Table Of ContentSimultaneous ordinary and type A -fold supersymmetries
N
in Schr¨odinger, Pauli, and Dirac equations
Choon-Lin Ho∗ and Toshiaki Tanaka†
Department of Physics, Tamkang University, Tamsui 25137, Taiwan, R.O.C.
We investigate physical models which possess simultaneous ordinary and type A N-fold super-
symmetries,whichwecalltypeA(N,1)-foldsupersymmetry. InequivalenttypeA(N,1)-foldsuper-
symmetric models with real-valued potentials are completely classified. Among them, we find that
a trigonometric Rosen–Morse type and its elliptic version are of physical interest. We investigate
various aspects of these models, namely, dynamical breaking and interrelation between ordinary
andN-foldsupersymmetries,shapeinvariance,quasi-solvability,andanassociated algebrawhichis
composedofonebosonicandfourfermionicoperatorsanddubbedtypeA(N,1)-foldsuperalgebra.
As realistic physical applications, we demonstrate how these systems can be embedded into Pauli
6
and Dirac equations in externalelectromagnetic fields.
0
0
PACSnumbers: 03.65.Ge, 03.65.Fd,03.65.Pm,11.30.Pb
2
Keywords: quantummechanics,N-foldsupersymmetry,quasi(-exact)solvability,shapeinvariance,relativis-
n ticwaveequations, algebraicmethods
a
J
3
2
I. INTRODUCTION
2
v
Eversince the formulationof supersymmetricquantummechanics [1], a muchdeeper understandingof structureof
0
varioussolvablepotentialsinnon-relativisticquantummechanicshasbeengainedthroughtheideasofsupersymmetry
2
(SUSY) (for a good review, see, e.g., [2, 3]). For instance, seemingly distinct potentials, e.g., an infinite square-well
0
9 and a cosec2 potential, are in fact supersymmetric partners, and their respective energy levels are connected by
0 supercharges. ThemethodsofSUSYhavealsobeenappliedtoobtainingexactspectraofquantumsystemswithmulti-
5 component wave functions, such as Pauli and Dirac equations. It is based on the fact that, for some electromagnetic
0 field configurations, Pauli and Dirac equations possess intrinsic SUSY structure, and different components of a wave
/
h function arerelatedby some supercharges[4, 5]. Morerecently, the method ofSUSY hasbeen employedto construct
t quasi-exactlysolvable potentials in the Pauliand Dirac equations [6, 7]. For quasi-exactlysolvable systems, it is only
-
p possible to determine algebraically a part of the spectrum but not the whole spectrum [8–12].
e Recently,theconceptsofSUSYandquasi-exactsolvabilitywerecombinedwithintheframeworkofso-called -fold
h N
supersymmetry [13], which is a natural generalization [14] of ordinary supersymmetric quantum mechanics. Among
:
v various nonlinear extensions of ordinary SUSY such as parasupersymmetry [15–17], fractional supersymmetry [18],
i and so on, -fold SUSY is characterized by the fact that anti-commutators of fermionic operators are polynomials
X N
of degree (at most) in bosonic operators. It has been proved in a generic way that -fold SUSY is essentially
r N N
equivalent to (weak) quasi-solvability (this latter term, in contrast to quasi-exact solvability, is used to include the
a
case where the system admits non-normalizable solutions in closed form) [13]. Up to now, three different families
of -fold supersymmetric systems have been found for arbitrary finite integer , namely, type A [19, 20], type B
N N
[21], and type C [22], which have correspondence with the classification of second-order linear differential operators
preserving a monomial-type vector space [23]. Of the three types, type A and type C under a specific condition have
been completely classified [20, 22].
While there are still a lot to be done in the mathematical developments of -fold SUSY, it is also interesting that
N
one looks for physical systems which possess such generalized SUSY. In view of the fact that so far all the -fold
N
SUSY potentials areonly one-dimensional,it is naturalthat one shouldlookfor physicalmodels whichareeffectively
one-dimensional. Experiencegainedinthe workin[6, 7]suggeststhatPauliandDiracequationsaregoodcandidates
to startwith. Inthis respectwe note thatthe authorsof[24]foundthatif the Pauliequationis generalizedsuchthat
the gyromagnetic ratio g =2 of the electron is changed to some unphysical values g =2 ( 2), then for certain
N N ≥
forms of magnetic fields, the generalized Pauli equation could possess type A -fold SUSY. The result is interesting
N
in a purely mathematical view point, but unfortunately it would not describe any existing physical systems.
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
2
Therefore, we would like to extend the realistic systems in [6, 7] to include -fold SUSY. Since the Pauli and
N
Dirac equations considered there all possess ordinary SUSY, it is therefore natural that we look for an ordinary su-
persymmetric system which has -fold supersymmetry as well, as a starting point of the aforementioned purpose.
N
This naturally led us further to consider a more general situation in which a system has simultaneous -fold super-
N
symmetry with two different values of . In our previous paper [25], we have succeeded in constructing a family of
N
such systems, which we have called type A ( , )-fold SUSY. These systems possess simultaneous type A - and
1 2 1
N N N
-foldSUSYs. Hence,theaimmentionedatthe beginningofthis paragraphisequivalenttofinding PauliandDirac
2
N
systems with type A ( ,1)-fold SUSY.
N
Inthispaper,weshallpresentadetailedstudyofSchr¨odinger,Pauli,andDiracsystemswhichpossessordinaryand
type A SUSYs simultaneously. It is amazing to realize that some well known shape-invariant potentials, namely, the
infinite square-well and the cosec2 potentials with specific strengths of the coupling constant, are indeed potentials
of such kind. We also find an elliptic generalization of these potentials which breaks shape invariance but preserves
the structure of type A ( ,1)-fold SUSY. We study various aspects of these models such as dynamical breaking and
N
interrelation between ordinary and -fold supersymmetries, shape invariance, quasi-solvability, and an associated
N
algebra which is composed of one bosonic and four fermionic operators and dubbed type A ( ,1)-fold superalgebra.
N
The organization of the paper is as follows. In Sect. II some of the main ideas of -fold SUSY are reviewed. We
N
then discuss in Sect. III the idea of simultaneous type A ( ,1)-fold SUSY. Section IV gives a detailed discussion of
N
trigonometric Rosen–Morse type potentials in Schr¨odinger equation. In particular, we show that shape invariance in
thecaseenablesustoestablisharelationbetweenordinaryand -foldsupercharges,andtoderivetypeA( ,1)-fold
N N
superalgebra in closed form for arbitrary . Similar discussion of elliptic potentials is given in Sect. V. Extensions
N
to the Pauli and the Dirac equations are presented in Sect. VI. Section VII concludes the paper. In the Appendix A,
we list the complete classification of real type A ( ,1)-fold SUSY. Appendix B gives a summary of the generalized
N
Bender–Dunne polynomials (GBDPs), and present some explicit forms of them for the shape-invariant and elliptic
potentials investigated in Sections IV and V.
II. N-FOLD SUPERSYMMETRY
A. Definition of N-fold Supersymmetry
We begin with defining -fold supersymmetry in one-dimensional quantum mechanics. Consider a matrix Hamil-
N
tonian H given by
N
H+ 0
H = N , (2.1)
N 0 H−
(cid:18) N (cid:19)
where H± are ordinary scalar Hamiltonians
N
1
H± = p2+V±(q), (2.2)
N 2 N
with p= id/dq. -fold supercharges Q± are defined by,
− N N
0 P− 0 0
Q− = N , Q+ = , (2.3)
N 0 0 N P+ 0
(cid:18) (cid:19) (cid:18) N (cid:19)
where the components P± are defined by,
N
P− =P , P+ =Pt , (2.4)
N N N N
in terms of a monic th-order linear differential operator P of the form,
N
N
dN dN−1 d
P = +w (q) + +w (q) +w (q). (2.5)
N dqN N−1 dqN−1 ··· 1 dq 0
In Eq. (2.4), the superscript t denotes the formaltransposition of operatorsin a linear space of functions of q defined
by (d/dq)t = d/dq. Note that when all the functions w (q) appearing in Eq. (2.5) are real-valued, the operator P+
− k N
defined by Eq. (2.4) is identical with the formal adjoint of P : P+ =P† .
N N N
3
A system (2.1) is said to be -fold supersymmetric with respect to Q± if it commutes with them:
N N
Q−,H = Q+,H =0. (2.6)
N N N N
In components, the commutation relations ((cid:2)2.6) are e(cid:3)xpre(cid:2)ssed as th(cid:3)e following intertwining relations:
P−H− H+P− =0, P+H+ H−P+ =0. (2.7)
N N − N N N N − N N
Therefore, the relations in (2.7) give the conditions for the system H to be -fold supersymmetric. Note that the
N
N
Hamiltonians (2.2)arealwayssymmetric under the formaltranspositionevenwhenthey arenotHermitian, andthus
each of the relations in Eq. (2.7) actually implies the other.
It was investigated in Ref. [13] that the -fold supersymmetric models defined above have several significant
N
properties similar to those of the ordinary supersymmetric models. In the following, we shall summarize some of the
most important aspects of -fold supersymmetry.
N
B. Quasi-solvability, mother Hamiltonian, and generalized Witten index
One ofthe mostimportant aspects of -fold supersymmetry is that the componentHamiltonians H− andH+ are
always weakly quasi-solvable with respecNt to the operators P− and P+, respectively [13, 20]. That is, H± leave the
N N
kernels of P± invariant:
N
H± ± ±, ± =kerP±. (2.8)
VN ⊂VN VN N
As a consequence, we can in principle diagonalize algebraically the Hamiltonians H± in the finite -dimensional
N
vector spaces ±, which are thus called solvable sectors of H±. If the space +(−) is a subspace of a Hilbert space
VN VN
L2 on which the Hamiltonian H+(−) is defined, the elements of +(−) provide a part of the exact eigenfunctions and
VN
thus H+(−) is called quasi-exactly solvable.
InordinarySUSY,theanti-commutatorofthesuperchargescorrespondstotheoriginalHamiltonianH . However,
N
it is not generally the case with -fold SUSY. This is because Q−,Q+ is now a 2 th-order differential operator.
N { N N} N
The half of the anti-commutator of Q− and Q+ is called mother Hamiltonian and is denoted by :
N N HN
1 1 P−P+ 0
= Q−,Q+ = N N . (2.9)
HN 2 N N 2 0 P+P−
(cid:18) N N (cid:19)
(cid:8) (cid:9)
An immediate consequence of the above definition is that the mother Hamiltonian always commutes with the -fold
N
supercharges,that is, it is -fold supersymmetric:
N
Q−, = Q+, =0. (2.10)
N HN N HN
Furthermore,ifthe originalHamiltonianH (cid:2) is -fold(cid:3) sup(cid:2)ersymme(cid:3)tric,the mother Hamiltonianalsocommutes with
N
N
H due to the relation (2.6):
N
,H =0. (2.11)
N N
H
We nowcometodiscussthecharacteristicsofthe(cid:2)spectrum(cid:3)ofan -foldsupersymmetricsystem. Aswithordinary
N
SUSY,thebosonic(lower)andthefermionic(upper)statesofsuchasystemareinone-to-onecorrespondenceprovided
that these states are eigenstates of the mother Hamiltonian with non-zero eigenvalues. This can be seen as follows.
Consider a normalized bosonic state Φ− satisfying
H−Φ− =E−Φ−. (2.12)
N
Since commutes with H , Φ− is also an eigenstate of :
N N N
H H
0 0
= . (2.13)
HN Φ− E Φ−
(cid:18) (cid:19) (cid:18) (cid:19)
If is non-zero, then there exists the following normalized state,
E
1
Φ+ = P−Φ−. (2.14)
√2 N
E
4
From Eq. (2.7), we have
H+Φ+ =E−Φ+, (2.15)
N
which shows that Φ+ is also an eigenstate of H+ with the same eigenvalue E−. Furthermore, this state is also the
N
eigenstate of the mother Hamiltonian with the same :
E
Φ+ Φ+
= , (2.16)
HN 0 E 0
(cid:18) (cid:19) (cid:18) (cid:19)
since commutes with Q+ and
HN N
Φ+ Q+ 0
= N . (2.17)
0 √2 Φ−
(cid:18) (cid:19) E (cid:18) (cid:19)
Similarly, one can show that a bosonic state is obtainable from a fermionic one at each energy level unless =0:
E
1
Φ− = P+Φ+. (2.18)
√2 N
E
One can generalize the Witten index of ordinary SUSY to -fold SUSY by
N
Tr( 1)F =dimkerQ− dimkerQ+, (2.19)
− N − N
where kerQ± are subspaces of a Hilbert space on which the super-Hamiltonian H is naturally defined. We note
N N
that only the physical states with energy = 0 of the mother Hamiltonian contribute to the index. The index
N
E H
thus takes an integer value as the number of these zero energy states is finite (2 at most). When the generalized
N
Witten index is non-zero, -fold supersymmetry is not broken dynamically.
N
C. Type A N-fold supersymmetry
As previously mentioned, at presentthree different families of -fold supersymmetric systems have been found for
N
arbitrary finite integer . We shall be concerned only with -fold supersymmetry of type A.
N N
The component of the type A -fold supercharge is defined by
N
d 1 d 3
P = +W(q) N − E(q) +W(q) N − E(q)
N
dq − 2 dq − 2 ×···
(cid:18) (cid:19)(cid:18) (cid:19)
d 3 d 1
+W(q)+ N − E(q) +W(q)+ N − E(q) . (2.20)
···× dq 2 dq 2
(cid:18) (cid:19)(cid:18) (cid:19)
According to Ref. [20], the necessary and sufficient condition for type A -fold supersymmetry is the following:
N
1 2 1
V±(q)= W(q)2 N − 2E′(q) E(q)2 NW′(q) R, (2.21)
N 2 − 24 − ± 2 −
(cid:0) (cid:1)
where R is a constant, and the functions W(q) and E(q) satisfy
d d d
E(q) +E(q) W(q)=0 for 2, (2.22)
dq − dq dq N ≥
(cid:18) (cid:19) (cid:18) (cid:19)
d d d d
2E(q) E(q) +E(q) E(q)=0 for 3. (2.23)
dq − dq − dq dq N ≥
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
Constructionof potentials possessingtype A -fold supersymmetry has been investigatedby analytic calculationsof
N
the intertwining relations, and by sl(2) construction based on quasi-solvability. We refer the readers to [20] for the
complete classification of the potentials.
The solvable sectors ± of the type A -foldsupersymmetric Hamiltonians,whichare the vectorspacespreserved
VN N
by H±, are given by the kernel of the type A -fold supercharges:
N N
VN± =kerPN± =e−WN± 1,z,...,zN−1 , (2.24)
(cid:10) (cid:11)
5
where the gauge functions ± are given by
WN
1
± = N − dqE dqW. (2.25)
WN 2 ∓
Z Z
Finally, we note here that, when =1, 1-fold SUSY is just ordinary SUSY, and P− and P+ are simply ordinary
N N N
supercharges A− = d +W and A+ = d +W, respectively.
dq −dq
III. SIMULTANEOUS ORDINARY AND N-FOLD SUSYS
Since various quantum systems of physical interest possess ordinary SUSY, an interesting question is therefore
whether some of these also possess -fold SUSY. This question has led us to investigate a more general problem,
N
namely, to classify systems which possess type A -fold SUSY with two different values of [25]. We have called
N N
such systems as type A ( , )-fold supersymmetric.
1 2
N N
A system H is said to have ( , )-fold supersymmetry if it commutes with two different -fold supercharges
1 2 i
N N N
(i=1,2) simultaneously, namely,
Q(1)±,H = Q(2)±,H =0. (3.1)
N1 N2
Without loss ofgenerality,we canassume (cid:2) . I(cid:3)n th(cid:2)is case,it(cid:3)is evidentthat the components H± of the system
1 2
N ≥N
H preserve two vector spaces (1)± = kerP(1)± and (2)± = kerP(2)± separately, where P(i)± are components of
VN1 N1 VN2 N2 Ni
Q(i)± (i=1,2). Hence, the solvable sectors ± of ( , )-fold supersymmetric Hamiltonians H± are generally
Ni VN1,N2 N1 N2
given by
± = (1)± (2)±. (3.2)
VN1,N2 VN1 ∪VN2
Classification of type A ( , )-fold supersymmetry has been presented in [25]. Here we shall focus on type A
1 2
N N
( ,1)-fold supersymmetric system, i.e., system possessing simultaneous ordinary and -fold SUSYs. For such a
N N
system, its potential must satisfy
1 2 1
V±(q)= W(q)2 N − 2E′(q) E(q)2 NW′(q) R
N 2 − 24 − ± 2 −
= 1 W (q)2 W′ (q(cid:0)) , (cid:1) (3.3)
2 N ± N
(cid:0) (cid:1)
where R is a constant, and W is the first derivative of a superpotential of ordinary SUSY, with the subscript
N
N
signifying that it is also related to -fold SUSY.
N
From Eq. (2.21) we immediately have
W′ =W′ , (3.4)
N N
2 1
W2 N − (2E′ E2) 2R=W2. (3.5)
− 12 − − N
The first condition (3.3) is easily integrated as
W = W +C, (3.6)
N
N
where C is a constant. Substituting Eq. (3.6) into Eq. (3.5), we obtain
2 1
2 1 W2+2 CW + N − (2E′ E2)+C2+2R=0. (3.7)
N − N 12 −
(cid:0) (cid:1)
To investigate Eq. (3.7), it is more convenient to gauge-transformthe type A Hamiltonians to [20]
d2 2 d ( 1)( 2) 1
H± =e−WN± A(z) + N − A′(z) Q(z) N − N − A′′(z) N − Q′(z) R eWN±, (3.8)
− dz2 2 ± dz − 12 ± 2 −
(cid:20) (cid:18) (cid:19) (cid:21)
6
where the gauge potentials ± are given by Eq. (2.25), and A(z) and Q(z) are polynomials of at most fourth- and
WN
second-degree, respectively, and related to E(q) and W(q) by
1
A(z)= (z′)2 =a z4+a z3+a z2+a z+a , (3.9)
4 3 2 1 0
2
A′(z)=z′′ =Ez′, (3.10)
Q(z)= Wz′ =b z2+b z+b . (3.11)
2 1 0
−
With the aid of these relations, we can show that the condition (3.7) is satisfied if
1 2(C2+2R) 2 8 2C2
Q2+ H[A]+ A = N AQ2, (3.12)
3 2 1 ( 2 1)2
(cid:18) N − (cid:19) N −
where
3
H[A]=AA′′ (A′)2, (3.13)
− 4
is an algebraic covariant called the Hessian of A [26, 27]. The solvable sectors ± of the type A ( ,1)-fold
VN,1 N
supersymmetric Hamiltonians are
± = ± ±, (3.14)
VN,1 VN ∪V1
where ± are given by Eq. (2.24), and ± consists of the states
VN V1
ψ± e±Cqexp dqW . (3.15)
0 ∝ ±N
(cid:18) Z (cid:19)
We note here that the condition (3.12) is expressed as an algebraically covariant form under the projective trans-
formations GL(2,K) (K =R or C) on A(z) and Q(z) (cf. Refs.[11, 20]):
αz+β
A(z) A(z)=∆−2(γz+δ)4A , (3.16)
7→ γz+δ
(cid:18) (cid:19)
αz+β
Q(z) Qb(z)=∆−1(γz+δ)2Q , (3.17)
7→ γz+δ
(cid:18) (cid:19)
b
where α,β,γ,δ K and ∆ = αδ βγ = 0. We can easily show that the polynomial A(z) of at most fourth-degree
is transformed t∈o one of the five−canon6ical forms listed in Table I by the GL(2,C) transformation. The complete
exhibition of each model for more general type A ( , )-fold SUSY in this complex classification scheme is found
1 2
N N
in Ref. [25].
Case Canonical Form H[A]
I 1/2 0
II 2z −3
III 2µz2 −4µ2z2
IV 2µ(z2−1) −4µ2(z2+2)
V 2z3−g z/2−g /2 −3z3−3g z2/2−6g z−3g2/16
2 3 2 3 2
TABLEI:CanonicalformsofA(z)andtheircorrespondingH[A]. Theparametersµ,g ,g ∈Csatisfyµ6=0andg3−27g2 6=0.
2 3 2 3
However,forphysicalapplicationswherethe Hamiltoniansshouldbe realandHermitian, itis moreconvenientand
suitable to expresspotentials interms ofrealfunctions. Forthis purpose,weshoulduse the realclassificationscheme
under the GL(2,R) transformation. In Appendix A, we list all the inequivalent models of type A ( ,1)-fold SUSY
N
in terms of real functions. Among these models, we find that only the Rosen–Morse type and some of the elliptic
models can have physical interest, which we shall discuss in the next two sections.
7
IV. SHAPE-INVARIANT POTENTIAL
In this section, we concentrate on the trigonometric version of Case IVa, Eqs. (A8)–(A10). The hyperbolic version
is notofphysicalinterest,asany elementofthe solvablesectors ± andψ± isnon-normalizableandthus unphysical.
Forconvenience,weshallhereafteradoptaunitsysteminwhich~VN=e=2m0 =c=1,sothattheresultsobtainedhere
for the Schr¨odingerequationcanbe readily carriedoverto the Pauliand Dirac equations. As such, the Hamiltonians
H± and potentials V± in this and the next two sections differ from those in the previous sections and in Appendix
N N
A by a multiplicative factor 2.
The trigonometric version of Case IVa can be obtained by applying a scale transformation Eqs. (A1) and (A2)
with ν < 0 to Eqs. (A8)–(A10). We take ν = 1/4 without loss of generality. With this choice, the functions which
−
characterizethe -fold( 2)andthe ordinarysuperchargesaregivenby (we hereafterchangethe variableq to x)
N N ≥
1
E(x)=cotx, W(x)= cotx, W (x)= N cotx. (4.1)
N
−2 − 2
The corresponding Hamiltonians read
d2 1
H± = +V±(x), V±(x)= ( 2)cosec2x 2 . (4.2)
N −dx2 N N 4 N N ± −N
(cid:2) (cid:3)
These potentials are periodic in x, but tend to positive infinity at x = nπ, n = 0, 1, 2.... Hence, the particle is
confined only in any one of the infinite cosec2 wells with width π in x. Without los±s of±generality, we shall take the
physical domain of interest to be 0<x<π. The solvable sectors are given by
± =(sinx)−N−21±1 1,cosx,...,(cosx)N−1 , ψ± (sinx)∓N2 . (4.3)
VN 0 ∝
Inwhatfollows,wewillinvestigateanddiscus(cid:10)svariousaspectsofsimu(cid:11)ltaneousordinaryand -foldsupersymmetries,
N
interplay of them, and their physical consequences.
A. Structure of ordinary SUSY
First of all, due to ordinary SUSY, the Hamiltonians H± are factorizable as
N
H− =A+A−, H+ =A−A+, (4.4)
N N N N N N
with the first-order operators
d d
A− +W , A+ +W . (4.5)
N ≡ dx N N ≡−dx N
It follows that
A−H− =H+A−, A+H+ =H−A+, (4.6)
N N N N N N N N
which corresponds to Eq. (2.7) in the case of 1-fold SUSY. If we introduce the supercharges A± in a matrix repre-
N
sentation by
0 A− 0 0
A− = N , A+ = , (4.7)
N 0 0 ! N A+N 0 !
the relations (4.4) and (4.6) are expressed as a Lie superalgebra:
A±,H =0, A−,A+ =H . (4.8)
N N N N N
Let us denote the normalizedeigenfunc(cid:2)tions of th(cid:3)e Hamilto(cid:8)nians H±(cid:9)by ψ(N)± with eigenvalues E(N)±, respectively.
N n n
Here the subscript n = 0,1,2,... denotes the number of nodes of the wave function. It is easily proved that V−
N
and V+, being partners of ordinary SUSY, have the same energy spectrum except for the ground state of V− with
N N
E(N)− =0,whichhasnocorrespondinglevelforV+. Moreexplicitly,wehavethefollowingsupersymmetricrelations:
0 N
E(N)+ =E(N)−, (4.9)
n n+1
8
ψ(N)+ = E(N)− −1/2A−ψ(N)−, (4.10)
n n+1 N n+1
ψ(N)− =(cid:0)E(N)+(cid:1)−1/2A+ψ(N)+. (4.11)
n+1 n N n
Hence A− converts an eigenfunction of H− into a(cid:0)n eigen(cid:1)function of H+ with the same energy, but with one less
N N N
number of nodes, while A+ does the reverse.
N
Thesupersymmetricelementψ−ofthesolvablesectorsinEq.(4.3)isnormalizableinthephysicaldomain0<x<π
0
we have considered, and corresponds to the ground state wave function ψ(N)− of V−:
0 N
ψ(N)− =ψ− (sinx)N2 , A−ψ(N)− =0. (4.12)
0 0 ∝ N 0
Hence, ordinary SUSY in this system is unbroken.
B. Two chains of shape-invariant potentials
Next, we note that the potentials V± satisfy
N
V+(x)=V− (x)+R , R = +1. (4.13)
N N+2 N N N
This implies
A−A+ =A+ A− +R , (4.14)
N N N+2 N+2 N
or, equivalently, H+ = H− +R . Potentials satisfying Eq. (4.13) are called shape-invariant, and are (exactly)
N N+2 N
solvable [2, 3]. Indeed, it turns out that V± belong to one of the well known shape-invariant potentials, namely, the
N
Rosen–Morse-Ipotentials [2, 3], only with parametersbeing relatedto the integer . In particular,the potentialV−
N N
corresponding to = 2 is simply an infinite square-well in the domain 0 < x < π. Below we give a brief discussion
of the relation betNween shape invariance and exact solvability, specializing to the potentials V± considered here.
N
ThewavefunctionandenergyofthegroundstateofVN−+2areψ0(N+2)− ∝(sinx)N2+2 (cf. Eq.(4.12))andE0(N+2)− =
0, respectively. The shape invariance (4.13) indicates that the ground state wave function ψ(N)+ of V+ is given by
0 N
ψ(N)+ =ψ(N+2)− withthe groundstateenergyE(N)+ =R . Onthe otherhand,fromthe supersymmetricrelations
0 0 0 N
(4.9)–(4.11), the first excited state ψ(N)− of V− is related to the ground state ψ(N)+ of V+ by ψ(N)− A+ψ(N)+,
1 N 0 N 1 ∝ N 0
withthe sameenergyE(N)− =E(N)+ =R . Thus,we haveψ(N)− A+ψ(N+2)−. By iteratingthis process,we can
1 0 N 1 ∝ N 0
obtain the wave function of the nth excited state of V− as
N
ψ(N)− A+A+ A+ A+ ψ(N+2n)−, (4.15)
n ∝ N N+2··· N+2(n−2) N+2(n−1) 0
and energy eigenvalues are given by
n−1
E(N)− = R =n( +n), n=0,1,2,.... (4.16)
n N+2k N
k=0
X
With these formulas,the energy andwavefunctions for allV± are alsoexactly calculable in closedform. We note
N+2k
here that since Eq. (4.13) requires the integers in V± to be differed by two, we therefore have two different chains
N N
of shape-invariant cosec2 potentials in which two neighboring potentials possess type A ( ,1)-fold SUSY, one for
N
even ’s, and the other for odd ’s.
ToNmake discussion of -fold NSUSY structure of V± transparent in the next subsection, we find it convenient to
N N
rewritethesymbolsofthepotentialsinthesamechainofshapeinvariancerelatedbysuperchargesA− andA+ ,
N+2k N+2k
(k =0,1,2,... and +2k 2). Let us write
N ≥
k−1
V(N+2k)(x)=V− (x)+ R , k =0,1,2,.... (4.17)
N+2k N+2l
l=0
X
The potentials V(m) and V(m+2) possess type A (m,1)-fold SUSY. Now let us denote by ψ(m) and E(m) (m =
n n
+ 2k, n,k = 0,1,2,...) the wave function and energy eigenvalue, respectively, of the nth energy level of the
N
9
potential V(m). According to Eq. (4.17), the ground state of V(m) is also the ground state of V− (with energy
m
E0(m)− = 0), hence we have ψ0(m) ∝ (sinx)m2 , and E0(m) = (m2 −N2)/4 from Eqs. (4.12) and (4.16). Then the
supersymmetric relations (4.4), (4.9)–(4.11) can be rewritten in terms of the eigenfunctions ψ(m) and eigenvalues
n
E(m) of the potential V(m) defined by Eq. (4.17) as follows:
n
A+A−ψ(m) =E(m)ψ(m), (4.18)
m m n n n
E(m) =E(m+2), (4.19)
n+1 n
A−ψ(m) ψ(m+2), (4.20)
m n ∝ n−1
A+ ψ(m) ψ(m−2). (4.21)
m−2 n ∝ n+1
The relations (4.20) and (4.21) are represented diagrammatically in Fig. 1 for odd .
N
C. Structure of type A N-fold SUSY
Now we examine how type A -fold SUSY manifests itself in the two chains of shape-invariant potentials. Es-
N
sentially three issues need be addressed, namely, (1) whether -fold SUSY is dynamically broken or not; (2) how
the supercharges P± and A± are related; and (3) what kind ofNsuperalgebra the five operators H ,A±,Q± which
N N N N N
characterizethesystemsatisfy. Weshalladdressthesequestionsbelowforthetwochainsofshape-invariantpotentials.
1. Dynamical N-fold SUSY breaking
If -fold SUSY is not broken, then the ground state, and perhaps some excited states, must belong to
N
− =(sinx)−N2+1 1,cosx,...,(cosx)N−1 . (4.22)
VN
In our present case, the ground state of V−, given by E(cid:10)q. (4.12), is arrangeda(cid:11)s
N
ψ(N) (sinx)N2 =(sinx)−N2+1(sinx)N−1. (4.23)
0 ∝
Now it is nothardto checkthat for odd , the factor(sinx)N−1 belongs to 1,cosx,...,(cosx)N−1 , while for even
N
it does not.
NNext, we shall consider the first excited states ψ(N). From the relation (4(cid:10).21), we find that ψ(N) (cid:11)can be obtained
1 1
by applying the supercharge A+ on ψ(N+2), i.e.,
N 0
ψ(N) A+ψ(N+2) = d N cotx (sinx)N2+2
1 ∝ N 0 −dx − 2
(cid:18) (cid:19)
(sinx)N2 cosx=(sinx)−N2+1(sinx)N−1cosx. (4.24)
∝
We easily see that ψ(N) does not belong to − for any . Similarly, one can check that no higher excited states
1 VN N
belong to −, as they pick up higher powers in sin and cos terms as increases.
VN N
In the same way, one can check that all the physical states of V+ = V(N+2) do not belong to + for any .
N VN N
Hence, from the generalizedWitten index, we conclude that -fold SUSY is preservedbarely by the ground state of
N
V− =V(N) for odd , and is completely broken for even .
N N N N
2. Actions of P± on eigenstates
N
AswehaveseeninEq.(4.1),wehaveE = 2W =cotxinourcase. Thissignificantpropertyenablesustoexpress
the -foldsuperchargesP± intermsoftheor−dinarysuperchargesA±. Infact,thetypeA -foldsuperchargedefined
N N N N
by Eq. (2.20) in this case becomes
10
d d 2 d 4 d 2
P = N cotx N − cotx + N − cotx + N − cotx . (4.25)
N
dx − 2 dx − 2 ··· dx 2 dx 2
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
Comparing each factor in P to A± defined by Eq. (4.5) with W = (cotx)/2, one has for odd = 2M +1
N N N −N N
(M =1,2,3,...)
P− =( 1)MA− A− A−A−A+A+ A+ A+ , (4.26a)
2M+1 − 2M+1 2M−1··· 3 1 1 3 ··· 2M−3 2M−1
P+ =( 1)MA− A− A−A−A+A+ A− A− , (4.26b)
2M+1 − 2M−1 2M−3··· 3 1 1 3 ··· 2M−1 2M+1
and for even =2M (M =1,2,3,...)
N
P− =( 1)MA− A− A−A−A+A+A+ A+ A+ , (4.27a)
2M − 2M 2M−2··· 4 2 0 2 4 ··· 2M−4 2M−2
P+ =( 1)MA− A− A−A−A−A+A+ A+ A+ . (4.27b)
2M − 2M−2 2M−4··· 4 2 0 2 4 ··· 2M−2 2M
We first discuss the action of P− on the eigenstates of V(N) for odd . As shown in the last section, for odd
N N
=2M +1 the ground state ψ(2M+1), which is the solution of A− ψ(2M+1) =0, belongs to the kernel − of
N 0 2M+1 0 V2M+1
P− , namely, P− ψ(2M+1) = 0. Hence, the ground state ψ(2M+1) is annihilated by both supercharges A−
2M+1 2M+1 0 0 2M+1
and P− . For the excited states ψ(2M+1) (n>0) we have
2M+1 n
P− ψ(2M+1) A− (A− A− A−)(A−A+)(A+ A+ A+ )ψ(2M+1). (4.28)
2M+1 n ∝ 2M+1 2M−1 2M−3··· 3 1 1 3 ··· 2M−3 2M−1 n
Applying Eq. (4.21) repeatedly, we have
(A+ A+ A+ )ψ(2M+1) ψ(3) , (4.29)
3 ··· 2M−3 2M−1 n ∝ n+M−1
since each A+ maps the nth excited state of V(m) to the (n+1)-th state of V(m−2). The action of the A−A+ on
1 1
ψ(3) is calculated using Eqs. (4.14) and (4.18) as
n+M−1
A−A+ψ(3) =(A+A−+R )ψ(3) = E(3) +R ψ(3) . (4.30)
1 1 n+M−1 3 3 1 n+M−1 n+M−1 1 n+M−1
(cid:16) (cid:17)
Hence A−A+ψ(3) ψ(3) . Finally, we note that all the A−’s grouped in the first bracket in the r.h.s. of
1 1 n+M−1 ∝ n+M−1
Eq. (4.28) simply reverse the actions of the A+’s grouped in the third bracket, and from Eq. (4.20) we have
(A− A− A−)ψ(3) ψ(2M+1). (4.31)
2M−1 2M−3··· 3 n+M−1 ∝ n
Taken together, Eq. (4.28) implies
P− ψ(2M+1) A− ψ(2M+1) ψ(2M+3). (4.32)
2M+1 n ∝ 2M+1 n ∝ n−1
Equations(4.28)and(4.32)canbe visualized,as shownin Fig.1, asa seriesof arrowsof A+’s (starting with A+ )
2M−1
pointing to the left, turning around with A−A+, and then pointing to the right with a series of A−’s (ending with
1 1
A− ).
2M+1
Hence, the state ψ(2M+3) is obtainable from the excited state ψ(2M+1) (n >0) either by the ordinary supercharge
n−1 n
A− , or by the -fold supercharge P− . By the same argument, it can be shown that the actions of A+
2M+1 N 2M+1 2M+1
and P+ are equivalent on the state ψ(2M+3). That allstates in the pair of potential V(N) andV(N+2) are related
2M+1 n−1
by both sets of superchargesis a manifestation ofthe underlying unbrokensimultaneous SUSY structures inherentin
the cosec2 potential with odd =2M +1.
Foreven =2M,typeA N-foldSUSYisbroken. Also,the presenceofA− preventsthe A− (m=2,4,..., 2)
N N 0 m N−
to reverse the actions of the corresponding A+ on ψ(N). Hence, the states in V(N) and V(N+2) are only related by
m n
the supercharges of unbroken ordinary SUSY, and not by those of broken type A -fold SUSY.
N
