tanaka). 0003-4916/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2009.08.002
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
2439
The original extension in Ref. [7] was considered by using higher-order intertwining operators which are expressed as products of first-order linear differential operators and then applying the ordinary SUSY results. Later in Refs. [10,11], through the analysis of the general second-order case, the concept of reducibility was introduced. In this regard, a higher-order intertwining operator is said to be reducible if it is factorized into a product of first-order differential operators such that with respect to each factor there exists an intermediate real Hamiltonian satisfying a (shifted) SUSY relation. Otherwise it is called irreducible. This concept, however, seems less useful in view of the current status where non-Hermitian quantum theories have been investigated intensively since the discovery of PT symmetry [12]. In fact, the SUSY method turned to be useful also in constructing a complex potential with real spectrum [13–15]. In addition to the usefulness of the reality constraint, there arises a natural question about the well-definiteness of the concept if we take into account the fact that in general factorization of higher-order linear differential operators is not unique. The latter fact indeed has been reported in the context of the factorization method and ordinary SUSY QM, that is, there exist several Schrödinger operators that admit different factorizations [16–22], see also Ref. [23] for a recent approach. Hence, a reducible higher-order intertwining operator may admit simultaneously another factorization for which there are no intermediate real Hamiltonians. On the other hand, the non-uniqueness in factorizing intertwining operators of arbitrary finite orders was, to the best of our knowledge, first reported in Ref. [24] for the well-known quasi-solvable sextic anharmonic oscillator potentials in the framework of type A N -fold supersymmetry (see, Eq. (47) in the latter reference). Later it was shown that the non-uniqueness of factorizations in type A N -fold SUSY is a consequence of the underlying GLð2; CÞ symmetry [25]. In a recent communication [26], the following two-parameter family of second-order supersymð1Þ ð2Þ b b BÞ, characterized by the two parameters A and B, satisfying metric (SSUSY) system ðh ; h ; A ð1Þ ð2Þ b b b b A Bh ¼ h A B was constructed: 2
ð1Þ
h
ð2Þ
h
c b þ c ¼ d þ V A;B ðxÞ e by B ¼B Eþ ; 2 2 dx2 2 bA b y c ¼ d þ V A;B;ext ðxÞ E c ; ¼A dx2 2 2
ð1Þ ð2Þ
where 2
V A;B ðxÞ ¼ ½B2 þ AðA þ 1Þcsch x Bð2A þ 1Þcsch x coth x;
ð3Þ
2ð2A þ 1Þ 2½4B2 ð2A þ 1Þ2 V A;B;ext ðxÞ ¼ V A;B ðxÞ þ ; 2B cosh x 2A 1 ð2B cosh x 2A 1Þ2
ð4Þ
and the constants E, e E, and c are given by
2 1 E¼ B ; 2
2 1 e E ¼ B ; 2
c ¼ 2B:
ð5Þ
b and B b are respectively given by The intertwining operators A
2B sinh x b ¼ d B 1 coth x A þ 1 cschx A ; dx 2 2 2B cosh x 2A 1 b ¼ d B 1 coth x A þ 1 cschx: B dx 2 2
ð6Þ ð7Þ
It was further shown in Ref. [26] that the system admits the intermediate Hamiltonians h given by 2 2 c b c ¼ d þ V A;B1 ðxÞ E c ¼ B by A bB b y þ c ¼ d þ V A;B1 ðxÞ e h¼A Eþ ; 2 2 2 2 dx2 dx2
ð8Þ
with 2
V A;B1 ðxÞ ¼ ½ðB 1Þ2 þ AðA þ 1Þcsch x ðB 1Þð2A þ 1Þcschx coth x;
ð9Þ
2440
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
b and Bh b ¼ hð2Þ A b ð1Þ ¼ h B. b We shall hereafter call the system which satisfy the ordinary SUSY relations Ah (1)–(9) the BQR SSUSY model. Taking into account the fact that type A 2-fold SUSY is the necessary and sufficient condition for the existence of (at least) two linearly independent analytic (local) solutions to Schrödinger equation of one degree of freedom [27], we immediately know that the above BQR SSUSY model, which is exactly solvable, also belongs to type A 2-fold SUSY. Therefore, it would be natural that it has (at least) two different factorizations of the second-order intertwining operator in view of the aforementioned GLð2; CÞ symmetry and has two different intermediate Hamiltonians correspondingly. Regarding the existence of intermediate Hamiltonians, we note the following fact shown in Ref. [24] that the most general form of type A N -fold SUSY quantum systems constructed directly from the N th-order intertwining operators of type A, namely, type A N -fold supercharge P N ¼ þ P N 1 . . . P NN (cf., Eq. (11)) by solving P N H ¼ H P N is more general than that of the systems constructed from the N repeated applications of the first-order intertwining operators P N k by solving ðk1Þ ð0Þ ¼ HðkÞ P and Hþ ¼ HðN Þ . It is apparent that P N kH N k ðk ¼ 1; . . . ; N Þ with the identification H ¼ H in the latter construction we automatically obtain a series of the intermediate Hamiltonians Hð1Þ ; . . . ; HðN 1Þ in addition to the N -fold SUSY pair Hamiltonians H at the cost of the generality. In the former construction, on the other hand, the existence of intermediate Hamiltonians is not guaranteed in general. Hence, the fact that the type A N -fold supercharge has the factorized form by definition does not necessarily imply the existence of intermediate Hamiltonians. In particular, the fact that type A N -fold supercharge admits different factorizations does not automatically mean that the most general type A N -fold SUSY system has different sets of intermediate Hamiltonians accordingly. Motivated by the backgrounds described above, we investigate in this article under what conditions type A N -fold SUSY systems admit intermediate Hamiltonians in the case of N ¼ 2. Furthermore, we also examine under the satisfaction of the conditions how many sets of such Hamiltonians are admissible for the type A 2-fold SUSY systems. In addition, we show that any such a system has another type of non-linear supersymmetries, namely, parasupersymmetry of order 2 [28]. We organize the article as follows. In the next section, we review the framework of type A N -fold SUSY by putting emphasis on the GLð2; CÞ symmetry. In Section 3, we investigate in details under what conditions a type A 2-fold SUSY quantum system has one or more intermediate Hamiltonians. In particular, we show that the maximum number of different intermediate Hamiltonians in type A 2-fold SUSY is two. In Section 4, we further show that when a type A 2-fold SUSY system admits (at least) one intermediate Hamiltonian, the system can have second-order parasupersymmetry. A novel generalization of 2-fold superalgebra is discussed briefly. As an application of the results, we construct in Section 5 a type A 2-fold SUSY system with two different intermediate Hamiltonians of generalized Pöschl–Teller type which includes the BQR SSUSY model as a particular case. Then, we close the article with discussion and perspectives of further developments in the last section. 2. Type A N -fold supersymmetry and GLð2; CÞ covariance Roughly speaking, type A N -fold SUSY quantum systems are composed of a pair of scalar Hamilto1 nians H and an N th-order linear differential operator P N of the following forms: 2
2
1 d 1 N 1 N þ WðxÞ2 ð2E0 ðxÞ EðxÞ2 Þ W 0 ðxÞ R; 2 dx2 2 24 2 N 1 Y d N 1 2k ¼ þ WðxÞ þ EðxÞ ; dx 2 k¼0
H ¼
ð10Þ
PN
ð11Þ
where R is a constant while EðxÞ and WðxÞ are analytic functions satisfying
1 We keep the original notations as far as possible. Thus, note that the function EðxÞ is different from the constant E in the BQR SSUSY model (1)–(9). Similarly, the function AðzÞ introduced later in (19) is different from the parameter A in the latter model.
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
d d d EðxÞ þ EðxÞ WðxÞ ¼ 0 for N P 2; dx dx dx d d d d 2EðxÞ EðxÞ þ EðxÞ EðxÞ ¼ 0 for N P 3: dx dx dx dx
2441
ð12Þ ð13Þ
The product of operators appeared in (11) is defined by n Y
Ak A n . . . A 1 A0 :
ð14Þ
k¼0
The operators H and P N satisfy an intertwining relation
PN H ¼ Hþ PN :
ð15Þ
One of the most important features of type A N -fold SUSY quantum systems is that the gauged Hame and Hþ introduced by iltonians H
e ¼ eW N H eW N ; H
W N ðxÞ ¼
N 1 2
Z
dx EðxÞ
Z
dx WðxÞ;
ð16Þ
e ðAÞ : preserve the so-called type A monomial space V N
eV e ðAÞ V e ðAÞ ; H N N
e ðAÞ ¼ h1; zðxÞ; . . . ; zðxÞN 1 i; V N
ð17Þ
where the new variable zðxÞ satisfies
z00 ðxÞ ¼ EðxÞz0 ðxÞ:
ð18Þ
Explicitly, they are given by
2 e ¼ AðzÞ d þ N 2 A0 ðzÞ Q ðzÞ d ðN 1ÞðN 2Þ A00 ðzÞ N 1 Q 0 ðzÞ þ R ; H 2 dz 12 2 dz2
ð19Þ
where the new functions AðzÞ and Q ðzÞ are defined by
2AðzÞ ¼ z0 ðxÞ2 ;
Q ðzÞ ¼ z0 ðxÞWðxÞ:
ð20Þ
The conditions (12) and (13) for type A N -fold SUSY are reduced to the following simple forms in terms of z: 3
d Q ðzÞ ¼ 0 for N P 2; dz3 5 d AðzÞ ¼ 0 for N P 3: dz5
ð21Þ ð22Þ
In particular, the condition (21) indicates that Q ðzÞ is a polynomial of at most second-degree in z for all N P 2:
Q ðzÞ ¼ b2 z2 þ b1 z þ b0 :
ð23Þ
In terms of zðxÞ, the potential terms V ðxÞ of type A N -fold SUSY Hamiltonians in (10) are expressed as
V ðxÞ ¼
1 3 2 ðN 1Þ AðzÞA00 ðzÞ A0 ðzÞ2 3Q ðzÞ2 3N ðA0 ðzÞQ ðzÞ 2AðzÞQ 0 ðzÞÞ 12AðzÞ 4
Rjz¼zðxÞ : ð24Þ The type A N -fold SUSY systems (10) and (11) have an underlying symmetry which, as we shall show, plays a central role in investigating the existence of intermediate Hamiltonians. It is GLð2; CÞ linear fractional transformations on the variable z introduced as
2442
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
z¼
aw þ b ða; b; c; d 2 C; D ad bc–0Þ: cw þ d
ð25Þ
Then, the type A monomial space is invariant under the GLð2; CÞ transformations induced by (25):
b e ðAÞ ½w ¼ ðcw þ dÞN 1 V e ðAÞ ½z # V e ðAÞ ½z V N N N
wþb z¼acwþd
e ðAÞ ½w: ¼V N
ð26Þ
e and Hþ are both such linear differential operators that preserve the type The gauged Hamiltonians H A monomial space, as was shown in (17). As a consequence, they are covariant under the following GLð2; CÞ transformations:
b e ½z # H e ½w ¼ ðcw þ dÞN 1 H e ½zðcw þ dÞðN 1Þ H
wþb z¼acwþd
ð27Þ
;
b b þ both have the same forms as given in (19) with z ree and H that is, the transformed operators H b b ðwÞ given by placed by w and with AðzÞ and Q ðzÞ replaced by the transformed functions AðwÞ and Q
b AðzÞ # AðwÞ ¼ D2 ðcw þ dÞ4 AðzÞ
wþb z¼acwþd
b ðwÞ ¼ D1 ðcw þ dÞ2 QðzÞ Q ðzÞ # Q
ð28Þ
;
z¼acwþb wþd
ð29Þ
:
b ðwÞ for arbitrary N P 2 is given by In particular, the explicit form of Q
^1 w þ b ^0 ; ^2 w2 þ b b ðwÞ ¼ b Q
ð30Þ
with
0
1 0 2 10 1 ^2 b2 a ac c2 b B^ C CB C 1 B @ b1 A ¼ D @ 2ab ad þ bc 2cd A@ b1 A: ^0 b0 bd d2 b2 b
ð31Þ
Utilizing the transformation (29) and the formulas
wðxÞ ¼
dzðxÞ b
czðxÞ a
;
w0 ðxÞ ¼
Dz0 ðxÞ ðczðxÞ aÞ2
¼ D1 ðcwðxÞ þ dÞ2 z0 ðxÞ;
ð32Þ
we obtain the transformations of WðxÞ and EðxÞ as
b c ðxÞ ¼ Q ðwÞ ¼ Q ðzÞ ¼ WðxÞ; WðxÞ # W z0 ðxÞ w0 ðxÞ 00 00 w ðxÞ z ðxÞ 2cz0 ðxÞ 2cz0 ðxÞ ¼ EðxÞ : ¼ 0 EðxÞ # b EðxÞ ¼ 0 w ðxÞ z ðxÞ czðxÞ a czðxÞ a
ð33Þ ð34Þ
The invariance of the pair of type A N -fold SUSY Hamiltonians H under the GLð2; CÞ transformations also follows from a direct application of (33) and (34):
c; b H ½W; E ¼ H ½ W E;
ð35Þ
c ðxÞ ¼ WðxÞ and from (18) and (34) we have since W
2b E 0 ðxÞ b EðxÞ2 ¼ 2E0 ðxÞ EðxÞ2 :
ð36Þ P N
On the other hand, the invariance of the type A N -fold supercharge is not manifest in the factorized form (11) and due to the fact that b EðxÞ – EðxÞ the factorized form is in appearance not invariant:
d c ðxÞ þ N 1 2k b EðxÞ þW dx 2 k¼0 N 1 Y d N 1 2k 2cz0 ðxÞ ¼ þ WðxÞ þ EðxÞ : dx 2 czðxÞ a k¼0
c; b PN ½ W E ¼
N 1 Y
ð37Þ
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
2443
The fact that P N is also invariant under the GLð2; CÞ transformations
c; b PN ½ W E ¼ PN ½W; E;
ð38Þ
proved in Ref. [25], despite the non-invariance in appearance for c – 0, indicates that the type A N -fold supercharge admits a one-parameter family of different factorizations characterized by the parameter a=c. 3. Intermediate Hamiltonians for N ¼ 2 From now on, we shall restrict ourselves to the case of N ¼ 2. The type A N -fold SUSY systems (10) and (11) for N ¼ 2 read 2
d E0 ðxÞ EðxÞ2 þ WðxÞ2 þ 2R 2W 0 ðxÞ; 2 2 dx 4 2 d d E0 ðxÞ EðxÞ2 P2 ¼ P 21 P22 ¼ 2 þ 2WðxÞ þ W 0 ðxÞ þ WðxÞ2 þ ; dx 2 dx 4 2H ¼
ð39Þ ð40Þ
where
P21 ¼
d EðxÞ þ WðxÞ ; dx 2
P22 ¼
d EðxÞ þ WðxÞ þ : dx 2
ð41Þ
The superHamiltonian H 2 and the type A 2-fold supercharges Q 2 introduced with the ordinary fermionic variables w as
H 2 ¼ H w wþ þ Hþ wþ w ;
Q 2 ¼ P2 w ;
ð42Þ
satisfy the type A 2-fold superalgebra [25]:
½Q 2 ; H 2 ¼ fQ 2 ; Q 2 g ¼ 0;
2
fQ 2 ; Q þ2 g ¼ 4ðH 2 þ RÞ2 þ 4b0 b2 b1 :
ð43Þ
In the expanded form of the type A 2-fold supercharge components (40), its invariance under the GLð2; CÞ transformations is now manifest by applying (33) and (36):
c; b P2 ½ W E ¼ P2 ½W; E:
ð44Þ
However, each factor of the type A N -fold supercharge in the factorized form is not invariant since b EðxÞ–EðxÞ as shown in (37), and thus we generally have
b P ½ W c; b P E–P21 ½W; E; 21 21
b P ½ W c; b P E–P22 ½W; E: 22 22
ð45Þ
i1
Next, we introduce another Hamiltonian H , which we shall call an intermediate Hamiltonian, as
P22 H ¼ Hi1 P22 ;
P21 Hi1 ¼ Hþ P21 ;
ð46Þ i1
which are compatible with (15). It is evident that H is in general not invariant under the GLð2; CÞ i1 transformation in contrast with H due to the fact that both of P 21 and P 22 which intertwine H with H have no invariance (45). Hence, we can expect a family of intermediate Hamiltonians for each given type A 2-fold SUSY system. Needless to say, however, we do not always have such an intermediate Hamiltonian for a given system. The necessary and sufficient conditions for its existence are that there exist two constants C 22 and C 21 such that H and Hi1 are expressed as (see, e.g., Refs. [2–4])
2H ¼ Pþ22 P22 þ 2C 22 ;
2Hþ ¼ P21 Pþ21 þ 2C 21 ;
2Hi1 ¼ P22 Pþ22 þ 2C 22 ¼ Pþ21 P21 þ 2C 21 ; 2 where Pþ ij are the transpositions of P ij , that is,
2
Note that we do not assume the reality of the functions WðxÞ and EðxÞ.
ð47Þ
2444
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
Pþ21 ¼
d EðxÞ þ WðxÞ ; dx 2
P þ22 ¼
d EðxÞ þ WðxÞ þ : dx 2
ð48Þ
More explicitly, the conditions (47) read 2
d EðxÞ2 E0 ðxÞ þ WðxÞ2 þ EðxÞWðxÞ þ W 0 ðxÞ þ 2C 22 ; 2 2 dx 4 2 d EðxÞ2 E0 ðxÞ 2Hi1 ¼ 2 þ WðxÞ2 þ EðxÞWðxÞ þ þ W 0 ðxÞ þ 2C 22 þ 2 dx 4 2 d EðxÞ2 E0 ðxÞ ¼ 2 þ WðxÞ2 EðxÞWðxÞ þ W 0 ðxÞ þ 2C 21 ; þ 2 dx 4 2 d EðxÞ2 E0 ðxÞ þ W 0 ðxÞ þ 2C 21 : 2Hþ ¼ 2 þ WðxÞ2 EðxÞWðxÞ þ 2 dx 4 2H ¼
ð49aÞ
ð49bÞ ð49cÞ
From Eqs. (39) and (49), the necessary and sufficient conditions reduce to
W 0 ðxÞ þ EðxÞWðxÞ ¼ 2R 2C 22 ¼ C 21 C 22 ¼ 2C 21 þ 2R:
ð50Þ
Noting the relation
W 0 ðxÞ þ EðxÞWðxÞ ¼ Q 0 ðzÞ;
ð51Þ
which easily follows from (18) and (20), we find that the latter conditions (50) are equivalent to
Q ðzÞ ¼ ðC 22 C 21 Þz þ b0 ;
2R ¼ C 22 þ C 21 ;
ð52Þ
with b0 being another constant. We recall that for the most general type A N -fold SUSY systems for all N P 2, Q ðzÞ is given by a polynomial of at most second-degree (23). Hence, a given type A 2-fold SUSY system (39) admits an intermediate Hamiltonian Hi1 satisfying Eq. (46) if and only if
b2 ¼ 0;
b1 ¼ C 22 C 21 ;
2R ¼ C 22 þ C 21 :
ð53Þ
The last two conditions in (53) just determine these constants for the given values of b1 and R. Hence, only the first condition in (53) is essential for the existence of an intermediate Hamiltonian. As was discussed previously, the type A 2-fold supercharge P 2 is invariant under the GLð2; CÞ trans formation (44) while its factors P 22 and P 21 are not (45) for c – 0. As a consequence, the necessary and sufficient conditions (53) for the existence of another intermediate Hamiltonian Hi2 after a GLð2; CÞ transformation are accordingly changed as
^2 ¼ 0; b
^1 ¼ C b 22 C b 21 ; b
b 22 þ C b 21 ; 2R ¼ C
ð54Þ
b 22 and C b 21 are another set of constants. Again, only the first condition in (54) is essential for where C the existence of another intermediate Hamiltonian Hi2 after the transformation. Under the fulfillment of the conditions (54), the original type A 2-fold SUSY Hamiltonians H and the new intermediate Hamiltonian Hi2 are expressed in terms of the transformed supercharges as
b b bþ P b P b bþ 2H ¼ P 2Hþ ¼ P 21 21 þ 2 C 21 ; 22 22 þ 2 C 22 ; i2 b b bþ b P bþ b 2H ¼ P 22 22 þ 2 C 22 ¼ P 21 P 21 þ 2 C 21 ;
ð55Þ
b þ are the transpositions of P b which were defined in (45), that is, where P ij ij
b b ¼ d þ W c ðxÞ EðxÞ ; P 21 2 dx
b b ¼ d þ W c ðxÞ þ EðxÞ : P 22 2 dx
ð56Þ
It is now clear that a type A 2-fold SUSY system which satisfies the conditions (53) and thus admits an intermediate Hamiltonian Hi1 also admits another different intermediate Hamiltonian Hi2 after a GLð2; CÞ transformation with c–0 if and only if the conditions (54) are simultaneously fulfilled in ^2 ¼ 0. From the transformation formula addition to (53). It essentially means the satisfaction of b2 ¼ b (31) we immediately see that it is only possible for the GLð2; CÞ transformation with c – 0 which satisfies
2445
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
ab1 þ cb0 ¼ 0:
ð57Þ
Let us first consider the case when b1 ¼ 0. In this case we can assume that b0 – 0; otherwise Q ðzÞ ¼ 0 since b2 ¼ 0 is already assumed in order to meet the condition (53), and from (24) V ðxÞ ¼ V þ ðxÞ which means that the system is trivial as 2-fold SUSY. But for b1 ¼ 0 and b0 – 0 there is no solution to Eq. (57) except for c ¼ 0. But for any GLð2; CÞ transformation with c ¼ 0, the function EðxÞ is invariant by Eq. (34) and so is the intermediate Hamiltonian Hi1 . Hence in the case of b1 ¼ 0 the factorization of type A 2-fold supercharge admitting intermediate Hamiltonians is unique. On the other hand, in the case of b1 – 0 the solution to Eq. (57) is given by
a=c ¼ b0 =b1 :
ð58Þ
As was shown previously, any type A 2-fold supercharge admits one-parameter family of factoriza tions P 2 ¼ P 21 P 22 characterized by the parameter a=c. In addition, any type A 2-fold SUSY system with b2 ¼ 0 has at least one intermediate Hamiltonian Hi1 . Then, the result (58) tells us that if the system further satisfies b1 – 0, it can admit a one and only one additional and different intermediate Hamiltonian Hi2 at the one point (58) in the parameter space of a=c 2 C. In Table 1, we summarize the results. 4. Second-order parasupersymmetry In the previous section, we have just verified that a type A 2-fold SUSY system ðH ; P 2 ¼ P 21 P 22 Þ admits (at least) one intermediate Hamiltonian Hi1 if and only if the condition b2 ¼ 0 holds. In this section, we shall further show that any such a system can possess an additional symmetry, namely, parasupersymmetry of order 2 introduced in Ref. [28]. Indeed, for a given such type A 2-fold SUSY system we can define a triple of operators ðH P ; Q P Þ by
H P ¼ H ðwP Þ2 ðwþP Þ2 þ Hi1 ðwþP wP ðwþP Þ2 ðwP Þ2 Þ þ Hþ ðwþP Þ2 ðwP Þ2 ;
ð59aÞ
Q P ¼ Pþ22 ðwP Þ2 wþP þ Pþ21 wþP ðwP Þ2 ;
ð59bÞ
Q þP ¼ P22 wP ðwþP Þ2 þ P21 ðwþP Þ2 wP ;
where w P are parafermions of order 2 satisfying [29]
ðwP Þ2 –0;
ðwP Þ3 ¼ 0;
fwP ; wþP g þ fðwP Þ2 ; ðwþP Þ2 g ¼ 2I:
ð60Þ
Then, using Eq. (47) and the parafermionic algebra of order 2 in Ref. [29] we can show that the triple ðH P ; Q P Þ defined as (59) satisfies the second-order paraSUSY relations in Ref. [28]:
ðQ P Þ2 –0;
ðQ P Þ3 ¼ 0;
ðQ P Þ2 Q P
Q P Q P Q P
þ
þ
½Q P ; H P ¼ 0; Q P ðQ P Þ2
¼
ð61Þ
4Q P H P ;
ð62Þ
if and only if the constants C ij in (47) satisfy
C 22 ¼ C 21 ¼ b1 =2;
ð63Þ
and thus in particular R ¼ 0 by (53). Hence, we conclude that any type A 2-fold SUSY quantum system with (at least) one intermediate Hamiltonian also has paraSUSY of order 2 when R ¼ 0. The additional restriction R ¼ 0 arises since one of the paraSUSY conditions (62) is not invariant under any constant shift of H P . Furthermore, as was shown in Ref. [29] this type of realization of second-order paraSUSY
Table 1 The admissible numbers of different intermediate Hamiltonians in type A 2-fold supersymmetry. Conditions
Number of Hi
b2 – 0 b2 ¼ b1 ¼ 0 b2 ¼ 0, b1 – 0
0 1 2
2446
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
systems admits an additional novel non-linear relation as the following (cf., Eq. (6.66) in the latter reference): 2
ðQ P Þ2 ðQ þP Þ2 þ Q P ðQ P Þ2 Q P þ ðQ þP Þ2 ðQ P Þ2 ¼ 4ðH P Þ2 b1 ;
ð64Þ
which can be regarded as a generalized (type A) 2-fold superalgebra. In fact, on the one hand we immediately have from the paraSUSY relations in (61)
fðQ P Þ2 ; ðQ P Þ2 g ¼ 0;
½ðQ P Þ2 ; H P ¼ 0;
ð65Þ
while on the other hand the non-linear relation (64) reduces, in the subsector with the parafermion number zero and two, to 2
fðQ P Þ2 ; ðQ þP Þ2 g ¼ 4ðH P Þ2 b1 :
ð66Þ
Then, the commutation and anti-commutation relations (65) and (66) are, under the assumed condition b2 ¼ 0 and R ¼ 0, entirely identical with the type A 2-fold superalgebra (43) with the trivial 2 identification of the type A 2-fold supercharges Q 2 with ðQ P Þ and with the observation that in the subsector H P is essentially identical with H 2 . The relation between type A 2-fold SUSY and secondorder paraSUSY was briefly referred to in Ref. [29]. Here we have firstly shown the necessary and sufficient conditions for a type A 2-fold SUSY system to admit simultaneously second-order paraSUSY, namely, Eqs. (53) and (63). Finally, it is evident that we can construct two sets of second-order paraSUSY systems whenever a type A 2-fold SUSY system has two different intermediate Hamiltonians.
5. An application to the generalized Pöschl–Teller potential As an application of the general framework discussed in the previous two sections, we shall reconstruct the BQR SSUSY model (1)–(9) and its generalization which preserves all the SUSY and SSUSY structure therein. To this end, let us first choose the change of variable z ¼ zðxÞ which determines the relation between physical Hamiltonians and gauged ones as
x 2Aþ1 x 2A2Bþ1 x ð2Aþ2Bþ1Þ zðxÞ ¼ ðsinh xÞ2B tanh ¼ 22B sinh cosh ; 2 2 2
ð67Þ
where A and B are both constants. The function AðzÞ ¼ z0 ðxÞ2 =2 defined in (20) and its derivatives with respect to z are in general transcendental functions of z. Explicitly, they read
2B cosh x 2A 1 ; sinh x zðxÞ2 ð2B cosh x 2A 1Þ2
z0 ðxÞ ¼ zðxÞ
ð68Þ
AðzÞ ¼
ð69Þ
; 2 x 2sinh aþ cosh x aþ2 A0 ðzÞ ¼ zðxÞ 4B2 1 ; 2 sinh x b cosh x b2 2A þ 1 ; A00 ðzÞ ¼ 4B2 1 2 2B cosh x 2A 1 sinh x
ð70Þ ð71Þ
where aþ i and bi are all constants given by
a1 ¼ ð2A þ 1Þð4B 1Þ; a2 ¼ ð2A þ 1Þ2 þ 2Bð2B 1Þ; b1 ¼ ð2A þ 1Þð4B þ 3Þ;
b2 ¼ ð2A þ 1Þ2 þ 2ðB þ 1Þð2B þ 1Þ:
ð72Þ ð73Þ
Substituting them into the most general form of a pair of type A 2-fold SUSY potentials, Eq. (24) with N ¼ 2, we obtain
2447
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451 2
V ðxÞ ¼
Q ðzðxÞÞ2 sinh x 2zðxÞ2 ð2B cosh x 2A 1Þ
4ð2A þ 1ÞB cosh x þ ð2A þ 1Þ2 12B2
þ 2
8ð2B cosh x 2A 1Þ2 2
ð2A þ 1ÞB cosh x AðA þ 1Þ B 2
2sinh x 2
B2 R 2
þ
2
Q ðzðxÞÞð4B sinh x aþ1 cosh x þ aþ2 Þ zðxÞð2B cosh x 2A 1Þ2
Q 0 ðzðxÞÞ;
ð74Þ
where Q ðzÞ is a polynomial of at most second-degree given as in (23). The function WðxÞ characterizing the type A system in this case reads
WðxÞ ¼
Q ðzðxÞÞ Q ðzðxÞÞ sinh x : ¼ z0 ðxÞ zðxÞð2B cosh x 2A 1Þ
ð75Þ
The other function EðxÞ defined through the relation (18), which also characterizes the type A system, is calculated as
EðxÞ ¼
z00 ðxÞ ð2B þ 1Þ cosh x 2A 1 2B sinh x þ : ¼ z0 ðxÞ sinh x 2B cosh x 2A 1
ð76Þ
Next, let us consider the case when the type A 2-fold SUSY system admits an intermediate Hamiltonian, namely, b2 ¼ 0. From (75) and (76) we have
W 0 ðxÞ ¼
Q ðzðxÞÞ ð2A þ 1Þ cosh x 2B Q ðzðxÞÞ; Q 0 ðzðxÞÞ zðxÞ zðxÞð2B cosh x 2A 1Þ2
E0 ðxÞ ¼
2ð2A þ 1ÞB cosh x 4B2 ð2B cosh x 2A 1Þ2
EðxÞ2 ¼ 4B2 þ
ð2A þ 1Þ2 4B2 ð2B cosh x 2A 1Þ
2
ð2A þ 1Þ cosh x 2B 1 2
sinh x
ð77Þ ð78Þ
;
2ð2A þ 1Þð2B þ 1Þ cosh x ð2A þ 1Þ2 ð2B þ 1Þ2 2
sinh x
:
ð79Þ
Substituting (75)–(79) into (49b), using the relations (53) among the constants, and noting that Q 0 ðzÞ ¼ b1 when b2 ¼ 0, we obtain the intermediate potential V i1 ðxÞ as 2
V i1 ðxÞ ¼
QðzðxÞÞ2 sinh x 2
2zðxÞ ð2B cosh x 2A 1Þ
2
4ð2A þ 1ÞB cosh x ð2A þ 1Þ2 4B2 8ð2B cosh x 2A 1Þ2
ð2A þ 1ÞðB þ 1Þ cosh x AðA þ 1Þ ðB þ 1Þ2 2
2sinh x
þ
B2 R: 2
ð80Þ
Next, we shall consider the case when the system admits another different intermediate Hamiltonian Hi2 , namely, b1 – 0. The GLð2; CÞ transformation which takes the type A 2-fold supercharge to another factorization for which Hi2 exists must satisfy the condition (58). The parameter d does not play an important role in our context, so we set d ¼ 0 without any loss of generality. But in this case b cannot be 0 otherwise D ¼ 0. Thus, we fix the parameters as
a=c ¼ b0 =b1 z0 ; b=c ¼ 1; d ¼ 0:
ð81Þ
In other words, we choose the following GLð2; CÞ transformation on the variable zðxÞ:
wðxÞ ¼
1 1 ¼ : zðxÞ þ z0 ðsinh xÞ2B ðtanh x=2Þ2Aþ1 z0
ð82Þ
The function WðxÞ is invariant under the transformation (see, Eq. (33)) while EðxÞ is transformed according to (34) as
2448
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
2z0 ðxÞ 2zðxÞ 2B cosh x 2A 1 b EðxÞ ¼ EðxÞ ¼ EðxÞ þ zðxÞ þ z0 zðxÞ þ z0 sinh x ¼
½ð2B 1ÞzðxÞ ð2B þ 1Þz0 cosh x ð2A þ 1ÞðzðxÞ z0 Þ 2B sinh x : þ ðzðxÞ þ z0 Þ sinh x 2B cosh x 2A 1
ð83Þ
Noting the relation Q ðzÞ ¼ b1 ðz þ z0 Þ when b2 ¼ 0, we have
b c ðxÞ ¼ EðxÞWðxÞ þ 2b1 : EðxÞ W
ð84Þ
From the second expression of b EðxÞ in (83), we obtain the following formulas:
2zðxÞ ð2A þ 1Þ cosh x 2B 2z0 zðxÞ ð2B cosh x 2A 1Þ2 b E 0 ðxÞ ¼ E0 ðxÞ þ ; 2 2 zðxÞ þ z0 ðzðxÞ þ z0 Þ2 sinh x sinh x
ð85Þ
8BzðxÞ 4zðxÞ cosh xð2B cosh x 2A 1Þ 4z0 zðxÞð2B cosh x 2A 1Þ2 b : ð86Þ EðxÞ2 ¼ EðxÞ2 þ 2 2 zðxÞ þ z0 ðzðxÞ þ z0 Þ2 sinh x ðzðxÞ þ z0 Þsinh x ^1 ¼ b1 by the transformation formula (31) in Substituting (83)–(86) into Eq. (55) and noting that b our choice of the parameters (81), we obtain for H the same potentials as the ones in (74), as they should be, while for the other intermediate Hamiltonian Hi2 the following form of the potential:
zðxÞ2 ½ð2A þ 1Þ cosh x 2B
V i2 ðxÞ ¼ V i1 ðxÞ þ
2
2
ðzðxÞ þ z0 Þ sinh x
z0 zðxÞðaþ1 cosh x aþ2 Þ
þ
2
2
ðzðxÞ þ z0 Þ sinh x
4B2 z0 zðxÞ ðz þ z0 Þ2
:
ð87Þ
If we substitute (80) for V i1 ðxÞ into the above, we finally obtain the full expression of V i2 ðxÞ as 2
V i2 ðxÞ ¼
Q ðzðxÞÞ2 sinh x 2
2
2zðxÞ ð2B cosh x 2A 1Þ
4ð2A þ 1ÞB cosh x ð2A þ 1Þ2 4B2 8ð2B cosh x 2A 1Þ2
ð2A þ 1ÞðB 1Þ cosh x AðA þ 1Þ ðB 1Þ2 2
2sinh x z20 ½ð2A
þ 1Þ cosh x 2B 2
ðzðxÞ þ z0 Þ
4B2 z0 zðxÞ ðzðxÞ þ z0 Þ2
þ
B2 z0 zðxÞða1 cosh x a2 Þ Rþ 2 2 ðzðxÞ þ z0 Þ2 sinh x ð88Þ
;
where a i are defined in (72). We are now in a position to show that the type A 2-fold system with the two intermediate Hamiltonians (74), (80), and (88) contains as a special case the BQR SSUSY model (1)–(9). For the purpose, let put b0 ¼ 0 and b1 ¼ bB. In this case, Q ðzÞ ¼ bBz and z0 ¼ b0 =b1 ¼ 0. Then, the 2-fold SUSY pair of the potentials (74) and the two intermediate potentials (80) and (88) reduce to, respectively, 2
V ðxÞ ¼
2
2
4ðb þ 1Þð2A þ 1ÞB cosh x ðb 1Þð2A þ 1Þ2 4ðb þ 3ÞB2 8ð2B cosh x 2A 1Þ2
ð2A þ 1ÞB cosh x AðA þ 1Þ B2 2
2sinh x
2
þ
b B2 ð2A þ 1Þ cosh x 2B ; þ R bB 8 2 ð2B cosh x 2A 1Þ2
ð89Þ
and 2
V i1 ðxÞ ¼ ðb 1Þ and
4ð2A þ 1ÞB cosh x ð2A þ 1Þ2 4B2 8ð2B cosh x 2A 1Þ2
ð2A þ 1ÞðB þ 1Þ cosh x AðA þ 1Þ ðB þ 1Þ2 2
sinh x
2
þ
b B2 þ R; 8 2
ð90Þ
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
2
V i2 ðxÞ ¼ ðb 1Þ
2449
4ð2A þ 1ÞB cosh x ð2A þ 1Þ2 4B2 8ð2B cosh x 2A 1Þ2
ð2A þ 1ÞðB 1Þ cosh x AðA þ 1Þ ðB 1Þ2 2
sinh x
2
þ
b B2 þ R: 8 2
ð91Þ
b The components of supercharges P ij and P ij given by (41), (48) and (56) in this case read
d ð2B þ 1Þ cosh x 2A 1 ðb 1ÞB sinh x þ þ ; dx 2 sinh x 2B cosh x 2A 1 d ð2B þ 1Þ cosh x 2A 1 ðb þ 1ÞB sinh x þ ; P22 ¼ dx 2 sinh x 2B cosh x 2A 1 b ¼ d ð2B 1Þ cosh x 2A 1 þ ðb 1ÞB sinh x ; P 21 dx 2 sinh x 2B cosh x 2A 1 d ð2B 1Þ cosh x 2A 1 ðb þ 1ÞB sinh x b þ : P 22 ¼ þ dx 2 sinh x 2B cosh x 2A 1
P21 ¼
ð92Þ ð93Þ ð94Þ ð95Þ
It is now easy to see that the BQR SSUSY model is realized when b ¼ 1. Indeed, we have the following correspondences when b ¼ 1:
1 1 þ B2 2R; 2V þ ðxÞ ¼ V A;B; ext ðxÞ þ þ B2 2R; 4 4 1 1 2V i1 ðxÞ ¼ V A;Bþ1 ðxÞ þ þ B2 2R; 2V i2 ðxÞ ¼ V A;B1 ðxÞ þ þ B2 2R; 4 4 b Pþ or P b y ; P or P b ¼ A; bþ ¼ A b ¼ B; b P þ or P bþ ¼ B by; P21 or P 21 21 22 22 21 22 22 2V ðxÞ ¼ V A;B ðxÞ þ
and in particular
P 2
¼
P 21 P 22
¼
b P b P 21 22
ð96Þ ð97Þ ð98Þ
b B. b The relations among the constants are given by ¼A
b 22 ¼ 4 C b 21 ¼ 2B; c ¼ 4C 22 ¼ 4C 21 ¼ 2B or c ¼ 4 C 1 c c e E þ ¼ E ¼ B2 þ ; R ¼ 0: 4 2 2
ð99Þ ð100Þ
The last equality R ¼ 0 means from the results in Section 4 that the BQR SSUSY model also has secondorder paraSUSY. We note that the reason why the BQR SSUSY model is realized as the particular case b2 ¼ b0 ¼ 0 of the most general type A 2-fold SUSY is the same as the one discussed in Ref. [27], Section 5. 6. Discussion and summary In this article, we have investigated in detail under what conditions type A N -fold SUSY systems can have intermediate Hamiltonians in the case of N ¼ 2. It turns out that although type A 2-fold supercharge admits a one-parameter family of factorization into product of two first-order linear differential operators due to the underlying GLð2; CÞ symmetry, at most two different intermediate Hamiltonians are admissible. As a by product of the studies, we have also obtained the necessary and sufficient conditions for a type A 2-fold SUSY system to possess paraSUSY of order 2 as well. When it is the case, the type A 2-fold superalgebra together with the second-order parasuperalgebra constitute a generalized 2-fold superalgebra. As a demonstration of the general arguments, we have constructed the generalized Pöschl–Teller potentials which are components of type A 2-fold SUSY with two intermediate Hamiltonians and reduce to the BQR SSUSY model in a particular case. As for the concept like the reducibility in Ref. [10], the present investigations indicate that it would be more natural and useful to classify higher-order intertwining operators according to the existence and the number of intermediate Hamiltonians as has been done in Table 1. After employing the latter classification scheme, we can further classify them according to the properties of the intermediate Hamiltonians such as Hermiticity, PT symmetry, and so on. Regarding the generalized Pöschl–Teller potentials constructed in Section 5, it is worth noticing that the framework of N -fold SUSY works well even when the function AðzÞ, which controls the
2450
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
change of variable z ¼ zðxÞ from the physical coordinate x to the variable z in the gauged space, is a transcendental function of z without destroying quasi-solvability. For all N P 3 cases type A N -fold SUSY requires the additional condition (22) so that AðzÞ is allowed to be at most a polynomial of fourth order in z, which results in the admissible change of variable to be at most an elliptic function, see, e.g., Ref. [25]. For the N ¼ 2 case, on the other hand, there are no such restrictions and, to the best of our knowledge, our generalized Pöschl–Teller potentials are the first quasi-solvable examples where AðzÞ is given by a transcendental function of z as Eq. (69). The analyses for N ¼ 2 carried out in this article are easily generalized to the cases N P 3, but we anticipate that richer structure could emerge for the higher N cases. In the case of N ¼ 3, for instance, according to the factorization of type A 3-fold supercharge P 3 ¼ P 31 P 32 P 33 we can consider not only the and P and between P case where intermediate Hamiltonians between P 31 32 32 and P 33 both exist, but also the cases where they exist only between the former place or only between the latter place exclusively. It is also interesting to study whether or not type A N -fold SUSY systems for higher N , when they have intermediate Hamiltonians, can admit another symmetry. The fact that in the case of N ¼ 2 they have second-order paraSUSY indicates that they could have higher-order paraSUSY [30,31] for N P 3. Indeed, it was shown in Ref. [32] that a certain realization of paraSUSY of order 3 also admits a generalized 3-fold superalgebra. Hence, at least in the case of N ¼ 3 we have a reasonable basis to expect paraSUSY as an additional symmetry. Other candidates might be quasi-paraSUSY introduced in Ref. [29] and N -fold paraSUSY in Ref. [33]. Acknowledgments This work (T.Tanaka) was partially supported by the National Cheng Kung University under the grant No. OUA:95-3-2-071. References [1] E. Witten, Dynamical breaking of supersymmetry, Nucl. Phys. B 188 (1981) 513–554. [2] F. Cooper, A. Khare, U. Sukhatme, Supersymmetry and quantum mechanics, Phys. Rep. 251 (1995) 267–385. arXiv:hep-th/ 9405029. [3] G. Junker, Supersymmetric Methods in Quantum and Statistical Physics, Springer, Berlin, 1996. [4] B.K. Bagchi, Supersymmetry in Quantum and Classical Mechanics, Chapman and Hall/CRC press, Florida, 2000. [5] E. Schrödinger, A method of determining quantum-mechanical eigenvalues and eigenfunctions, Proc. R. Irish Acad. A 46 (1940) 9–16. [6] G. Darboux, Comput. Rend. Acad. Sci. 94 (1882) 1456. [7] A.A. Andrianov, M.V. Ioffe, V.P. Spiridonov, Higher-derivative supersymmetry and the Witten index, Phys. Lett. A 174 (1993) 273–279. arXiv:hep-th/9303005. [8] H. Aoyama, M. Sato, T. Tanaka, N -fold supersymmetry in quantum mechanics: general formalism, Nucl. Phys. B 619 (2001) 105–127. arXiv:quant-ph/0106037. [9] A.A. Andrianov, A.V. Sokolov, Nonlinear supersymmetry in quantum mechanics: algebraic properties and differential representations, Nucl. Phys. B 660 (2003) 25–50. arXiv:hep-th/0301062. [10] A.A. Andrianov, M.V. Ioffe, F. Cannata, J.P. Dedonder, Second order derivative supersymmetry, q deformations and the scattering problem, Int. J. Mod. Phys. A 10 (1995) 2683–2702. arXiv:hep-th/9404061. [11] A.A. Andrianov, M.V. Ioffe, D.N. Nishnianidze, Polynomial supersymmetry and dynamical symmetries in quantum mechanics, Theoret. Math. Phys. 104 (1995) 1129–1140. [12] C.M. Bender, S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998) 5243–5246. arXiv:physics/9712001. [13] F. Cannata, G. Junker, J. Trost, Schrödinger operators with complex potential but real spectrum, Phys. Lett. A 246 (1998) 219–226. arXiv:quant-ph/9805085. [14] B. Bagchi, R. Roychoudhury, A new PT -symmetric complex hamiltonian with a real spectrum, J. Phys. A: Math. Gen. 33 (2000) L1–L3. arXiv:quant-ph/9911104. [15] M. Znojil, F. Cannata, B. Bagchi, R. Roychoudhury, Supersymmetry without hermiticity within PT symmetric quantum mechanics, Phys. Lett. B 483 (2000) 284–289. arXiv:hep-th/0003277. [16] B. Mielnik, Factorization method and new potentials with the oscillator spectrum, J. Math. Phys. 25 (1984) 3387–3389. [17] D.J. Fernández C, New hydrogen-like potentials, Lett. Math. Phys. 8 (1984) 337–343. [18] D. Zhu, A new potential with the spectrum of an isotonic oscillator, J. Phys. A: Math. Gen. 20 (1987) 4331–4336. [19] C.N. Kumar, Isospectral Hamiltonians: generation of the soliton profile, J. Phys. A: Math. Gen. 20 (1987) 5397–5401. [20] N.A. Alves, E.D. Filho, The factorization method and supersymmetry, J. Phys. A: Math. Gen. 21 (1988) 3215–3225. [21] E.D. Filho, The morse oscillator generalised from supersymmetry, J. Phys. A: Math. Gen. 21 (1988) L1025–L1028. [22] A. Mitra, P.K. Roy, A. Lahiri, B. Bagchi, Nonuniqueness of the factorization scheme in quantum mechanics, Int. J. Theoret. Phys. 28 (1989) 911–916.
B. Bagchi, T. Tanaka / Annals of Physics 324 (2009) 2438–2451
2451
[23] C. Quesne, Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry, J. Phys. A: Math. Theor. 41 (2008) 392001. arXiv:0807.4087 [quant-ph]. [24] H. Aoyama, M. Sato, T. Tanaka, General forms of a N -fold supersymmetric family, Phys. Lett. B 503 (2001) 423–429. arXiv:quant-ph/0012065. [25] T. Tanaka, Type A N -fold supersymmetry and generalized Bender–Dunne polynomials, Nucl. Phys. B 662 (2003) 413–446. arXiv:hep-th/0212276. [26] B. Bagchi, C. Quesne, R. Roychoudhury, Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of PT symmetry, Pramana J. Phys. 73 (2009) 337–347. arXiv:0812.1488 [quant-ph].. [27] A. González-López, T. Tanaka, Nonlinear pseudo supersymmetry in the framework of N -fold supersymmetry, J. Phys. A: Math. Gen. 39 (2006) 3715–3723. arXiv:quant-ph/0602177. [28] V.A. Rubakov, V.P. Spiridonov, Parasupersymmetric quantum mechanics, Mod. Phys. Lett. A 3 (1988) 1337–1347. [29] T. Tanaka, Parasupersymmetry and N -fold supersymmetry in quantum many-body systems I. General formalism and second order, Ann. Phys. 322 (2007) 2350–2373. arXiv:hep-th/0610311. [30] M. Tomiya, Comment on generalized parasupersymmetric quantum mechanics, J. Phys. A: Math. Gen. 25 (1992) 4699– 4704. [31] A. Khare, Parasupersymmetric quantum mechanics of arbitrary order, J. Phys. A: Math. Gen. 25 (1992) L749–L754. [32] T. Tanaka, Parasupersymmetry and N -fold supersymmetry in quantum many-body systems II. Third order, Ann. Phys. 322 (2007) 2682–2702. arXiv:hep-th/0612263. [33] T. Tanaka, N -fold parasupersymmetry, Mod. Phys. Lett. A 22 (2007) 2191–2200. arXiv:hep-th/0611008.