On General Form of N-fold Supersymmetry

TH-1472 On General F orm of N -fold Sup ersymmetry T oshiaki T anak a ∗ Institute of Particle and Nucle ar Stu di es, High Ener gy A c c e ler ator R e se ar ch Or ganization (KEK), 1-1 Oho, Tsukub a, Ib ar aki 305-0801, Jap an Abstract W e analyze general structure of N -fold sup ersymmetry whic h pr o vides a systematic framework to construct we akly qu asi-solv able qu an tum mec hanical systems. Main ingredien ts of our analysis are dimensional analysis and in tro duction of an equiv alen t class of linear d ifferen tial oper ators asso ciated with N - fold su p ersym metry fo r eac h N . T o illustrate ho w they wo r k , we construct the most general form of N -fold s up er s ymmetric systems for N = 2, 3, and 4. P A CS num b e rs: 02.30.Hq; 03.65.Ca ; 03.65.Ge; 11.30.P b Keywords: N -fold s upe r symmetry; W eak quasi-s o lv a bilit y ; Linea r differ ent ia l op erator s; Intert wining r ela- tions; Dimensional analysis ; Equiv alent classe s ∗ Electronic addres s: toshiaki@p os t.kek.jp 1 I. INTR ODUCT ION N -fold sup ersymmetry (SUSY) [1 – 3] is one of the most p ow erful framew o rks fo r construct- ing a one-dimensional quan tum mec hanical (QM) system whic h admits analytic solutions in closed form in a certain sense. This is due to the fact that N - fold SUSY is essen tia lly equiv alen t to w eak quasi-solv abilit y whic h is until no w the least restrictiv e concept ab out the a v a ilabilit y of solutions in closed form. The la tter crucial fact was first prov ed in a general fashion in Ref. [2] and w as la ter complemen ted slightly in Ref. [4]. F or a review, see, e.g., Ref. [5]. In general, construction of an N -f old SUSY system get more difficult as the num b er N ∈ N increases since w e m ust solv e coupled nonlinear differen tial equations fo r N unkno wn functions (see the next section). T o b ypass the latter difficult y , a systematic algorithm for constructing an N -fold SUSY system based on quasi-solv abilit y (in the strong sense) was prop osed in Ref. [6 ]. A key ingredien t of the algorithm is to ch o ose first an N -dimensional linear space of sp ecific functions suc h that it can b e preserv ed b y a second-order linear differen tial op erator. It has b een prov ed to b e quite efficien t and so far four inequiv a lent t yp es of N - fold SUSY, namely , t yp e A [4, 7], type B [8 ], type C [6], and t yp e X 2 [9] are success fully constructed with the a lgorithm. W e note that almost a ll the mo dels ha ving essen tially the same symmetry a s N -fold SUSY but called with other terminolog ies in the literature, suc h as P¨ osc hl–T eller and Lam ´ e p oten tials, a re actually particular cases of t yp e A N -fold SUSY. It is eviden t, how ev er, that the algor it hm is helpless to construct a we akly quasi-solv able system whic h only a dmits a finite-dimens ional inv arian t subspace determined b y another differen tial equation. The framew o r k of N -fold SUSY cov ers suc h systems a nd thus prov ides a more general formalism than higher-deriv ativ e generalizations of D arb oux t r a nsformation suc h as the Crum’s metho d [10] whic h relies on a set of exact eigenfunctions of a regular Sturm–Liouville system. T o construct a w eakly quasi-solv able system, we m ust in general treat directly the aforementioned coupled nonlinear differen tial equations. F or the simplest case of N = 2 , the general result w as already studied a nd rep orted in Refs. [2, 11, 1 2]. On the other hand, for the cases of N > 2 un til no w on there is, at the b est of our know ledge, only one pap er [13] whic h studied the N = 3 case. This fact w o uld reflect the difficult y a nd complexit y o f t he pro blems for larger N . In this work, w e in v estigate general strucu ture of N -fold SUSY systems to extract relev ant clues to construct them whic h hav e in part icular w eak quasi-solv ability . F or this purp o se, we first employ dimensional analysis whic h is a w ell-kno wn p o w erful to ol in general ph ysics. It turns out that it is also quite efficien t in acquiring deep er understandings of N -fold SUSY. W e then in tro duce equiv alen t classes of linear differen tial op erators asso ciated with N -fold SUSY. W e find that it enables us to deal with op erator equalities a pp eared in N - fold SUSY more systematically and transparently . W e organize the pap er as follo ws. In the next se ction, w e first briefly review the in- gredien ts o f N - fold SUSY. Then, w e in tro duce t w o k ey concepts for analyzing its general structure, namely , dimensional analysis and equiv alen t classes of linear differential o p erators asso ciated with N -f o ld SUSY. In Sections I II– IV, w e apply the general argumen ts to obtain general form of N -fold SUSY for N = 2, 3, and 4, resp ectiv ely . W e sho w ho w dimensional analysis enables us to reduce the complexit y o f the problems on solving the conditions for N -fold SUSY and on finding inte gr al constan ts of the systems. In the last sec tio n, w e summarize the pap er and provide commen ts on the future issues. 2 I I. GENERAL CONSIDE RA TION T o b egin with, w e shall briefly review ingredien ts of N -fold SUSY as preliminaries. F or details, see the review [5]. An N -fo ld SUSY QM system in one-dimension is comp osed of a pair of Hamiltonians H ± and a pair of N th-or der linear differen tial op erators P ± N H ± = − 1 2 d 2 d q 2 + V ± ( q ) , P − N = d N d q N + N − 1 X k =0 w [ N ] k ( q ) d k d q k , P + N = ( P − N ) T , (2.1) where the sup erscript T denotes the tr a nsp osition of a linear op erator [3 ], which satisfy the in tertw ining relation P − N H − − H + P − N = N X k =0 I [ N ] k d k d q k = 0 , (2.2) and its transp osed relation H − P + N − P + N H + = 0 . The op erators P ± N are actually comp onen ts of N -fold sup ercharges. One of the most significan t consequences of the in tertwin ing relation (2.2) is we ak quasi- solvability [2, 4 ]. That is, each N - fold SUSY Hamiltonian H ± preserv es the linear space k er P ± N : H ± k er P ± N ⊂ k er P ± N . (2.3) If the differen tial equation P − N φ = 0 and/or P + N φ = 0 admits a n um b er of analytic solutions in close d form, H − and/or H + is not o nly w eakly quasi-solv a ble but also quasi-solvable in the strong sense. But in general, an N -fold SUSY Hamilto nia n is merely w eakly quasi-solv able and do es not admit any analytic lo cal solutions. W e also note that k er P ± N is not necessarily a subspace of the linear space, whic h is usually the Hilb ert space L 2 ( S ) ( S ⊂ R ), in whic h the op erator H ± acts. Another p eculiar feature o f an N -fold SUSY system is that the pro duct P ∓ N P ± N whic h arises as a comp onent of the anti-comm uta tor of N -fo ld supercharges is an N th-degree p olynomial in the Hamiltonian H ± [2, 3 ] and th us has the fo llo wing form: P ∓ N P ± N = 2 N " ( H ± + C 0 ) N + N − 1 X k =1 C k ( H ± + C 0 ) N − k − 1 # , (2.4) where C k ( k = 0 , . . . , N − 1) are all constan ts. The N zeros of the p olynomial in the r.h.s. of (2.4) correspo nd to the spectrum of H ± in the space k er P ± N . Hence, they are actually a part of the eigen v alues o f H ± if k er P ± N ⊂ L 2 ( S ). In the latter case, w e can calculate the part of the eigen v alues algebraically from (2.4) eve n though the corresp onding eigenfunctions cannot b e o bta ined in closed form. W e note that the N energy sp ectra w ould exhibit the characteristic features suc h a s the disapp earance of nonp erturbative corrections due to the generalized non-renormalization theorem in N - f old SUSY, which w ere rep orted on the realistic ph ysical systems suc h as the asymmetric double-w ell p oten tial in [14] a nd the p erio dic and t he symmetric t r iple-w ell p oten tials in [15]. One o f the most difficult problems on N -fold SUSY is t o analyze the condition (2.2 ) for N -fold SUSY. It is compo sed o f coupled no nlinear differen tial equations for the so far 3 undetermined f unctions w [ N ] k ( k = 0 , . . . , N − 1). As is easily expected, its complexit y g ets terrible as the in teger N increases. Hence, it is quite difficult to solv e directly the condition (2.2) for la r ger N . In the subsequen t t wo sections, w e shall in ve stigat e general asp ects of the complicated structure of N -fo ld SUSY systems whic h w ould pro vide us a clearer view on them. A. Dimensional Analysis Dimensional ana lysis is one of the p o w erful metho ds to mak e ph ysical consideration and in particular to estimate a ph ysical quan tity under consideration without solving equations directly , cf., an y textb o o k on general ph ysics. In this section, w e shall see t ha t it supplies us with a v aluable guiding principle in solving t he condition for N -fold SUSY. T o mak e dimensional analysis on our system (2.1), w e first note that we ha v e implicitly emplo y ed the unit system where the ratio o f square action (the Planc k constant) to mass ~ 2 /m is dimensionless. The only relev a n t phys ical dimension is then the length , denoted b y [L], whic h is carried b y the ph ysical p osition v aria ble q . It is easy to see from (2.1 ) and (2.4) that the ph ysical dimensions of V ± , C k , and w [ N ] k in terms of the length are give n by V ± [L − 2 ] , C k [L − 2( k +1) ] ( k = 0 , . . . , N − 1) , w [ N ]( m ) k [L k −N − m ] ( k = 0 , . . . , N − 1 , m = 0 , 1 , 2 , . . . ) , (2.5) where w [ N ]( m ) k ( q ) is the m th deriv at ive o f w [ N ] k ( q ) with resp ect to q . Dimensional analysis relies on the ob vious fact that all the terms whic h appear in a single formula under consideration m ust ha v e the same phys ical dimension. F or instance, a p oten tia l has the ph ysical dimension [L − 2 ] and th us m ust b e expressed a s a sum of t erms all of whic h hav e the same ph ysical dimension [L − 2 ]. Hence, if we only consider a p olynomial of C k and w [ N ]( m ) k ( m = 0 , 1 , 2 , . . . ), it must hav e the follow ing f orm V = α 0 w [ N ] N − 2 + α 1 w [ N ] ′ N − 1 + α 2  w [ N ] N − 1  2 − C 0 [L − 2 ] , (2.6) where α k ( k = 0 , 1 , 2) are all dimensionless parameters. In fact, w e can see t hat a pair of N -fold SUSY p otentials V ± satisfying (2.2) do es hav e the fo rm (2 .6). The l.h.s. of (2.2) is a linear differential op erator of at most N th order, a s is indicated in (2.2), and it is eviden t that the iden tity (2.2) holds if and only if all the co efficien ts I [ N ] k of ∂ k = d k / d q k ( k = 0 , . . . , N ) v anish. The latter requiremen t f or the co efficien ts of ∂ N and ∂ N − 1 reads as I [ N ] N = w [ N ] ′ N − 1 − ( V + − V − ) = 0 [L − 2 ] , (2.7a) 2 I [ N ] N − 1 = w [ N ] ′′ N − 1 + 2 w [ N ] ′ N − 2 + 2 N V −′ − 2 w [ N ] N − 1 ( V + − V − ) = 0 [L − 3 ] . (2.7b) The set of conditions (2.7) can b e easily solv ed as V ± = − 1 N w [ N ] N − 2 +  N − 1 2 N ± 1 2  w [ N ] ′ N − 1 + 1 2 N  w [ N ] N − 1  2 − C 0 [L − 2 ] , (2.8) whic h indeed has the form of (2.6). The fo rm ula (2.8) pro vides a general expression f or a pair of N -fold SUSY p oten tials V ± for an arbitrary N ∈ N . One of its c haracteristic features 4 is that they a r e expressible solely in terms of the tw o functions w [ N ] N − 1 and w [ N ] N − 2 irresp ectiv e of what additional conditions w [ N ] k ( k = 0 , . . . , N − 1) should satisfy . The remaining conditions for N - fold SUSY coming f rom the co efficien ts of ∂ k for k = 0 , . . . , N − 2 in (2.2) are in g eneral algebraic equations consisting of w [ N ]( m ) k ( m = 0 , 1 , 2 , . . . ) after the substitution o f (2.8) into (2.2). Dimensional analysis t ells us that the op erat o r in the l.h.s. of (2.2) has the ph ysical dimension [L −N − 2 ] and th us I [ N ] k ( k = 0 , . . . , N − 2) has the ph ysical dimension [L k −N − 2 ]. F or instance, I [ N ] N − 2 and I [ N ] N − 3 are calculated as − 4 N I [ N ] N − 2 = N ( N − 1 ) w [ N ] ′′′ N − 1 + 2 N ( N − 2) w [ N ] ′′ N − 2 − 4 N w [ N ] ′ N − 3 − 2( N − 1) 2 w [ N ] N − 1 w [ N ] ′′ N − 1 − 2 N ( N − 1)  w [ N ] ′ N − 1  2 + 4 N w [ N ] ′ N − 1 w [ N ] N − 2 + 4( N − 1) w [ N ] N − 1 w [ N ] ′ N − 2 − 4( N − 1 )  w [ N ] N − 1  2 w [ N ] ′ N − 1 [L − 4 ] , (2.9) − 12 N I [ N ] N − 3 = N ( N − 1 )( N − 2)  w [ N ] ′′′′ N − 1 + 2 w [ N ] ′′′ N − 2  − 6 N  w [ N ] ′′ N − 3 + 2 w [ N ] ′ N − 4  − ( N − 1)( N − 2) h (2 N − 3) w [ N ] N − 1 w [ N ] ′′′ N − 1 + 6 N w [ N ] ′ N − 1 w [ N ] ′′ N − 1 i + 6( N − 2) h w [ N ] ′′ N − 1 w [ N ] N − 2 + ( N − 1) w [ N ] N − 1 w [ N ] ′′ N − 2 i + 12 N w [ N ] ′ N − 1 w [ N ] N − 3 + 12( N − 2) w [ N ] N − 2 w [ N ] ′ N − 2 − 6( N − 1 )( N − 2) h  w [ N ] N − 1  2 w [ N ] ′′ N − 1 + w [ N ] N − 1  w [ N ] ′ N − 1  2 i − 12( N − 2) w [ N ] N − 1 w [ N ] ′ N − 1 w [ N ] N − 2 [L − 5 ] , (2.10) and consist of the terms whic h are consisten t with the dimensional a na lysis. Ideally , o ne can obtain a general for m of N -fold SUSY systems if one succeeds in express- ing a ll the N functions w [ N ] k ( k = 0 , . . . , N − 1), whic h c haracterize the systems, in terms of a single function, sa y , u and its deriv ativ es u ′ , u ′′ , . . . b y solving the set of the N − 1 constrain ts I [ N ] k = 0 ( k = 0 , . . . , N − 1). I f it is eve ntually the case, w e ha ve a set o f N functionals u [ N ] k ( k = 0 , . . . , N − 1) such that w [ N ] k = u [ N ] k [ u ] [L k −N ] ( k = 0 , . . . , N − 1) . (2.11) One of the most imp ortant asp ects of (2.1 1) is that eac h u [ N ] k ( k = 0 , . . . , N − 1) has the same phys ical dimension as the one of w [ N ] k . It in particular means tha t there w ould b e a set of transformations w [ N ] k → u [ N ] k whic h preserv e all the phy sical dimensions. Con v ersely , if w e can find a set of dimension-preserving transformations w [ N ] k → u [ N ] k , w e may solv e the set of the constrain ts more easily . In Sections I I I–V, w e will emplo y this strategy to see how drastically w e can reduce the complexit y of the constrain ts. T o solv e the constrain ts to get (2.11) is in principle p ossible unless some of the constraints automatically imply others since we ha ve N − 1 constrain ts for the N unk nown functions. Ho w ev er, the ta sk would get dra stically harder as the in teger N increases. One of the clues to circum v en t the situation is in (2.4 ). All the co efficien ts of deriv a tiv e op erators (except for the highest 2 N th- o rder and including the low est 0t h- order) in the l.h.s. o f (2.4) are quadratic forms of w [ N ]( m ) k while the r.h.s. dep ends o n, in a dditio n to the p oten tials V ± , the N constants C k ( k = 0 , . . . , N − 1) whic h are absen t in the l.h.s. The latter fa ct indicates the 5 existence of N in tegral constan ts of any N -fo ld SUSY system whic h w ould b e functionals of w [ N ] j ( j = 0 , . . . , N − 1) and V ± whose ph ysical dimensions are the same as the ones of C k : C k = J k [ w [ N ] , V ] [L − 2( k +1) ] ( k = 0 , . . . , N − 1) . (2.12) They m ust emerge from t he integration of the set o f differential equations d d q J k [ w [ N ] , V ] = 0 [L − (2 k +3) ] ( k = 0 , . . . , N − 1) . (2.13) The latter equations are another set of constraints . On the other hand, the set o f equalities I [ N ] k = 0 ( k = 0 , . . . , N ) are the o nly constraints whic h come from t he condition for N - fold SUSY. Hence, the differen tia l equations (2.13) m ust b e suc h equations that hold whenev er all the conditions I [ N ] k = 0 ( k = 0 , . . . , N ) are satisfied. This means that all the quantities d J k / d q ( k = 0 , . . . , N − 1) in the l.h.s. of (2.13 ) w ould b e expressible in terms of I [ N ] j in a w ay suc h that the iden tities I [ N ] j = 0 ( j = 0 , . . . , N ) apparen tly imply (2.13). The most general form of suc h a kind w ould b e d J k d q = N X j =0 L k j I [ N ] j [L − (2 k +3) ] ( k = 0 , . . . , N − 1) , (2.14) where L k j are a ll linear differen tia l op erators whose co efficien ts consist of only w [ N ] k , V ± , and their deriv ativ es. Eac h L k j m ust ha s the phys ical dimension [ L N − 2 k − j − 1 ] since the ones of d J k / d q and I [ N ] j are [L − (2 k +3) ] and [L N − 2 − j ], resp ectiv ely . T o see the v alidit y of the ab ov e a r g umen t, let us consider the constan t C 0 . F rom (2.8), w e immediately know the fo rm of J 0 as C 0 = J 0 [ w [ N ] , V ] = − V − − 1 N w [ N ] N − 2 + 1 2 N w [ N ] ′ N − 1 + 1 2 N  w [ N ] N − 1  2 [L − 2 ] . (2.15) Hence, the differen tial equation which leads to the la t t er equation is 0 = d J 0 d q = − V −′ − 1 N w [ N ] ′ N − 2 + 1 2 N w [ N ] ′′ N − 1 + 1 N w [ N ] N − 1 w [ N ] ′ N − 1 [L − 3 ] . (2.16) W e then c hec k by using (2.7) that the r.h.s. of (2.16 ) can b e in fa ct expressed in terms of I [ N ] j as d J 0 d q = 1 N w [ N ] N − 1 I [ N ] N − 1 N I [ N ] N − 1 [L − 3 ] , (2.17) whic h indeed has the fo rm of (2.1 4) with L 0 N = 1 N w [ N ] N − 1 [L − 1 ] , L 0 N − 1 = − 1 N [L 0 ] , L 0 k = 0 ( k = 0 , . . . , N − 2) , (2.18) all having the correct ph ysical dimensions [L N − j − 1 ] for L 0 j ( j = 0 , . . . , N ). In practice, w e already solv ed t he tw o conditio ns (2.7) to obtain the general for m of V ± as (2.8), and thus w e can totally eliminate V ± in the remaining conditions I [ N ] k = 0 6 ( k = 0 , . . . , N − 2 ). As a r esult, the remaining in tegral constan ts C k ( k = 1 , . . . , N − 1) w ould b e functionals o f o nly w [ N ] j ( j = 0 , . . . , N − 1): C k = J k [ w [ N ] , V [ w [ N ] ]] := J k [ w [ N ] ] [L − 2( k +1) ] ( k = 1 , . . . , N − 1) . (2.19) Accordingly , d J k / d q ( k = 1 , . . . , N − 1) w ould b e expresse d in terms of the remaining I [ N ] j ( j = 0 , . . . , N − 2) as 0 = d J k d q = N − 2 X j =0 L k j I [ N ] j [L − (2 k +3) ] ( k = 1 , . . . , N − 1) . (2.20) W e will lat er see in Sections I I I–V that the a b ov e a na lysis is actually v alid and helps us to obtain general forms of N -fold SUSY for N = 2, 3, and 4. T o summarize, the existence of N constants C k ( k = 0 , . . . , N − 1) in the r.h.s. of (2.4) has the dir ect relation to the existence of N constraints I [ N ] k = 0 ( k = 0 , . . . , N − 1) whic h are differential equations after eliminating the algebraic constraint I [ N ] N = 0. The ph ysical dimensions of the relev ant quan tities are I [ N ] k [L k −N − 2 ] , u [ N ] k [L k −N ] , J k [L − 2( k +1) ] , L k j [L N − 2 k − j − 1 ] . (2.21) B. Equiv alen t Classes of Linear Differen tial O p erators The existence o f N − 1 constraints I [ N ] j = 0 ( j = 0 , . . . , N − 2) after the determination of the p oten t ia l pair (2.8) r esults in the existence of null op erators asso ciated with the N -fold SUSY system under consideration. Let us first in tro duce a linear space of linear differen tial o p erators, denoted by K [ N ] , whos e co efficien ts are all functionals of only w [ N ] k ( k = 0 , . . . , N − 1). Let K ij [ w [ N ] ] ∈ K [ N ] ( i = 0 , 1 , 2 , . . . ) and define a set of functionals f [ N ] i [ w [ N ] ] b y f [ N ] i [ w [ N ] ] = N − 2 X j =0 K ij [ w [ N ] ] I [ N ] j . (2.22) W e t hen define a subspace of K [ N ] , denoted b y K [ N ] 0 , whic h consists of linear differen tial op erators whose co efficien ts are all giv en b y f [ N ] i in tro duced in (2.22). That is, K 0 ∈ K [ N ] 0 means that there exists a set of linear differen tial op erators K ij [ w [ N ] ] ∈ K [ N ] suc h that K 0 = X i f [ N ] i [ w [ N ] ] ∂ i = X i N − 2 X j =0 K ij [ w [ N ] ] I [ N ] j ! ∂ i . (2.23) It is ob vious by definition that an y elemen t of K [ N ] 0 is a n ull op erator so long as all the N - f old SUSY constraints I [ N ] j = 0 ( j = 0 , . . . , N − 2) a re satisfied. Hence, the linear space in whic h an N -fold SUSY system is considered is actually the quotien t space K [ N ] / K [ N ] 0 . It naturally leads us to in tro duce an equiv alence class of linear differential o erato rs in K [ N ] . W e shall sa y that t wo linear differen tial op erators L 1 , L 2 ∈ K [ N ] b elong to e quivalen t class asso ciate d 7 with N -fold sup ersymmetry and express the equiv a lence as L 2 N ∼ L 1 if L 2 − L 1 ∈ K [ N ] 0 . An y equalit y b et wee n op erators L 2 = L 1 app eared in an N -fold SUSY system should b e t hus regarded as an equiv alen t relation L 2 N ∼ L 1 of the latter equiv alent class. In particular, an y explicit ex pression for a sp ecific op erator suc h as H ± and P ± N should b e considered a s a represen ta tiv e of it with resp ect to the equiv alen t class. In what follows, w e will emplo y the equiv alence relation N ∼ only when w e w ould like t o stress that the left a nd the right hand sides of the form ula under consideration is iden tical if and o nly if (some of ) the constrain ts I [ N ] j = 0 ( j = 0 , · · · , N − 2 ) are satisfied. I I I. 2-F O L D SUPERSYMMETR Y It is quite instructiv e to see ho w the general consideration in the previous section mak e sense in the case of 2-fold SUSY though its general form w as already o btained b y the direct in tegrations of the constraints [2, 11, 12 ]. Components of 2- f old sup erc harg es are giv en b y P − 2 = ∂ 2 + w 1 ∂ + w 0 , P + 2 = ∂ 2 − w 1 ∂ + w 0 − w ′ 1 , (3.1) where and hereafter w e shall omit the superscript [ N ] of w [ N ] k etc. fo r the simplicit y unless the omission w ould not cause any ambiguit y or confusion. The conditio n for 2-fold SUSY P − 2 H − − H + P − 2 = 0 is satisified if and only if the following three equalities hold: V + − V − = w ′ 1 , (3.2) w ′′ 1 + 2 w ′ 0 + 4 V −′ − 2 w 1 ( V + − V − ) = 0 , (3.3) w ′′ 0 + 2 V −′′ + 2 w 1 V −′ − 2 w 0 ( V + − V − ) = 0 . (3.4) Substituting ( 3 .2) into (3.3) and (3.4), and in tegrating the resulting equation from (3.3 ), we obtain 4 V + = 3 w ′ 1 − 2 w 0 + ( w 1 ) 2 − 4 C 0 , (3.5) 4 V − = − w ′ 1 − 2 w 0 + ( w 1 ) 2 − 4 C 0 , (3.6) − 4 I 0 = w ′′′ 1 − w 1 w ′′ 1 − 2( w ′ 1 ) 2 + 4 w ′ 1 w 0 + 2 w 1 w ′ 0 − 2( w 1 ) 2 w ′ 1 = 0 , (3.7) where C 0 is an in tegral constan t. T o integrate the third equation (3.7 ), we shall first mak e a dimension-preserving transformatio n w 0 → u 0 . W e choose it suc h that it will con ve rt sim ul- taneously bo t h the pairs P ± 2 and V ± in to symmetric forms. The most g eneral transformation of p olynomial ty p e preserving the ph ysical dimension [L − 2 ] of w 0 w ould b e w 0 = u 0 + 1 2 w ′ 1 − α 0 ( w 1 ) 2 [L − 2 ] , (3.8) where u 0 [L − 2 ] and α 0 is a dimensionless parameter. W e not e that t he la t t er t ransformation indeed renders b oth the pair of 2 -fold sup erc harge comp onents and the pair of p oten tials of symmetric forms as P ± 2 = ∂ 2 ∓ w 1 ∂ + u 0 − α 0 ( w 1 ) 2 ∓ 1 2 w ′ 1 , (3.9) 4 V ± = − 2 u 0 + (2 α 0 + 1)( w 1 ) 2 ± 2 w ′ 1 − 4 C 0 . (3.10) 8 With the transformation (3.8), the condition (3.7) r eads as − 4 I 0 = w ′′′ 1 + 4 w ′ 1 u 0 + 2 w 1 u ′ 0 − 2(4 α 0 + 1)( w 1 ) 2 w ′ 1 = 0 [L − 4 ] . (3.11) Hence, it gets simplest when α 0 = − 1 / 4 . (3.12) The next t ask w e should do is to construct the total differen t ia l d J 1 / d q in (2.20). W e note that d J 1 / d q and I 0 ha v e the phy sical dimensions [L − 5 ] and [L − 4 ], respective ly . Hence, the op erator L 10 defined in (2 .20) in this case m ust ha v e the ph ysical dimension [L − 1 ]. Except for the differen tial op erator d / d q , there is essen tially only one m ultiplicativ e op erator o f p olynomial type which hav e that dimension, na mely , L 10 ∝ w 1 [L − 1 ]. In f act, w e can easily c hec k that w 1 I 0 is of a total differen tial form a nd th us w e put 16 d J 1 d q = − 8 w 1 I 0 = 2 w 1 w ′′′ 1 + 8 w 1 w ′ 1 u 0 + 4( w 1 ) 2 u ′ 0 = 0 [L − 5 ] . (3.13) The latter differen tial equation is integrated to yield 16 J 1 [ w ] = 2 w 1 w ′′ 1 − ( w ′ 1 ) 2 + 4( w 1 ) 2 u 0 = 1 6 C 1 [L − 4 ] , (3.14) where C 1 is another in tegral constan t ha ving the correct ph ysical dimension [L − 4 ] listed in (2.5). Hence, we can express u 0 , and th us w 0 as w ell, in terms of w 1 as u 0 = w 0 − w ′ 1 2 − ( w 1 ) 2 4 = − w ′′ 1 2 w 1 + ( w ′ 1 ) 2 4( w 1 ) 2 + 4 C 1 ( w 1 ) 2 . (3.15) Substituting it in to (3.9) and (3.1 0), w e finally get the general form of 2-fold SUSY systems as P ± 2 2 ∼ ∂ 2 ∓ w 1 ∂ + ( w 1 ) 2 4 − w ′′ 1 2 w 1 + ( w ′ 1 ) 2 4( w 1 ) 2 + 4 C 1 ( w 1 ) 2 ∓ w ′ 1 2 , (3.16) V ± 2 ∼ ( w 1 ) 2 8 + w ′′ 1 4 w 1 − ( w ′ 1 ) 2 8( w 1 ) 2 − 2 C 1 ( w 1 ) 2 ± w ′ 1 2 − C 0 . (3.17) Finally , pro ducts of the comp o nen ts of 2- f old sup erc harg es P ∓ 2 P ± 2 are calculated as P − 2 P + 2 = 4 [( H + + C 0 ) 2 + C 1 ] − 2 I 0 , (3.18) P + 2 P − 2 = 4 [( H − + C 0 ) 2 + C 1 ] + 2 I 0 , ( 3 .19) where I 0 and C 1 are giv en b y (3 .7) and (3.14), respective ly . Hence, w e obtain the equalit y (2.4) for N = 2 as an equiv alent relation asso ciated with 2- fold SUSY: P ∓ 2 P ± 2 2 ∼ 4[( H ± + C 0 ) 2 + C 1 ] . (3.20) 9 IV. 3-F OLD SUPERSYMMETR Y Next, w e shall reexamine the case of 3- f old SUSY, whic h w as once in v estigated briefly in Ref. [1 3 ], by utilizing our general analysis. Comp onents of 3-fold sup erc harg es are given b y P − 3 = ∂ 3 + w 2 ∂ 2 + w 1 ∂ + w 0 , P + 3 = − ∂ 3 + w 2 ∂ 2 − ( w 1 − 2 w ′ 2 ) ∂ + w 0 − w ′ 1 + w ′′ 2 . (4.1) The condition for 3-fold SUSY P − 3 H − − H + P − 3 = 0 is satisfied if and only if the follo wing four equalities hold: V + − V − = w ′ 2 , (4.2) w ′′ 2 + 2 w ′ 1 + 6 V −′ − 2 w 2 ( V + − V − ) = 0 , (4.3) w ′′ 1 + 2 w ′ 0 + 6 V −′′ + 4 w 2 V −′ − 2 w 1 ( V + − V − ) = 0 , (4.4) w ′′ 0 + 2 V −′′′ + 2 w 2 V −′′ + 2 w 1 V −′ − 2 w 0 ( V + − V − ) = 0 . (4.5) Substituting (4.2) in to (4.3)–(4.5 ) and in tegrating the resulting equation from (4.3), w e obtain 6 V + = 5 w ′ 2 − 2 w 1 + ( w 2 ) 2 − 6 C 0 , (4.6) 6 V − = − w ′ 2 − 2 w 1 + ( w 2 ) 2 − 6 C 0 , (4.7) − 6 I 1 = 3 w ′′′ 2 + 3 w ′′ 1 − 6 w ′ 0 − 4 w 2 w ′′ 2 − 6( w ′ 2 ) 2 + 6 w ′ 2 w 1 + 4 w 2 w ′ 1 − 4( w 2 ) 2 w ′ 2 = 0 , (4.8) − 6 I 0 = w ′′′′ 2 + 2 w ′′′ 1 − 3 w ′′ 0 − w 2 w ′′′ 2 − 6 w ′ 2 w ′′ 2 + w ′′ 2 w 1 + 2 w 2 w ′′ 1 + 6 w ′ 2 w 0 + 2 w 1 w ′ 1 − 2( w 2 ) 2 w ′′ 2 − 2 w 2 ( w ′ 2 ) 2 − 2 w 2 w ′ 2 w 1 = 0 , (4.9) where C 0 is a n integral constant. T o integrate the remaining equations (4.8) and (4.9), w e shall first construct a set o f dimension-preserving transformations w k → u k ( k = 0 , 1) whic h will con v ert sim ultaneously b oth the pairs P ± 3 and V ± in to symmetric forms, lik e (3.8) in the case of 2- fold SUSY. The most general transformations of p olynomial type preserving the ph ysical dimensions [L k − 3 ] of w k w ould b e w 1 = 6 u 1 + w ′ 2 − α 1 ( w 2 ) 2 [L − 2 ] , w 0 = u 0 + 3 u ′ 1 − β 1 w ′′ 2 − α 1 w 2 w ′ 2 − 6 β 2 w 2 u 1 − β 3 ( w 2 ) 3 [L − 3 ] , (4.10) where u k [L k − 3 ] and α 0 , β k ( k = 1 , 2 , 3) are all dimens ionless parameters. They a ctually render b oth the pair o f 3-fold sup erc harge comp onen ts a nd the pair o f p oten tials of symmetric forms as P ± 3 = ∓ ∂ 3 + w 2 ∂ 2 ∓ [6 u 1 − α 1 ( w 2 ) 2 ∓ w ′ 2 ] ∂ + u 0 − β 1 w ′′ 2 − 6 β 2 w 2 u 1 − β 3 ( w 2 ) 3 ∓ (3 u ′ 1 − α 1 w 2 w ′ 2 ) , (4.11 a ) 6 V ± = − 12 u 1 + (2 α 1 + 1)( w 2 ) 2 ± 3 w ′ 2 − 6 C 0 . (4.11b) 10 With the tr ansformations (4.10), the tw o constraints ( 4 .8) and (4.9) ar e equiv alent to the follo wing new set o f conditio ns: − 3 ¯ I 1 = 9 I 1 = 3( β 1 + 1) w ′′′ 2 − 3 u ′ 0 + 18( β 2 + 1) w ′ 2 u 1 + 6(3 β 2 + 2) w 2 u ′ 1 − (7 α 1 − 9 β 3 + 2)( w 2 ) 2 w ′ 2 = 0 [L − 4 ] , (4.12) 3 ¯ I 0 = − 9(2 I 0 − ∂ I 1 ) = 3 u ′′′ 1 − ( α 1 − 1) w 2 w ′′′ 2 − 3(2 β 1 + α 1 + 1) w ′ 2 w ′′ 2 + 6 w ′ 2 u 0 + 72 u 1 u ′ 1 − 12(2 α 1 + 3 β 2 + 1) w 2 w ′ 2 u 1 − 12 α 1 ( w 2 ) 2 u ′ 1 + 2[2( α 1 ) 2 + α 1 − 3 β 3 ]( w 2 ) 3 w ′ 2 = 0 [L − 5 ] . (4.13) Hence, they would get simplest when α 1 = 1 , β 1 = − 1 , β 2 = − 1 , β 3 = 1 . (4.14) With the la t ter c hoice of para meter v alues, the new set of conditions (4.12) and (4.13) reads as ¯ I 1 = u ′ 0 + 2 w 2 u ′ 1 = 0 [L − 4 ] , (4.15) ¯ I 0 = u ′′′ 1 + 2 w ′ 2 u 0 + 24 u 1 u ′ 1 − 4( w 2 ) 2 u ′ 1 = 0 [L − 5 ] . (4.16) The first equation (4.15 ) enables us to express u 0 in terms of u 1 and w 2 as an indefinite in tegral u 0 = − 2 Z d q w 2 u ′ 1 [L − 3 ] . (4.17) Next, let us construct the total differential d J 1 / d q in (2.20). W e note that d J 1 / d q and ¯ I k ( k = 1 , 2) ha ve the ph ysical dimensions [L − 5 ] and [L k − 5 ], resp ectiv ely . Hence, t he o p erators L 1 k defined in (2.2 0 ) in this case m ust ha v e the ph ysical dimension [L − k ]. Th us, L 10 is just a dimensionless constan t while L 11 ∝ w 2 if w e restrict L 1 k to m ultiplicative op erat o rs of p olynomial t yp e. Indeed, w e can easily find that one of the latter c hoices leads to a total differen tial − 4 d J 1 d q = ¯ I 0 + 2 w 2 ¯ I 1 = u ′′′ 1 + 2( w ′ 2 u 0 + w 2 u ′ 0 ) + 24 u 1 u ′ 1 = 0 [L − 5 ] , (4.18) whic h can b e easily integrated as − 4 J 1 [ w ] = u ′′ 1 + 2 w 2 u 0 + 12( u 1 ) 2 = − 4 C 1 [L − 4 ] , (4.19) where C 1 is another in tegral constan t ha ving the correct ph ysical dimension [L − 4 ] listed in (2.5). F rom (4 .16) and (4.19), w e can express u 0 in terms of u 1 and w 2 without recourse to an y indefinite integral as u 0 = − u ′′′ 1 + 24 u 1 u ′ 1 − 4( w 2 ) 2 u ′ 1 2 w ′ 2 = − u ′′ 1 + 12( u 1 ) 2 + 4 C 1 2 w 2 . (4.20) T o construct the second integral C 2 of 3-fold SUSY systems , w e first note that d J 2 / d q and ¯ I k ( k = 0 , 1) ha ve the ph ysical dimensions [L − 7 ] and [L k − 5 ], resp ectiv ely . Hence, t he o p erators 11 L 2 k defined in (2.20) in this case m ust hav e the phy sical dimension [L − k − 2 ]. Th us, candidates for L 20 are w ′ 2 , u 1 , and ( w 2 ) 2 , while t hose for L 21 are w ′′ 2 , u ′ 1 , u 0 , w 2 w ′ 2 , w 2 u 1 , and ( w 2 ) 3 , if w e restrict L 2 k to m ultiplicativ e op erators of p olynomial t yp e. With the c hoice of L 20 = 0 and L 21 ∝ u 0 , we can construct a t otal differen tial as 4 d J 2 d q = u 0 ¯ I 1 = u 0 u ′ 0 + 2 w 2 u ′ 1 u 0 = u 0 u ′ 0 − u ′ 1 u ′′ 1 − 12( u 1 ) 2 u ′ 1 − 4 C 1 u ′ 1 = 0 [L − 7 ] , (4.21) where (4.19) has b een used. The latter relation is indeed easily inte g r a ted as 8 J 2 [ w ] = ( u 0 ) 2 − ( u ′ 1 ) 2 − 8( u 1 ) 3 − 8 C 1 u 1 = 8 C 2 [L − 6 ] , (4.22) with another integral constan t C 2 ha ving the correct ph ysical dimension [L − 6 ] listed in (2.5). With the use of (4.2 2 ) we can express u 0 solely in terms of u 1 . The n, substituting the obtained expression for u 0 in to (4.15), we can also express w 2 solely in terms of u 1 . Hence, w e can ev en tually ha v e an expression fo r V ± and P ± 3 in terms of only a single arbitr ary function u 1 . Ho w ev er, the latt er expression is relativ ely complicated and thus it w ould b e more con ve nient in practice to express t hem in terms of tw o of the t hree functions w 2 , u 1 , and u 0 . If we eliminate u 0 in (4.11 ) by using (4.20), w e hav e the expression in terms of w 2 and u 1 as P ± 3 3 ∼ ∓ ∂ 3 + w 2 ∂ 2 ∓  6 u 1 − ( w 2 ) 2 ∓ w ′ 2  ∂ + w ′′ 2 + 6 w 2 u 1 − ( w 2 ) 3 − u ′′ 1 2 w 2 − 6( u 1 ) 2 w 2 − 2 C 1 w 2 ∓ (3 u ′ 1 − w 2 w ′ 2 ) , (4.23a) V ± = − 2 u 1 + 1 2 ( w 2 ) 2 ± 1 2 w ′ 2 − C 0 . (4.23b) On the other hand, if w e eliminate w 2 in (4.11) b y using (4 .1 5), w e hav e the express ion in terms of u 1 and u 0 as P ± 3 3 ∼ ∓ ∂ 3 − u ′ 0 2 u ′ 1 ∂ 2 ∓  6 u 1 − ( u ′ 0 ) 2 4( u ′ 1 ) 2 ±  u ′′ 0 2 u ′ 1 − u ′′ 1 u ′ 0 2( u ′ 1 ) 2  ∂ + u 0 − u ′′′ 0 2 u ′ 1 + u ′′ 1 u ′′ 0 ( u ′ 1 ) 2 +  u ′′′ 1 2( u ′ 1 ) 2 − ( u ′′ 1 ) 2 ( u ′ 1 ) 3 − 3 u 1 u ′ 1  u ′ 0 + ( u ′ 0 ) 3 8( u 1 ) 3 ∓  3 u ′ 1 − u ′ 0 u ′′ 0 4( u ′ 1 ) 2 + u ′′ 1 ( u ′ 0 ) 2 4( u ′ 1 ) 3  , (4.24a) V ± 3 ∼ − 2 u 1 + ( u ′ 0 ) 2 8( u ′ 1 ) 2 ∓  u ′′ 0 4 u ′ 1 − u ′′ 1 u ′ 0 4( u ′ 1 ) 2  − C 0 . (4.24b) Finally , pro ducts of the comp o nen ts of 3- f old sup erc harg es P ∓ 3 P ± 3 are calculated as P − 3 P + 3 = 8[( H + + C 0 ) 3 + C 1 ( H + + C 0 ) + C 2 ] + 3 I 1 ∂ 2 + 2[(2 ∂ + w 2 ) I 1 − I 0 ] ∂ + (2 ∂ 2 + 2 w 2 ∂ − 2 w ′ 2 + w 1 ) I 1 − 2( ∂ + w 2 ) I 0 , (4.25) P + 3 P − 3 = 8[( H − + C 0 ) 3 + C 1 ( H − + C 0 ) + C 2 ] − 3 I 1 ∂ 2 − 2[( ∂ − w 2 ) I 1 + I 0 ] ∂ − w 1 I 1 − 2( ∂ − w 2 ) I 0 , (4.26) where I 1 , I 0 , C 1 , and C 2 are give n by (4.8), ( 4.9), ( 4 .19), and (4.22), resp ectiv ely . Hence , w e obtain the equalit y (2.4) for N = 3 as an equiv alen t relation asso ciated with 3-fold SUSY: P ∓ 3 P ± 3 3 ∼ 8[( H ± + C 0 ) 3 + C 1 ( H ± + C 0 ) + C 2 ] . (4.27) 12 V. 4-F O LD SUPERSYMMETR Y In this section, w e shall study the case of 4- f old sup ersymmetry . Comp onents of 4- fold sup erc harges are g iven by P − 4 = ∂ 4 + w 3 ∂ 3 + w 2 ∂ 2 + w 1 ∂ + w 0 , P + 4 = ∂ 4 − w 3 ∂ 3 + ( w 2 − 3 w ′ 3 ) ∂ 2 − ( w 1 − 2 w ′ 2 + 3 w ′′ 3 ) ∂ + w 0 − w ′ 1 + w ′′ 2 − w ′′′ 3 . (5.1) The condition for 4-fold sup ersymmetry P − 4 H − − H + P − 4 = 0 is satisfied if and o nly if the follo wing five equalities hold: V + − V − = w ′ 3 , (5.2) w ′′ 3 + 2 w ′ 2 + 8 V −′ − 2 w 3 ( V + − V − ) = 0 , (5.3) w ′′ 2 + 2 w ′ 1 + 12 V −′′ + 6 w 3 V −′ − 2 w 2 ( V + − V − ) = 0 , (5.4) w ′′ 1 + 2 w ′ 0 + 8 V −′′′ + 6 w 3 V −′′ + 4 w 2 V −′ − 2 w 1 ( V + − V − ) = 0 , (5.5) w ′′ 0 + 2 V −′′′′ + 2 w 3 V −′′′ + 2 w 2 V −′′ + 2 w 1 V −′ − 2 w 0 ( V + − V − ) = 0 . (5.6) Substituting (5.2) in to (5.3)–(5.6 ) and in tegrating the resulting equation from (5.3), w e obtain 8 V + = 7 w ′ 3 − 2 w 2 + ( w 3 ) 2 − 8 C 0 , (5.7) 8 V − = − w ′ 3 − 2 w 2 + ( w 3 ) 2 − 8 C 0 , (5.8) − 8 I 2 = 6 w ′′′ 3 + 8 w ′′ 2 − 8 w ′ 1 − 9 w 3 w ′′ 3 − 12( w ′ 3 ) 2 + 8 w ′ 3 w 2 + 6 w 3 w ′ 2 − 6( w 3 ) 2 w ′ 3 = 0 , (5.9) − 8 I 1 = 4 w ′′′′ 3 + 8 w ′′′ 2 − 4 w ′′ 1 − 8 w ′ 0 − 5 w 3 w ′′′ 3 − 24 w ′ 3 w ′′ 3 + 2 w ′′ 3 w 2 + 6 w 3 w ′′ 2 + 8 w ′ 3 w 1 + 4 w 2 w ′ 2 − 6( w 3 ) 2 w ′′ 3 − 6 w 3 ( w ′ 3 ) 2 − 4 w 3 w ′ 3 w 2 = 0 , (5.10) − 8 I 0 = w ′′′′′ 3 + 2 w ′′′′ 2 − 4 w ′′ 0 − w 3 w ′′′′ 3 − 8 w ′ 3 w ′′′ 3 − 6( w ′′ 3 ) 2 + w ′′′ 3 w 2 + 2 w 3 w ′′′ 2 + w ′′ 3 w 1 + 8 w ′ 3 w 0 + 2 w 2 w ′′ 2 + 2 w ′ 2 w 1 − 2( w 3 ) 2 w ′′′ 3 − 6 w 3 w ′ 3 w ′′ 3 − 2 w 3 w ′′ 3 w 2 − 2( w ′ 3 ) 2 w 2 − 2 w 3 w ′ 3 w 1 = 0 . (5.11) where C 0 is an in tegral constan t. T o in tegra te the remaining equations (5.9)–(5 .1 1), let us first lo ok for a set o f dimension-preserving transformatio ns w k → u k ( k = 0 , 1 , 2) whic h will con v ert sim ultaneously b oth the pairs P ± 4 and V ± in to symmetric forms as in the cases of 2- and 3- fold SUSY. The most general transformatio ns of p olynomial t yp e preserving t he ph ysical dimensions [L k − 4 ] of w k w ould b e w 2 = u 2 + 3 2 w ′ 3 − α 1 ( w 3 ) 2 [L − 2 ] , (5.12) w 1 = u 1 + u ′ 2 − β 1 w ′′ 3 − 2 α 1 w 3 w ′ 3 − β 2 w 3 u 2 − β 3 ( w 3 ) 3 [L − 3 ] , (5.13) w 0 = u 0 + 1 2 u ′ 1 − γ 1 u ′′ 2 −  β 1 2 + 1 4  w ′′′ 3 − γ 2 w 3 w ′′ 3 − γ 3 ( w ′ 3 ) 2 − β 2 2 ( w 3 u 2 ) ′ − γ 4 w 3 u 1 − γ 5 ( u 2 ) 2 − 3 β 3 2 ( w 3 ) 2 w ′ 3 − γ 6 ( w 3 ) 2 u 2 + γ 7 ( w 3 ) 4 [L − 4 ] , (5.14) where u k [L k − 4 ] and α 0 , β k ( k = 1 , 2 , 3), and γ k ( k = 1 , . . . , 7) a re all dimensionless param- eters. They indeed make b oth the pa ir of 4-fold sup erc harge comp onen ts and the pair of 13 p oten tia ls symmetric as P ± 4 = ∂ 4 ∓ w 3 ∂ 3 +  u 2 − α 1 ( w 3 ) 2 ∓ 3 2 w ′ 3  ∂ 2 ∓  u 1 − β 1 w ′′ 3 − β 2 w 3 u 2 − β 3 ( w 3 ) 3 ∓ ( u ′ 2 − 2 α 1 w 3 w ′ 3 )  ∂ + u 0 − γ 1 u ′′ 2 − γ 2 w 3 w ′′ 3 − γ 3 ( w ′ 3 ) 2 − γ 4 w 3 u 1 − γ 5 ( u 2 ) 2 − γ 6 ( w 3 ) 2 u 2 − γ 7 ( w 3 ) 4 ∓  1 2 u ′ 1 −  β 1 2 + 1 4  w ′′′ 3 − β 2 2 ( w 3 u 2 ) ′ − 3 β 3 2 ( w 3 ) 2 w ′ 3  , (5.15a) 8 V ± = − 2 u 2 + (2 α 1 + 1)( w 3 ) 2 ± 4 w ′ 3 − 8 C 0 . (5.15b) With the tr a nsformations (5.1 4 ), the remaining three constrain ts (5.9)– ( 5.11) are equiv alent to the follo wing new set of conditions: ¯ I 2 = 4 I 2 = − (4 β 1 + 9) w ′′′ 3 + 4 u ′ 1 − 4( β 2 + 1) w ′ 3 u 2 − (4 β 2 + 3) w 3 u ′ 2 + (10 α 1 − 12 β 3 + 3)( w 3 ) 2 w ′ 3 = 0 [L − 4 ] , (5.16) 4 ¯ I 1 = 4( I 1 − ∂ I 2 ) = − 2(2 γ 1 + 1) u ′′′ 2 + 4 u ′ 0 − (4 β 1 γ 4 − 4 α 1 + 4 γ 2 + 9 γ 4 + 2) w 3 w ′′′ 3 + 2(6 α 1 + 2 β 1 − 2 γ 2 − 4 γ 3 + 3) w ′ 3 w ′′ 3 − 4( γ 4 + 1) w ′ 3 u 1 − 2(4 γ 5 + 1) u 2 u ′ 2 − 2(2 β 2 γ 4 − 2 α 1 − 2 β 2 + 2 γ 4 + 4 γ 6 − 1) w 3 w ′ 3 u 2 − (4 β 2 γ 4 − 2 α 1 + 3 γ 4 + 4 γ 6 )( w 3 ) 2 u ′ 2 − [4( α 1 ) 2 − 10 α 1 γ 4 + 12 β 3 γ 4 + 2 α 1 − 3 γ 4 − 4 β 3 + 16 γ 7 ]( w 3 ) 3 w ′ 3 = 0 [L − 5 ] , (5.17) ¯ I 0 = − 4(4 I 0 − 2 ∂ I 1 + ∂ 2 I 2 ) = w ′′′′′ 3 + 4 w ′′′ 3 u 2 − (4 β 1 + 3) w ′′ 3 u ′ 2 − 4(4 γ 1 + 1) w ′ 3 u ′′ 2 + w 3 u ′′′ 2 + 16 w ′ 3 u 0 + 4 u ′ 2 u 1 − (6 α 1 + 1)( w 3 ) 2 w ′′′ 3 + 4(2 α 1 β 1 + 2 α 1 + β 1 − 4 γ 2 ) w 3 w ′ 3 w ′′ 3 + 8( α 1 − 2 γ 3 )( w ′ 3 ) 3 − 4(2 α 1 + 4 γ 4 + 1) w 3 w ′ 3 u 1 − 16 γ 5 w ′ 3 ( u 2 ) 2 − 4 β 2 w 3 u 2 u ′ 2 + 4(2 α 1 β 2 + β 2 − 4 γ 6 )( w 3 ) 2 w ′ 3 u 2 − 4 β 3 ( w 3 ) 3 u ′ 2 + 4(2 α 1 β 3 + β 3 − 4 γ 7 )( w 3 ) 4 w ′ 3 = 0 [L − 6 ] . (5.18) Hence, they would get simplest when 1 α 1 = 3 / 2 , β 1 = − 9 / 4 , β 2 = − 1 , β 3 = 3 / 2 , γ 1 = − 1 / 2 , γ 2 = 1 , γ 3 = 11 / 8 , γ 4 = − 1 , γ 5 = − 1 / 4 , γ 6 = 1 / 2 , γ 7 = − 3 / 8 . (5.19) 1 Another p o s sible choice co uld be α 1 = 0, β 1 = − 9 / 4 , β 2 = − 3 / 4, β 3 = 1 / 4, γ 1 = − 1 / 2, γ 2 = − 1 / 2, γ 3 = − 1 / 8, γ 4 = − 1, γ 5 = − 1 / 4, γ 6 = 0, γ 7 = 1 / 16. With the latter choice, it turns o ut that the third int eg ral C 3 admits a simpler expr ession tha n (A1) but the expr ession for the second integral C 2 contains terms whic h a re linea r in u 1 , in contrast with (5.26) and (5.28). 14 With the latt er c hoice of pa rameter v alues, the new set of conditions (5.16 ) –(5.18) reads as ¯ I 2 = 4 u ′ 1 + w 3 u ′ 2 = 0 [L − 4 ] , (5.20) ¯ I 1 = u ′ 0 = 0 [L − 5 ] , (5.21) ¯ I 0 = w ′′′′′ 3 + 4 w ′′′ 3 u 2 + 6 w ′′ 3 u ′ 2 + 4 w ′ 3 u ′′ 2 + w 3 u ′′′ 2 + 16 w ′ 3 u 0 + 4 u ′ 2 u 1 − 10( w 3 ) 2 w ′′′ 3 − 40 w 3 w ′ 3 w ′′ 3 − 10( w ′ 3 ) 3 + 4 w ′ 3 ( u 2 ) 2 + 4 w 3 u 2 u ′ 2 − 24( w 3 ) 2 w ′ 3 u 2 − 6( w 3 ) 3 u ′ 2 + 30( w 3 ) 4 w ′ 3 = 0 [L − 6 ] . (5.22) The first equation (5.20 ) enables us to express u 1 in terms of u 2 and w 3 as an indefinite in tegral 4 u 1 = − Z d q w 3 u ′ 2 [L − 3 ] . (5.23) The second equation (5.21) j ust me ans that u 0 is a constan t. On the other hand, C k ( k = 0 , . . . , N − 1) are the only constan ts whic h app ear in general form of N -fold SUSY. The ph ysical dimension o f u 0 is [L − 4 ] and th us we can put u 0 = 2 C 1 [L − 4 ] , (5.24) where C 1 is an in tegra l constant hav ing the correct ph ysical dimension [L − 4 ] listed in (2.5). W e hav e no w obtained the first integral C 1 and th us can skip ov er the step to construct the quan tity J 1 . Next, to construct t he second in tegra l C 2 of 4-fold SUSY sys tems, w e first note that d J 2 / d q and ¯ I k ( k = 0 , 1 , 2) ha ve the ph ysical dimensions [L − 7 ] and [L k − 6 ], resp ectiv ely . Hence, the op erators L 2 k defined in (2 .20) in this case must hav e the ph ysical dimens io n [L − k − 1 ]. Th us, candidat es for L 20 are w 3 , those f o r L 21 are w ′ 3 , u 2 , and ( w 3 ) 2 , while those for L 22 are w ′′ 3 , u ′ 2 , u 1 , w 3 w ′ 3 , w 3 u 2 , and ( w 3 ) 3 , if w e restrict L 2 k to m ultiplicative op erators of p o lynomial ty p e. With the choice of L 20 ∝ w 3 and L 21 = L 22 = 0 , w e can construct a total differen tial as − 128 d J 2 d q = 2 w 3 ¯ I 0 = 0 [L − 7 ] , (5.25) where w e hav e omitted the explicit expression. The in tegration o f the latt er yields − 128 J 2 [ w ] = 2 w 3 w ′′′′ 3 − 2 w ′ 3 w ′′′ 3 + ( w ′′ 3 ) 2 − 16( u 1 ) 2 + 8 w 3 w ′′ 3 u 2 − 4( w ′ 3 ) 2 u 2 + 4 w 3 w ′ 3 u ′ 2 + 2( w 3 ) 2 u ′′ 2 − 20( w 3 ) 3 w ′′ 3 − 10( w 3 ) 2 ( w ′ 3 ) 2 + 4( w 3 ) 2 ( u 2 ) 2 − 12( w 3 ) 4 u 2 + 10( w 3 ) 6 + 32 C 1 ( w 3 ) 2 = − 128 C 2 [L − 6 ] , (5.26) where (5.24 ) has b een applied, and C 2 is another in tegr a l constant ha ving the correct phy sical dimension [L − 6 ] listed in (2 .5 ). F rom (5 .22) and (5.24 ), we can express u 1 in terms of u 2 and w 3 without recourse to an y indefinite integral a s u 1 = [ − w ′′′′′ 3 − 4 w ′′′ 3 u 2 − 6 w ′′ 3 u ′ 2 − 4 w ′ 3 u ′′ 2 − w 3 u ′′′ 2 + 10( w 3 ) 2 w ′′′ 3 + 40 w 3 w ′ 3 w ′′ 3 + 10( w ′ 3 ) 3 − 4 w ′ 3 ( u 2 ) 2 − 4 w 3 u 2 u ′ 2 + 24( w 3 ) 2 w ′ 3 u 2 + 6( w 3 ) 3 u ′ 2 − 30( w 3 ) 4 w ′ 3 − 32 C 1 w ′ 3 ] / (4 u ′ 2 ) . (5.27) 15 Instead, using (5.26), w e can express u 1 in terms of u 2 and w 3 as a solution to the follow ing quadratic equation 16( u 1 ) 2 = 2 w 3 w ′′′′ 3 − 2 w ′ 3 w ′′′ 3 + ( w ′′ 3 ) 2 + 8 w 3 w ′′ 3 u 2 − 4( w ′ 3 ) 2 u 2 + 4 w 3 w ′ 3 u ′ 2 + 2( w 3 ) 2 u ′′ 2 − 20( w 3 ) 3 w ′′ 3 − 10( w 3 ) 2 ( w ′ 3 ) 2 + 4( w 3 ) 2 ( u 2 ) 2 − 12( w 3 ) 4 u 2 + 10( w 3 ) 6 + 32 C 1 ( w 3 ) 2 + 128 C 2 . (5.28) T o o bt a in the third in tegral C 3 of 4-f o ld SUSY systems, w e note that d J 3 / d q and ¯ I k ( k = 0 , 1 , 2) hav e the phys ical dimensions [L − 9 ] and [L k − 6 ], resp ectiv ely . Hence, the op era t o rs L 3 k defined in (2.20) in this case mus t ha ve the phys ical dimension [L − k − 3 ]. With a similar dimensonal analysis, we find that the choice of L 30 ∝ w ′′ 3 + 4 u 1 + 2 w 3 u 2 − 2( w 3 ) 3 and L 31 = L 32 = 0 leads t o a total differen tial: 512 d J 3 d q = [ w ′′ 3 + 4 u 1 + 2 w 3 u 2 − 2( w 3 ) 3 ] ¯ I 0 = 0 [L − 9 ] , (5.29) where w e ha v e omitted the explicit expression. The in tegral o f the latter equation reads as J 3 [ w ] = C 3 with another in tegral constan t C 3 ha ving the correct ph ysical dimension [L − 8 ] listed in (2 .5 ). The explicit fo r m ula fo r J 3 is giv en b y (A1) in App endix A. By using (at least) t w o of the eq ualities (5 .27), (5.28), and (A1), w e can eliminate u 1 to obtain the relation b et we en w 3 and u 2 . Hence, w e are now, in principle, able to express a 4-f old SUSY system solely in terms of a single arbit r a ry function. As in the case of 3 -fold SUSY, how ev er, it w ould b e more con ven ient for a pra ctical purp ose to ha v e an expression in terms of t w o of the four functions w 3 , u 2 , u 1 , and u 0 . F or example, if w e eliminate u 1 and u 0 in the system b y using (5.27) and (5 .24), resp ectiv ely , the system (5.15) with the parameter v alues (5.19) can b e represen ted in terms o f w 3 and u 2 as (A2). Finally , pro ducts of t he comp onen ts of 4-fold sup erch ar ges P ∓ 4 P ± 4 are calculated as P − 4 P + 4 = 16[( H + + C 0 ) 4 + C 1 ( H + + C 0 ) 2 + C 2 ( H + + C 0 ) + C 3 ] + 4 X i =0 f + i [ w ] ∂ i , (5.30) P + 4 P − 4 = 16[( H − + C 0 ) 4 + C 1 ( H − + C 0 ) 2 + C 2 ( H − + C 0 ) + C 3 ] + 4 X i =0 f − i [ w ] ∂ i , (5.31) where C 1 , C 2 , and C 3 are giv en b y (5.24), (5.26), and (A1 ), resp ectiv ely . Both of the second terms in the ab ov e are elemen ts of K [4] 0 in tro duced in Section I I B and ha ve the form of (2.23). The explicit forms o f f ± i are giv en b y (A3) and (A4). Hence , we obtain the equalit y (2.4) for N = 4 as an equiv alent relation asso ciated with 4- fold SUSY: P ∓ 4 P ± 4 4 ∼ 16[( H ± + C 0 ) 4 + C 1 ( H ± + C 0 ) 2 + C 2 ( H ± + C 0 ) + C 3 ] . (5.32) VI. DISCUSSION AND SUMMAR Y In this w o r k, w e hav e clarified general structure of N -fold SUSY systems by considering dimensional analysis a nd in tro ducing the equiv alen t classes of linear differen tial op erators asso ciated with t hem. W e ha ve t hen sho wn t ha t the latter general consideration is in f act effectiv e in constructing the most general N -fold SUSY systems and their in tegral constants 16 for N = 2, 3, a nd 4. Application to systems for N > 4 w o uld b e straigh tforward and the problems would get mo r e transparent ev en though still remain highly complicated. Finally , some r emarks on the future issues are in order. 1. Although w e hav e only considered ordinary one-dimensional Schr¨ odinger op erato rs, generalization to other op erators w ould b e p ossible. In phys ical a pplications, one of the in teresting extensions is to a quan tum system with p osition-dep enden t mass for whic h N -fold SUSY w a s succe ssfully formulated in R ef. [16]. In the latter case, there is an additional freedom of mass function and it is part icularly in teresting to see how its existence w ould f o rce us to mo dify o r g eneralize the general considerations made in Section I I. 2. In this work, w e ha ve restricted the dimension-prese rving transformations (3.8), (4.10), and (5 .14) to p o lynomial type, namely , transformations whic h are p olynomials in w [ N ]( m ) k ( m = 0 , 1 , 2 , . . . ). On the other hand, the o bt a ined results suc h as (3.15), (4.20), and (5.27) indicate that the relations amo ng w [ N ]( m ) k are in general expressed b y rational functions. Hence, we ma y b e able to reduce f urther the complexity of the conditions for N -fo ld SUSY b y extending the t yp e of transformations from p olynomial to rational function. How eve r, the n um b er of admissible forms of rational functions whic h preserv e ph ysical dimension w ould drastically increase. F or instance, an y sum of r ational functions of the form f [ N ] n [ w ] /g [ N ] n [ w ] ( n ∈ Z ) w [ N ] k = u [ N ] k + X n ∈ Z f [ N ] n [ w ] g [ N ] n [ w ] [L k −N ] , where f [ N ] n [ w ] and g [ N ] n [ w ] are p olynomials in w [ N ]( m ) k ha ving the ph ysical dimensions [L n + k −N ] and [L n ], resp ectiv ely , can serv e as a (par t of ) tra nsformation whic h preserv es the ph ysical dimension [L k −N ] of w [ N ] k . As a result, w e ma y need additional guidelines to restrict the fo rms of tra nsformations to mak e an efficien t analysis. W e are curious to kno w ho w to get suc h guidelines systematically for the purp ose. 3. As w as p oin ted out in Section I I, an N - fold SUSY system is in general only we akly quasi- solv able but is not quasi-solv able in the strong sense and th us do es not necess a r ily admit analytic lo cal solutions in closed form. In the case of N = 2, it was pro v ed in Ref. [17] that t yp e A 2-fold SUSY is a necessary and sufficien t condition for a one-dimensional quantum mec hanical sy stem to ha v e qu a si- solv abilit y in the strong sense with tw o independen t analytic lo cal solutions. So, it is interes ting to see what kind of condition is necess ar y and sufficien t for a quan tum system to admit three or four indep enden t analytic lo cal solutions. W e will rep ort on the latter sub jects in subsequen t publications. App endix A: F orm ulas In this App endix, we presen t the length y form ulas app eared in 4- fold SUSY in Section V. 17 The third in tegral: 1024 J 3 [ w ] = 2 w ′′ 3 w ′′′′ 3 − ( w ′′′ 3 ) 2 + 8 w ′′′′ 3 u 1 + 4 w 3 w ′′′′ 3 u 2 − 4 w ′ 3 w ′′′ 3 u 2 + 6( w ′′ 3 ) 2 u 2 − 2 w 3 w ′′′ 3 u ′ 2 + 6 w ′ 3 w ′′ 3 u ′ 2 + 2 w 3 w ′′ 3 u ′′ 2 + 32 w ′′ 3 u 2 u 1 + 24 w ′ 3 u ′ 2 u 1 + 8 w 3 u ′′ 2 u 1 + 32 u 2 ( u 1 ) 2 − 4( w 3 ) 3 w ′′′′ 3 + 12( w 3 ) 2 w ′ 3 w ′′′ 3 − 16( w 3 ) 2 ( w ′′ 3 ) 2 − 24 w 3 ( w ′ 3 ) 2 w ′′ 3 + ( w ′ 3 ) 4 − 80( w 3 ) 2 w ′′ 3 u 1 − 80 w 3 ( w ′ 3 ) 2 u 1 + 16 w 3 w ′′ 3 ( u 2 ) 2 − 4( w ′ 3 ) 2 ( u 2 ) 2 + 8 w 3 w ′ 3 u 2 u ′ 2 + 4( w 3 ) 2 u 2 u ′′ 2 − ( w 3 ) 2 ( u ′ 2 ) 2 + 32 w 3 ( u 2 ) 2 u 1 − 56( w 3 ) 3 w ′′ 3 u 2 − 20( w 3 ) 2 ( w ′ 3 ) 2 u 2 − 4( w 3 ) 4 u ′′ 2 − 64( w 3 ) 3 u 2 u 1 + 8( w 3 ) 2 ( u 2 ) 3 + 40( w 3 ) 5 w ′′ 3 + 10( w 3 ) 4 ( w ′ 3 ) 2 + 48( w 3 ) 5 u 1 − 28( w 3 ) 4 ( u 2 ) 2 + 36( w 3 ) 6 u 2 − 15( w 3 ) 8 + 32 C 1 ( w ′ 3 ) 2 + 256 C 1 w 3 u 1 + 64 C 1 ( w 3 ) 2 u 2 − 32 C 1 ( w 3 ) 4 + 256( C 1 ) 2 = 1 0 24 C 3 [L − 8 ] . (A1) P ± 4 and V ± in terms of w 3 and u 2 : P ± 4 4 ∼ ∂ 4 ∓ w 3 ∂ 3 +  u 2 − 3 2 ( w 3 ) 2 ∓ 3 2 w ′ 3  ∂ 2 ∓  3 4 w ′′ 3 − w ′′′′′ 3 4 u ′ 2 − w ′′′ 3 u 2 u ′ 2 − w ′ 3 u ′′ 2 u ′ 2 − w 3 u ′′′ 2 4 u ′ 2 + 5( w 3 ) 2 w ′′′ 3 2 u ′ 2 + 10 w 3 w ′ 3 w ′′ 3 u ′ 2 + 5( w ′ 3 ) 3 2 u ′ 2 − w ′ 3 ( u 2 ) 2 u ′ 2 + 6( w 3 ) 2 w ′ 3 u 2 u ′ 2 − 15( w 3 ) 4 w ′ 3 2 u ′ 2 − 8 C 1 w ′ 3 u ′ 2 ∓ ( u ′ 2 − 3 w 3 w ′ 3 )  ∂ + 1 2 u ′′ 2 − 5 2 w 3 w ′′ 3 − 11 8 ( w ′ 3 ) 2 + 1 4 ( u 2 ) 2 − 3 2 ( w 3 ) 2 u 2 + 15 8 ( w 3 ) 4 − w 3 w ′′′′′ 3 4 u ′ 2 − w 3 w ′′′ 3 u 2 u ′ 2 − w 3 w ′ 3 u ′′ 2 u ′ 2 − ( w 3 ) 2 u ′′′ 2 4 u ′ 2 + 5( w 3 ) 3 w ′′′ 3 2 u ′ 2 + 10( w 3 ) 2 w ′ 3 w ′′ 3 u ′ 2 + 5 w 3 ( w ′ 3 ) 3 2 u ′ 2 − w 3 w ′ 3 ( u 2 ) 2 u ′ 2 + 6( w 3 ) 3 w ′ 3 u 2 u ′ 2 − 15( w 3 ) 5 w ′ 3 2 u ′ 2 − 8 C 1 w 3 w ′ 3 u ′ 2 + 2 C 1 ∓  7 8 w ′′′ 3 + 1 2 w ′ 3 u 2 + 3 8 w 3 u ′ 2 − 9 4 ( w 3 ) 2 w ′ 3  , (A2a) V ± = − 1 4 u 2 + 1 2 ( w 3 ) 2 ± 1 2 w ′ 3 − C 0 . (A2b) The functions f ± i [ w ] in (5.30) and (5.31): f + 4 = − 4 I 2 , f + 3 = 4 I 1 − 4(3 ∂ + w 3 ) I 2 , f + 2 = − 4 I 0 + (8 ∂ + 3 w 3 ) I 1 − 1 2 [28 ∂ 2 + 15 w 3 ∂ − 9 w ′ 3 + 4 w 2 + ( w 3 ) 2 ] I 2 , f + 1 = − 2(2 ∂ + w 3 ) I 0 + 2(3 ∂ 2 + 2 w 3 ∂ − 2 w ′ 3 + w 2 ) I 1 − 2[4 ∂ 3 + 3 w 3 ∂ 2 − 2(2 w ′ 3 − w 2 ) ∂ − 4 w ′′ 3 + w 1 − 2 w 3 w ′ 3 ] I 2 , f + 0 = − 2( ∂ 2 + w 3 ∂ − 2 w ′ 3 + w 2 ) I 0 + [2 ∂ 3 + 2 w 3 ∂ 2 − 2(2 w ′ 3 − w 2 ) ∂ − 4 w ′′ 3 + w 1 − 2 w 3 w ′ 3 ] I 1 − 1 16 [32 ∂ 4 + 32 w 3 ∂ 3 − 32(2 w ′ 3 − w 2 ) ∂ 2 − 4(31 w ′′ 3 − 2 w ′ 2 − 6 w 1 + 14 w 3 w ′ 3 ) ∂ − 60 w ′′′ 3 + 8 w ′′ 2 + 16 w 0 − 68 w 3 w ′′ 3 + 7( w ′ 3 ) 2 − 4 w ′ 3 w 2 + 8 w 3 w ′ 2 − 4( w 2 ) 2 − 22( w 3 ) 2 w ′ 3 + 4( w 3 ) 2 w 2 − ( w 3 ) 4 ] I 2 , (A3) 18 and f − 4 = 4 I 2 , f − 3 = 4 I 1 + 4( ∂ − w 3 ) I 2 , f − 2 = 4 I 0 + (4 ∂ − 3 w 3 ) I 1 + 1 2 [4 ∂ 2 − 3 w 3 ∂ − 3 w ′ 3 + 4 w 2 − ( w 3 ) 2 ] I 2 , f − 1 = 2(2 ∂ − w 3 ) I 0 + 2( ∂ 2 − w 3 ∂ − w ′ 3 + w 2 ) I 1 + 2( w ′′ 3 + 2 w ′ 2 − w 1 − 2 w 3 w ′ 3 ) I 2 , f − 0 = 2( ∂ 2 − w 3 ∂ − w ′ 3 + w 2 ) I 0 + ( w ′′ 3 + 2 w ′ 2 − w 1 − 2 w 3 w ′ 3 ) I 1 − 1 16 [4( w ′′ 3 + 2 w ′ 2 − 2 w 1 − 2 w 3 w ′ 3 ) ∂ − 12 w ′′′ 3 − 24 w ′′ 2 + 16 w 0 + 28 w 3 w ′′ 3 + 23( w ′ 3 ) 2 − 4 w ′ 3 w 2 + 8 w 3 w ′ 2 − 4( w 2 ) 2 − 6( w 3 ) 2 w ′ 3 + 4( w 3 ) 2 w 2 − ( w 3 ) 4 ] I 2 , (A4) where I 2 , I 1 , and I 0 are given by (5.9), (5.10), and (5 .11), resp ectiv ely . 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