Supersymmetrical Separation of Variables in Two-Dimensional Quantum Mechanics

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 6 (2010), 075, 10 pages Sup ersymmetrical Separation of V ariables in Tw o -Dimensio nal Quan tum Me c hanics ⋆ Mikhail V. IOFFE Saint-Petersbur g State Unive rsity, St.-Petersbur g , 198504 Russia E-mail: m.ioffe@p ob ox .spbu.ru URL: http:// hep.niif .spbu.ru/staff/ioffe_e.htm Received August 24, 2010, in f ina l form Septem be r 19, 20 10; Publishe d o nline September 2 4, 201 0 doi:10.38 42/SIGMA.20 10.075 Abstract. Two dif ferent approa ches ar e formulated to analyze tw o-dimensional qua ntum mo dels which are not a menable to standard separation o f v a riables. Bo th metho ds ar e essen- tially based o n supersymmetr ic a l s econd order intert wining relations and shap e inv arianc e – t wo main ingr edients of the sup ersymmetrica l quantum mechanics. The f irst metho d ex - plores the o ppo rtunity to separate v ariables in the sup ercharge, and it allows to f ind a pa rt of sp ectr um of the Schr¨ odinger Hamiltonian. The seco nd metho d works when the standard separatio n of v ariables pr o cedure can b e applied for one of the partner Hamiltonians. Then the sp ectrum a nd wa ve functions of the second par tner ca n b e found. Both metho ds are illustrated by the example of tw o-dimensiona l genera liz ation of Mo rse p otential for dif fer ent v alues of parameter s. Key wor ds: s upe r symmetry; separa tio n of v ariables; integrabilit y; solv abilit y 2010 Mathematics Subje ct Classific atio n: 8 1Q60 1 In tro duction The exactly solv ab le m o dels in quantum mec hanics are of sp ecial interest during man y y ears b oth by metho dological and practical reasons. By no w, the m ain ac hiev emen ts we re r elated to one-dimensional Shr ¨ odinger equation. Indeed, a list of exactly solv able one-dimensional prob- lems (Harmonic oscillat or, Coulom b, Morse, P¨ osc hl–T elle r p otent ials e tc.) w as o btained b y an algebraic p ro cedure in the fr amework of factorization metho d [1] in the middle of last cen- tury . This metho d wa s repro du ced r ather recen tly in sup ersymmetrical quan tum mec h anics approac h [2] initiated by the semin al pap ers of E. Witten [3]. More of th at, this appr oac h gav e man y new exactly solv able p oten tials whic h were obtained as sup erpartners of “old” exac tly solv able mo d els. It is necessa ry to men tion also th e imp ortant p ap er of L. Gendens tein [4], where the new fr uitful notion of shap e inv ariance w as introduced. F or the s ake of tru th, more than a century ago the so called Darb oux transf ormation [5] for Sturm –Liouville equation wa s w ell kn o wn among mathematicians. Its app lication to a sp ecif ic S c hr¨ odinger-lik e equation is actually equiv alent [6, 7] to the factorization metho d . The situation is muc h w orse for t w o-dimensional quant um mec hanics. The only r egular metho d to solv e analytical ly the Schr¨ odinger equation is we ll kno wn metho d of sep aration of v ariables [8]. This metho d replaces the tw o-dimen s ional problem by a pair of one-dimensional problems. It can b e u sed for v ery restrictive class of mo d els. F ull c lassif ication o f mod els whic h allo w ed separation of v ariables was giv en by L.P . Eisenhart [9]: four p ossibilities exist – Cartesian, p olar, elliptic and parab olic co ordinates. The general form of p oten tials amenab le to ⋆ This paper is a contribution t o the Proceedin gs of th e W orkshop “Sup ersymmetric Quantum Me- chanics and Sp ectral Design” (July 18–30, 2010, Benasqu e, Sp ain). The full collection is av ailable at http://w ww.emis.de/j ournals/SIGMA/SUSYQM2010.html 2 M.V. Iof fe separation of v ariables is known explicitly up to arbitrary functions of one v ariable. And analyti- cal solution is p ossible only if these fu nctions b elong to the list of exactly solv able p oten tials. All these Hamiltonians H are in tegrable: the symmetry op erator R of second ord er in deriv ative s (in momen ta) exists: [ H , R ] = 0. Besides mo d els amenable to separation of v ariables, th e class of s o called Calogero-lik e m o dels [10] is known as well. Th ey describ e the sp ecif ic forms of pairw ise in teraction of N particles on a lin e, and they are solv able b y means of sp ecial transformation of v ariables wh ich leads to a separation of v ariables. The p roblem is in analogous state for higher dimensions of space. In termediate class of m o dels – quasi-exactly-solv able (QES) p oten tia ls (or, partially sol- v able) – b ecame interesting durin g last years. This notion concerns m o dels for whic h only a part of sp ectrum and corresp ondin g wa ve f unctions can b e f ound analyticall y . In one-dimensional quan tum mec hanics a lot of su c h mo d els w ere built with some hidd en algebraic stru cture [11]. The sup er s ymmetrical appr oac h also ga v e some new QES p oten tials [12]. Th us, the searc h of new appr oac hes to solution of non trivial tw o- dimensional quant um me- c hanical m o dels seems to b e of curr en t imp ortance. It wa s already mentio ned ab ov e that su p er- symmetrical quantum mec hanics provides b oth new w a ys to deriv e s ome old r esults and in ter- esting metho d to obtain new ones. In this p ap er w e sh all fo cus on the case of t w o-dimensional Sc hr¨ odinger equation. Namely , we sh all pr esen t t w o pr o cedures of usin g of the sup ersymmet- rical in tert win ing relations with s up ercharges of second order in deriv ativ es as pro cedu res of SUSY-separation of v ariables. In S ection 2 the general f orm of tw o-dimensional sup ersymmetrical quantum mec hanics with second order sup erc harges will b e formulate d. Section 3 p resen ts the f irst pro cedur e of SUSY separation of v ariables wh ere v ariables are separated in th e sup erc harge. It leads to QES mo dels, and the sp ecif ic m o del of t w o-dimensional Morse p oten tial illustrates th is metho d. In Section 4 the second pro cedu re of SUSY separation of v ariables is giv en wh ere v ariables are separated in one of partner Hamiltonians. In the case of the s ame Morse mo del, but with particular v alues of parameter, it allo ws to solv e the mo del completely , i.e. to f ind analytica lly the whole sp ectrum and all w a v e functions. 2 Tw o-dimensional SUSY quan tum mec hanics Direct generalizatio n of one-dimensional Witten’s SUSY quantum mec hanics to the arbitrary dimensionalit y d of space was formulate d in [6, 13]. The Sup erhamiltonian included ( d + 1) matrix comp on ents of d if f er ent matrix d im en sionalit y , and these comp onents are in tert wined by comp onent s of s up ercharge – op erators linear in deriv ati v es. In particular, in the case of d = 2 t w o scalar Hamiltonians and a 2 × 2 matrix Hamiltonian are intert w ined H (0) ⇐ ⇒ H ik ⇐ ⇒ e H (0) q ± i p ∓ i where q ± i = ∓ ∂ i + ( ∂ i W ( ~ x )) , p ± i = ε ik q ∓ k , H (0) q + i = q + k H (1) k i , e H (0) p + i = p + k H (1) k i . Some physical p roblems were consid er ed in this f r amew ork. F or example, the sp ectrum of the P auli op erator describing spin 1 / 2 ferm ion in the external electrostatic and magnetic f ield was in v estiga ted [14, 15, 16]. Ho w ev er, the follo wing natur al qu estion arises: is it p ossible to a v oid matrix Hamiltonians fr om the sc heme? A n y attempt to in tert wine tw o scalar Hamiltonians b y means of f irst order op erators leads to p oten tial s with standard separation of v ariables [17] whic h are not in teresting f or u s here. Sup ersymmetrical Separation of V ariables in Two-Dime nsional Qu an tum Mec hanics 3 The nont rivial w a y to a v oid matrix Hamiltonians lies in the framewo rk of p olynomial SUSY. The latte r wa s used for the f irst time [18, 7 ] in one-dimensional SUSY quantum mec hanics. In tw o-dimensional conte xt it was prop osed in [19], where a pair of scalar tw o-dimensional Hamiltonians H (0) , H (1) w as in tert wined b y second order op erators Q ± H (0) Q + = Q + H (1) , Q − H (0) = H (1) Q − , where the Hamiltonians ha v e the S c hr¨ odinger form H (0 , 1) = − ∂ 2 i + V (0 , 1) ( ~ x ) . As f or th e intert w in ing sup erc harges Q ± , the f irst naiv e idea is to c ho ose redu cible (factorized) sup erc harge Q + = q + i e q − i q ± i p ∓ i H (0) ⇐ ⇒ H ik ⇐ ⇒ e H (0) k H (1) ⇐ ⇒ H ik ⇐ ⇒ e H (1) e q ± i e p ∓ i It is too naiv e [19], since th is construction leads to Hamilto nians amenable to separation of v ariables in p olar co ordinates V ( ~ x ) = a 2 ρ 2 + 1 ρ 2 F ( ϕ ) . The second idea is to c ho ose Q + = q + i U ik e q − k with some u nitary twist b y constant m atrix U ik q ± i p ∓ i H (0) ⇐ ⇒ H ik ⇐ ⇒ e H (0) k H (1) ⇐ ⇒ U il H lm U † mk ⇐ ⇒ e H (1) e q ± i e p ∓ i Some QES mo dels w ere obtained b y this trick [20 ]. The most general form of second ord er sup ercharges Q + = g ik ( ~ x ) ∂ i ∂ k + C i ( ~ x ) ∂ i + B ( ~ x ) , Q − ≡ ( Q + ) † leads to a complicate system of nonlinear second order dif ferenti al equ ations for fun ctions g ik , C i , B , and p oten tials V (0 , 1) ( ~ x ). Its general solution is imp ossible, bu t some particular solutions w ere f ound [19, 21]. The simplest c hoice g ik ( ~ x ) = δ ik giv es the separation of v aria bles in p olar co ordinates. The Loren tz form g ik ( ~ x ) = diag (1 , − 1) does not lead to separation of v ariables, and some particular s olutions w ere found [19]. Here w e fo cus on g ik ( ~ x ) = diag(1 , − 1). In this case, the system is simplif ied essen tially . New v ariables x ± = x 1 ± x 2 are u seful together with x 1 , x 2 . Using the intert win ing relations, one can pro v e that n ew functions C ± dep end on one v ariable only C + ≡ C 1 − C 2 = C + ( x + ) , C − ≡ C 1 + C 2 = C − ( x − ) , x ± = x 1 ± x 2 . The general s olution for Loren tz metric can b e p ro vided b y solving the only equation ∂ − ( C − F ) = − ∂ + ( C + F ) , 4 M.V. Iof fe where new useful function is F = F 1 ( x + + x − ) + F 2 ( x + − x − ). Thus, the equation is the functional dif feren tial equation, and no regular pro cedur e of its solution is kn o wn. The r equired p otenti als V (0 , 1) ( ~ x ) and th e function B ( ~ x ) are expressed in terms of C ± and F 1 , 2 V (0 , 1) = ± 1 2 ( C ′ + + C ′ − ) + 1 8  C 2 + + C 2 −  + 1 4 ( F 2 ( x + − x − ) − F 1 ( x + + x − )) , B = 1 4 ( C + C − + F 1 ( x + + x − ) + F 2 ( x + − x − )) . A v ariet y of such pairs of p oten tials w as found in [19]. 3 SUSY-separation of v ariables I: QES mo dels The f irst v ariant of SUS Y-separation of v ariables is realized w hen the Hamiltonian H do es not allo w stand ard sep aration of v ariables, but the s u p ercharge Q + do es allo w [22, 23]. The general sc heme is the f ollo w ing. Let’s supp ose that we kno w zero mo des of Q + Q + Ω n ( ~ x ) = 0 , n = 0 , 1 , . . . , N , Q + ~ Ω( ~ x ) = 0 . The in tert wining relation H (0) Q + = Q + H (1) ob ey the imp ortant prop ert y: the space of zero mo d es is closed und er the action of H (1) : H (1) ~ Ω( ~ x ) = ˆ C ~ Ω( ~ x ) . If the matrix ˆ C is kno wn, and if it can b e diagonalized ˆ B ˆ C = ˆ Λ ˆ B , ˆ Λ = d iag( λ 0 , λ 1 , . . . , λ N ) , the eigen v alues of H (1) can b e f ound algebraica lly H (1) ( ˆ B ~ Ω( ~ x )) = ˆ Λ( ˆ B ~ Ω( ~ x )) . Th us, for realizati on of this s c heme w e n eed – to f in d zero mo des Ω n ( ~ x ); – to f in d constan t matrix B , su c h that ˆ B ˆ C = ˆ Λ ˆ B . As for zero mo des, they can b e ob tained b y using the sp ecial similarity transformation (not unitary!), whic h remo v es the terms linear in d eriv ativ es from Q + q + = e − χ ( ~ x ) Q + e + χ ( ~ x ) = ∂ 2 1 − ∂ 2 2 + 1 4 ( F 1 (2 x 1 ) + F 2 (2 x 2 )) , χ ( ~ x ) = − 1 4  Z C + ( x + ) dx + + Z C − ( x − ) dx −  . No w, q + allo w s separation of v ariables for arbitrary solution of intert wining relations, and we obtain the f irst v a rian t of n ew pro cedur e – SUSY-separation of v ariables. S imilarly to the con v en tio nal separation of v ariables, separation of v ariables in the op erator q + itself do es not guaran tee solv abilit y of the problem. The next task is to solve tw o one-dimen sional problems  − ∂ 2 1 − 1 4 F 1 (2 x 1 )  η n ( x 1 ) = ǫ n η n ( x 1 ) ,  − ∂ 2 2 + 1 4 F 2 (2 x 2 )) ρ n ( x 2  = ǫ n ρ n ( x 2 ) . Three remarks are appropriate no w. Sup ersymmetrical Separation of V ariables in Two-Dime nsional Qu an tum Mec hanics 5 Remark 1. Th e same similarit y transformation of H (1) do es n ot lead to op erator amenable to separation of v ariables. Remark 2. The norm alizabilit y of Ω n has to b e stud ied atten tiv ely d u e to non-unitarity of the similarit y transformation. Remark 3. W e h a v e no r easons to exp ect exact s olv abilit y of the mo del, but quasi-exact- solv abilit y can b e pr edicted. As for the matrix ˆ B , it must b e found by s ome sp ecif ic pro cedure. Suc h p r o cedure was us ed in example whic h w ill b e presente d b elo w. In pr inciple, the f irst sc heme of SUSY-separation of v aria bles can b e used for arb itrary mo dels satisfying inte rt wining relations by sup erc harges with Loren tz metrics. The list of solutions of in tert wining relations is already rather long, and it m ay increase in fu tu re. Th e main obstacle is analytical solv abilit y of one-dimens ional equations, obtained after sep aration of v ariables in the op erator q + . Belo w we describ e brief ly such a mod el whic h can b e considered as the generalized tw o- dimensional Morse p otenti al C + = 4 aα, C − = 4 aα coth αx − 2 , f i ( x i ) ≡ 1 4 F i (2 x i ) = − A  e − 2 αx i − 2 e − αx i  , i = 1 , 2 , V (0) , (1) = α 2 a (2 a ∓ 1) sinh − 2  αx − 2  + 4 a 2 α 2 + A  e − 2 αx 1 − 2 e − αx 1 + e − 2 αx 2 − 2 e − αx 2  , where A > 0, α > 0, a is real. T o explain the name, w e present the p otent ial in the form V ( ~ x ) = V Morse ( x 1 ) + V Morse ( x 2 ) + v ( x 1 , x 2 ) , where f ir st t w o terms are just one-dimens ional Morse p otent ials, and the last term mixes v ari- ables x 1 , x 2 . The solutions of one-dimensional Sc hr¨ odinger equations are w ell known [24], and the zero mo des can b e written [22, 23] as Ω n ( ~ x ) =  α √ A ξ 1 ξ 2 | ξ 2 − ξ 1 |  2 a exp  − ξ 1 + ξ 2 2  ( ξ 1 ξ 2 ) s n F ( − n, 2 s n + 1; ξ 1 ) F ( − n, 2 s n + 1; ξ 2 ) , ξ i ≡ 2 √ A α exp( − αx i ) , s n = √ A α − n − 1 2 > 0 . The conditions of n ormalizabilit y and of absen ce of the “fall to the cen ter” are a ∈  −∞ , − 1 4 − 1 4 √ 2  , s n = √ A α − n − 1 2 > − 2 a > 0 . T o obtain th e m atrix ˆ C explicitly , one m ust act by H (1) on Ω n . The matrix tur ns ou t to b e triangular, and therefore, the energy eigen v alues coincide with its diagonal elemen ts E k = c k k = − 2  2 aα 2 s k − ǫ k  . T o f ind a v ariety of wa ve functions is a more dif f icult task. F or that it is necessary to f in d all elemen ts of ˆ C and all elemen ts of matrix ˆ B . Th e recurrent pro cedur e f or the case of tw o- dimensional Morse p oten tial w as giv en in [22, 23]. This v ariet y can b e enlarged by means of shap e inv ariance pr op ert y [25] of the mo d el H (0) ( ~ x ; a ) = H (1) ( ~ x ; ˜ a ) + R ( a ) , ˜ a = a − 1 / 2 , R ( a ) = α 2 (4 a − 1) . 6 M.V. Iof fe Similarly to one-dimensional sh ap e inv ariance, eac h wa ve f unction constru cted by SUSY-sepa- ration of v ariables leads to a set of additional w a v e f unctions H (0) ( a )  Q − ( a ) Q −  a − 1 2  · · · Q −  a − M − 1 2  Ψ  a − M 2  =  E 0  a − M 2  + R  a − M − 1 2  + · · · + R ( a )  ×  Q − ( a ) Q −  a − 1 2  · · · Q −  a − M − 1 2  Ψ  a − M 2  . Analogous approac h works for the t w o-dimensional generalizatio n of P¨ osc hl–T eller mo d el [20] and for some t w o-dimensional p erio dic p otentia ls [26]. 4 SUSY-separation of v ariables I I: exact solv abilit y Among all known solutions of tw o- dimensional in tert wining relations with second order sup er- c harges a su b class exists [27], wh ere one of int ert wined Hamiltonians is amenable to standard separation of v ariables d u e to sp ecif ic c hoice of p arameters of the mo del. Its sup erpartner still do es not allo w separation of v ariables. The sc heme will b e describ ed b elo w for the same sp ecif ic mo d el whic h is t w o-dimensional generalizat ion of Morse p otentia l V (0) , (1) = α 2 a (2 a ∓ 1) sinh − 2  αx − 2  + 4 a 2 α 2 + A  e − 2 αx 1 − 2 e − αx 1 + e − 2 αx 2 − 2 e − αx 2  . Let’s c ho ose a 0 = − 1 / 2 in order to v a nish the mixed term in V (1) . Then H (1) allo w s the con v en tio nal separatio n of v ariables. Moreo ver, after sep aration of v ariables eac h of obtained one-dimensional problems is exactly solv able. W e met just this one-dimensional problem ab o v e in a dif feren t conte xt. The discrete sp ectrum of this one-dimensional mo del is ǫ n = − α 2 s 2 n , s n ≡ √ A α − n − 1 2 > 0 , n = 0 , 1 , 2 , . . . . W a v e functions are expressed in terms of degenerate h yp ergeometric fun ctions η n ( x i ) = exp  − ξ i 2  ( ξ i ) s n F ( − n, 2 s n + 1; ξ i ) , ξ i ≡ 2 √ A α exp( − αx i ) . Due to separation of v ariables, the t w o-dimensional problem with H (1) ( ~ x ) is exactly solv able. Its en ergy eigen v alues are E n,m = E m,n = ǫ n + ǫ m , b eing t w o-fold degenerate f or n 6 = m . The corresp ond ing eigenfunctions can b e c hosen as symmetric or (for n 6 = m ) ant isymmetric combinatio ns Ψ (1) S,A E n,m ( ~ x ) = η n ( x 1 ) η m ( x 2 ) ± η m ( x 1 ) η n ( x 2 ) . Our aim here is to solv e completely the p roblem for H (0) ( ~ x ) w ith a 0 = − 1 / 2. The m ain to ol is again the SUSY int ert wining relations, i.e. isosp ectralit y of H (0) and H (1) but up to zero mo des and singular prop erties of Q ± . In general, w e ma y exp ect three kind s of levels of H (0) ( ~ x ): Sup ersymmetrical Separation of V ariables in Two-Dime nsional Qu an tum Mec hanics 7 (i) T he lev els, whic h coincide w ith E nm . Their wa ve functions can b e obtained f rom Ψ (1) b y means of Q + . (ii) T h e leve ls, whic h w ere absen t in the sp ectrum of H (1) ( ~ x ), if some w a v e fu nctions of H (0) ( ~ x ) are simultaneously th e zero mo des of Q − . Then the second in tert wining relation w ould not giv e an y partner s tate among b ound states of H (1) ( ~ x ). (iii) The lev els, whic h w ere al so absen t in the s p ectrum of H (1) ( ~ x ), if some w a v e functions of H (0) ( ~ x ) b ecome nonnormalizable after action of op erator Q − . W e hav e to analyze these thr ee classes of p ossible b ound states of H (0) one after another. (i) Th e f irst SUSY in tert wining r elation giv es the tw o- fold degenerate wa ve fu nctions of H (0) with energies E nm : Ψ (0) E nm = Q + Ψ (1) E nm . But Q + includes singularity on the line x 1 = x 2 , therefore the normalizabilit y of Ψ (0) E n,m dep end s cru cially on the b eha vior of Ψ (1) E n,m on the line ξ 1 = ξ 2 . On e can c hec k th at only antisymmetric functions Ψ (1) survive , i.e. only symmetric Ψ (0) survive . Th is fact can b e demonstrated [27] b oth by direct calculation and by indir ect metho d - by means of symmetry op erator R (0) . The indir ect metho d explores that th e symmetry op erator R (0) = Q − Q + for a 0 = − 1 / 2 can b e written in terms of one-dimensional Morse Hamiltonians h 1 ( x 1 ), h 2 ( x 2 ) R (0) = ( h 1 ( x 1 ) − h 2 ( x 2 )) 2 + 2 α 2 ( h 1 ( x 1 ) + h 2 ( x 2 )) + α 4 . Therefore, R (0) Ψ (0) A E n,m ( ~ x ) = r n,m Ψ A E n,m ( ~ x ) , r n,m = α 4  ( n − m ) 2 − 1][( s n + s m ) 2 − 1  , and k Ψ (1) S E n,m k 2 = h Ψ (0) A E n,m | Q − Q + | Ψ (0) A E n,m i = r n,m k Ψ (0) A E n,m k 2 . F or n = m , w a v e fu nctions Ψ (0) S E n,n v anish identic ally by trivial r easons. It is clear n ow that w a v e fu nctions Ψ (0) S E n,n ± 1 also v anish. F or all other n , m , fun ctions Ψ (0) S E n,m ha v e p ositiv e and f inite norm, and there is no degeneracy of th ese lev els. (ii) These p ossible b ound states of H (0) are the normalizable zero mo des of Q − . The v ariet y of suc h zero mo des is kn o wn from [22]: they exist only for p ositive v alues of a a ∈  1 4 + 1 4 √ 2 , + ∞  , whic h d o es n ot conta in the v al ue a 0 = − 1 / 2. Thus, no normalizable b ound states of this class exist for H (0) . (iii) W e ha v e to s tu dy an op p ortunity that Q − destro ys n ormalizabilit y of some eigenfunc- tions of H (0) . It could o ccur due to s in gular c haracter of Q − at x 1 = x 2 . Th e analysis w as p erformed [27] in suitable co ordin ates. It sho ws that Q − is not able to tr ansform normalizable w a v e f unction to nonnorm alizable. Therefore, the third class of p ossible wa ve functions H (0) do es not exist to o. Summing u p, the sp ectrum of H (0) with a 0 = − 1 / 2 consists only of the b oun d s tates w ith energies E nm for | n − m | > 1. Th is sp ectrum is b ounded from ab o v e by the condition of p ositivit y of s n , s m : n, m < √ A/α − 1 / 2 . The corresp onding w a v e fu nctions are obtained analytically [27]. The results abov e can b e expanded to the whole hierarch y of Morse p oten tial s w ith a k = − ( k + 1) / 2 with k = 0 , 1 , . . . by means of s h ap e inv ariance prop ert y . Let’s denote element s of 8 M.V. Iof fe the hierarc h y as H (0) ( ~ x ; a k ), H (1) ( ~ x ; a k ). All these Hamiltonians are also exactly solv able du e to shap e in v ariance of the mo del H (0) ( ~ x ; a k − 1 ) = H (1) ( ~ x ; a k ) , k = 1 , 2 , . . . . This means that the follo wing chain (hierarch y) of Hamiltonians can b e bu ilt H (1) ( ~ x ; a 0 ) ÷ H (0) ( ~ x ; a 0 ) = H (1) ( ~ x ; a 1 ) ÷ H (0) ( ~ x ; a 1 ) = · · · ÷ H (1) ( ~ x ; a k − 1 ) = H (0) ( ~ x ; a k ) ÷ H (0) ( ~ x ; a k ) , where the sign ÷ d enotes in tert wining b y Q ± ( a i ). In the general case, the functions Ψ (0) E n,m ( ~ x ; a k ) = Q + ( a k )Ψ (1) E n,m ( ~ x ; a k ) = Q + ( a k ) Q + ( a k − 1 ) · · · Q + ( a 0 )Ψ (1) A E n,m ( ~ x ; a 0 ) (if normalizable) are the w a v e fu nctions of H (0) ( ~ x ; a k ) w ith energies E n,m = − α 2 ( s 2 n + s 2 m ). T he symmetries of wa ve functions alternate and dep end on th e length of c hain. Th is is true but up to zero m o des of op erators Q + . It is necessary to k eep u nder the control normalizabilit y of Ψ and zero m o des of Q + . This con trol is p er f ormed algebraically b y means of identit y , w h ic h m ust b e fulf illed up to a f u nction of H R (1) ( a k ) = R (0) ( a k − 1 ) . Actually , the follo wing equation holds: Q − ( a k ) Q + ( a k ) = Q + ( a k − 1 ) Q − ( a k − 1 ) + α 2 (2 k + 1)  2 H (0) ( ~ x ; a k − 1 ) + α 2 (2 k 2 + 2 k + 1)  . These relations allo wed to ev aluate the norm s of w a v e functions. Th e r esult is the follo wing. 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