N-fold Supersymmetry in Quantum Mechanical Matrix Models

TH-1479 N -fold Sup ersymmetry in Quan tum Mec hanical Matrix Mo dels T oshiaki T anak a ∗ Institute of Particle and Nu cl e ar Studies, High Ener gy A c c eler ator R ese ar ch Or gani zation (KEK), 1-1 Oho, Tsukub a, Ib ar ak i 305-08 01, Jap an Abstract W e form ulate N -fold sup ersymm etry in quantum mechanica l mat r ix mo d els. As an example, w e constru ct general t wo -by-t wo Hermitian matrix 2-fold sup er s ymmetric quan tum mec hanical systems. W e find that th er e are tw o inequ iv alen t such systems, b oth of wh ic h are c haracterized b y tw o arb itrary scalar fun ctions, and o n e of whic h does not reduce to the scalar s y s tem. The obtained systems are all w eakly quasi- solv able. P ACS nu mber s: 02.30.Hq; 03.6 5.Ca; 03.65.Fd; 11.3 0.Pb Keywords: N -fold sup ersymmetry ; Quasi-solv a bilit y; In tertwining r elations; Matrix models ; Matrix linear differential opera tors ∗ Electronic a ddress: to shiak i@pos t.kek.jp 1 I. INTR ODUCT ION Recen tly , sup ersymmetry (SUSY) and shape in v a riance in quan tum mec hanical matrix mo d- els ha ve attra cted m uch atten tion in t he lit era t ur e, e.g., Refs. [1– 14] in v arious phy sical con- texts. T o av oid confusion, w e no t e that w e here mean SUSY b etw een t w o matrix Sc hr¨ odinger op erators in tertw ined by a matrix linear differen tial op erator but n ot SU SY b et w een t w o scalar Sc hr¨ odinger op erato rs in a matrix sup erHamiltonian intert wined by a scalar linear differen tial op erator. On the other hand, the framew ork of N -fold SUSY [15–17] has been shown to b e quite fruitful among sev eral generalizations of o r dinary SUSY especially since the establishmen t of its equiv alence with w eak quasi-solv abilit y in Ref. [16], f o r a review see R ef. [1 8]. Due to the facts that the N = 1 case corresp onds t o ordinary SUSY and that shap e in v a r iance automatically implies weak quasi-solv abilit y , N -fold SUSY con tains b oth ordinary SUSY and s hap e inv aria nce as its particular cases. Hence, it is quite natura l to ask whether a for mulation of N -fo ld SUSY is p o ssible for matrix systems. T o the b est of our kno wledge, there was so f ar only one suc h an attempt whic h corresp onds to the N = 2 case fo r 2 × 2 matrix mo dels [4]. Ho wev er, the analysis there w as quite restrictiv e, w as dev oted mostly to the particular cases where the systems were a v a ilable by tw o success ive SUSY transformations, and r esorted to the inv olve d assumptions and ansatz. As a consequence, its formu la t io n do es not hav e a form which is suitable for discussing g eneral asp ects, esp ecially those whic h w ere established and a ppreciated la ter after that w ork. In t his art icle, we for m ulate for the first time N -fold SUSY for a system comp o sed of matrix Sc hr¨ odinger op erators for all p ositiv e in tegral N in suc h a general fashion that t he recen t crucial deve lo pments in the field are fully incorp ora t ed in the fo r ma lism. T o see such a system actually exists, w e construct a s an illustratio n general 2 × 2 Hermitian matrix 2- f old SUSY system s without an y assumption or ansatz. W e fin d that there are t wo ine quiv alen t systems , b oth of whic h are c hara cterized by t w o arbitrary real scalar functions. In trig uingly , one of the systems do es not admit reduction to the most general 2-fold SUSY scalar system. W e o r g anize this article a s follow s. In the next section, we generically define N -f o ld SUSY in quan tum mec hanical matrix mo dels. Then, w e in v estigate in detail 2 × 2 Hermitian matrix systems for N = 2 in Section I I I. W e explicitly solve a ll the conditions for 2 - fold SUSY to obtain general form of the latter systems. In the last section, w e refer to sev eral future issues to b e follow ed after t his w ork. I I. GENERAL SETTIN G A quan tum mec hanical system w e shall consider here is a pair of n × n matr ix Sc hr¨ odinger op erators H ± = − 1 2 I n d 2 d q 2 + V ± ( q ) , (1) 2 where I n is an n × n unit matrix a nd eac h V ± ( q ) is an n × n matrix-v alued complex function. Let us in tro duce a pair of n × n m a t r ix linear differen tial op erators of order N P − N = I n d N d q N + N − 1 X k =0 w k ( q ) d k d q k , (2a) P + N = ( − 1) N I n d N d q N + N − 1 X k =0 ( − 1) k d k d q k w k ( q ) , (2b) where w k ( q ) ( k = 0 , . . . , N − 1 ) are n × n matrix-v a lued complex functions. Then, t he system (1) is said to b e N -fold su p ersymmetric with resp ect to (2) if t he fo llowing relations are all satisfied: P ∓ N H ∓ − H ± P ∓ N = 0 , (3a) P ∓ N P ± N = 2 N " ( H ± + C 0 ) N + N − 1 X k =1 C k ( H ± + C 0 ) N − k − 1 # , (3b) where C k ( k = 0 , . . . , N − 1 ) a re n × n constan t matrices. It is ev ident that in the case of n = 1 the abov e definition of N -fold SUS Y reduces to the ordinary one for a pa ir of scalar Sc hr¨ odinger op erators. As in the scalar case, t he first relation (3a ) immediately implies almost isosp ectrality of H ± and we ak quasi-solvability H ± k er P ± N ⊂ k er P ± N . W e note that in contrast to t he formulation in R ef. [4] where its fo cus w as on hidden symmetry o p erators c haracterized as the deviation from the se cond algebraic relatio n (3b) w e ha v e treated it as a part of the definition of N -fold SUSY. See also the discussion in Section IV, item 1 for its relev ance. I I I. TW O-BY-TW O HERMITI AN 2 -F OLD SUPERSYMMETR Y As an example, let us construct the most general 2 × 2 Hermitian matrix 2- f old SUSY systems whic h a re defined in t he Hilb ert space L 2 of tw o-comp onen t functions equipp ed with the inner pro duct de fined b y ( φ, ψ ) = Z S d q φ † ( q ) ψ ( q ) , φ, ψ ∈ L 2 ( S ) , S ⊂ R , (4) where the Hermitian conjuga t e † is as usual the com bination o f complex conjugate and transp osition. The most general form of a pair of 2 × 2 Hermitian matrix Sc hr¨ odinger op erators is g iv en b y H ± = − 1 2 I 2 d 2 d q 2 + 3 X µ =0 V ± µ ( q ) σ µ , (5) where σ 0 = I 2 is a 2 × 2 unit matrix, σ i ( i = 1 , 2 , 3) are the P auli ma t r ices, and V µ ( q ) ( µ = 0 , . . . , 3) are all real scalar functions. Comp onen ts of 2 × 2 matrix 2-fold sup erc har g es 3 ha v e the follo wing for m P − 2 = I 2 d 2 d q 2 + 3 X µ =0 w 1 µ ( q ) σ µ ! d d q + 3 X µ =0 w 0 µ ( q ) σ µ , (6a) P + 2 = I 2 d 2 d q 2 − d d q 3 X µ =0 w 1 µ ( q ) σ µ ! + 3 X µ =0 w 0 µ ( q ) σ µ , (6b) where w 1 µ ( q ) and w 0 µ ( q ) ( µ = 0 , . . . , 3) are all real scalar functions. T o inv estigate the N -fold SUSY condition (3) fo r the N = 2 case under consideration, w e first note that P − 2 H − − H + P − 2 = 0 implies P + 2 H + − H − P + 2 = 0 and vice ve rsa since they are Hermitian conjugate with each other with resp ect to the inner pr o duct (4), thanks to the choices (5) and (6). Hence, it is sufficien t to study only the former. A direct calculation show s that the in tertw ining relation P − 2 H − − H + P − 2 = 0 holds if a nd only if the following set of conditions is satisfied: V + µ − V − µ = w ′ 1 µ , (7) w ′′ 10 + 2 w ′ 00 + 4 V −′ 0 − 2 3 X µ =0 w 1 µ ( V + µ − V − µ ) = 0 , (8) w ′′ 1 i + 2 w ′ 0 i + 4 V −′ i − 2 w 1 i ( V + 0 − V − 0 ) − 2 w 10 ( V + i − V − i ) = 0 , (9) 3 X j,k =1 ǫ ij k w 1 j ( V + k + V − k ) = 0 , (10) w ′′ 00 + 2 V −′′ 0 + 2 3 X µ =0  w 1 µ V −′ µ − w 0 µ ( V + µ − V − µ )  = 0 , (11) w ′′ 0 i + 2 V −′′ i + 2 w 1 i V −′ 0 + 2 w 10 V −′ i − 2 w 0 i ( V + 0 − V − 0 ) − 2 w 00 ( V + i − V − i ) = 0 , (12) 3 X j,k =1 ǫ ij k  w 1 j V −′ k + w 0 j ( V + k + V − k )  = 0 . (13) On the o t her hand, w e find tha t the 2-fo ld sup eralgebra P ∓ 2 P ± 2 = 4 [( H ± + C 0 ) 2 + C 1 ] with Hermitian constan t matr ices C k = P 3 µ =0 C k µ σ µ ( C k µ ∈ R , k = 0 , 1) holds for the upp er sign 4 if and only if 4 V + 0 = 3 w ′ 10 − 2 w 00 + 3 X µ =0 ( w 1 µ ) 2 − 4 C 00 , (14) 4 V + i = 3 w ′ 1 i − 2 w 0 i + 2 w 10 w 1 i − 4 C 0 i , (15) 3 X j,k =1 ǫ ij k w 1 j ( w ′ 1 k − w 0 k ) = 0 , (16) 2 V + ′′ 0 − 4 3 X µ =0 ( V + µ + C 0 µ ) 2 − 4 C 10 = w ′′′ 10 − w ′′ 00 + 3 X µ =0  w 1 µ w ′′ 1 µ + w ′ 1 µ w 0 µ − w 1 µ w ′ 0 µ − ( w 0 µ ) 2  , (17) 2 V + ′′ i − 8( V + 0 + C 00 )( V + i + C 0 i ) − 4 C 1 i = w ′′′ 1 i − w ′′ 0 i + w ′′ 10 w 1 i + w 10 w ′′ 1 i − w ′ 00 w 1 i + w 00 w ′ 1 i + w ′ 10 w 0 i − w 10 w ′ 0 i − 2 w 00 w 0 i , (18) 3 X j,k =1 ǫ ij k ( w ′′ 1 j w 1 k + w ′ 1 j w 0 k + w 1 j w ′ 0 k ) = 0 , (19) and for the low er sign if and only if 4 V − 0 = − w ′ 10 − 2 w 00 + 3 X µ =0 ( w 1 µ ) 2 − 4 C 00 , (20) 4 V − i = − w ′ 1 i − 2 w 0 i + 2 w 10 w 1 i − 4 C 0 i , (21) 3 X j,k =1 ǫ ij k w 1 j w 0 k = 0 , (22) 2 V −′′ 0 − 4 3 X µ =0 ( V − µ + C 0 µ ) 2 − 4 C 10 = − w ′′ 00 + 3 X µ =0  w ′ 1 µ w 0 µ + w 1 µ w ′ 0 µ − ( w 0 µ ) 2  , (23) 2 V −′′ i − 8( V − 0 + C 00 )( V − i + C 0 i ) − 4 C 1 i = − w ′′ 0 i + w ′ 00 w 1 i + w 00 w ′ 1 i + w ′ 10 w 0 i + w 10 w ′ 0 i − 2 w 00 w 0 i , (24) 3 X j,k =1 ǫ ij k ( w ′ 1 j w 0 k + w 1 j w ′ 0 k ) = 0 . (25) The form ulas (14), (15), (20), and (21) determine the form of the p oten tials V ± µ and are compatible with the conditions (7), (8), and (9). Then, t he conditions (10), (16), and (22) are iden tical with 3 X j,k =1 ǫ ij k w 1 j w 0 k = 3 X j,k =1 ǫ ij k w 1 j w ′ 1 k = 3 X j,k =1 ǫ ij k w 1 j C 0 k = 0 . (26) The most g eneral solutions to the latter set of conditions are given by w 1 i = C 0 i v 1 , w 0 i = C 0 i v 0 , (27) 5 where v 1 and v 0 are at presen t arbitra ry scalar functions. Substituting (27) in to (14), (15), (20), a nd ( 2 1), we obtain 4 V + 0 = 3 w ′ 10 − 2 w 00 + ( w 10 ) 2 + C 2 ( v 1 ) 2 − 4 C 00 , (28a) 4 V + i = C 0 i (3 v ′ 1 − 2 v 0 + 2 w 10 v 1 − 4) , (28b) 4 V − 0 = − w ′ 10 − 2 w 00 + ( w 10 ) 2 + C 2 ( v 1 ) 2 − 4 C 00 , (28c) 4 V − i = C 0 i ( − v ′ 1 − 2 v 0 + 2 w 10 v 1 − 4) , (28d) where C 2 = P 3 i =1 ( C 0 i ) 2 . The solutions (27) automatically satisfy (13), (19), and (25 ) . With the substitution of (27) and ( 2 8) into t he remaining conditions, (11) and (12) ar e resp ectiv ely iden tical with w ′′′ 10 − w 10 w ′′ 10 − 2( w ′ 10 ) 2 + 4 w ′ 10 w 00 + 2 w 10 w ′ 00 − 2( w 10 ) 2 w ′ 10 − C 2  v 1 v ′′ 1 + 2( v ′ 1 ) 2 − 4 v ′ 1 v 0 − 2 v 1 v ′ 0 + 2 w ′ 10 ( v 1 ) 2 + 4 w 10 v 1 v ′ 1  = 0 , (29) v ′′′ 1 − w ′′ 10 v 1 − 4 w ′ 10 v ′ 1 − w 10 v ′′ 1 + 4 w ′ 10 v 0 + 2 w 10 v ′ 0 + 2 w ′ 00 v 1 + 4 w 00 v ′ 1 − 4 w 10 w ′ 10 v 1 − 2( w 10 ) 2 v ′ 1 − 2 C 2 ( v 1 ) 2 v ′ 1 = 0 , (30) (17) a nd ( 1 8) ar e resp ectiv ely with 2 w ′′′ 10 − 5( w ′ 10 ) 2 + 8 w ′ 10 w 00 + 4 w 10 w ′ 00 − 6( w 10 ) 2 w ′ 10 + 4( w 10 ) 2 w 00 − ( w 10 ) 4 − C 2  5( v ′ 1 ) 2 − 8 v ′ 1 v 0 − 4 v 1 v ′ 0 + 6 w ′ 10 ( v 1 ) 2 + 12 w 10 v 1 v ′ 1 − 8 w 10 v 1 v 0 − 4 w 00 ( v 1 ) 2 + 6( w 10 ) 2 ( v 1 ) 2  − C 4 ( v 1 ) 4 − 16 C 10 = 0 , (31) C 0 i  v ′′′ 1 − 5 w ′ 10 v ′ 1 + 4 w ′ 10 v 0 + 2 w 10 v ′ 0 + 2 w ′ 00 v 1 + 4 w 00 v ′ 1 − 6 w 10 w ′ 10 v 1 − 3( w 10 ) 2 v ′ 1 + 2( w 10 ) 2 v 0 + 4 w 10 w 00 v 1 − 2( w 10 ) 3 v 1 − C 2 ( v 1 ) 2 (3 v ′ 1 − 2 v 0 + 2 w 10 v 1 )  − 8 C 1 i = 0 , (32) and (23) a nd (24) are resp ectiv ely with 2 w ′′′ 10 − 4 w 10 w ′′ 10 − 3( w ′ 10 ) 2 + 8 w ′ 10 w 00 + 4 w 10 w ′ 00 − 2( w 10 ) 2 w ′ 10 − 4( w 10 ) 2 w 00 + ( w 10 ) 4 − C 2  4 v 1 v ′′ 1 + 3( v ′ 1 ) 2 − 8 v ′ 1 v 0 − 4 v 1 v ′ 0 + 2 w ′ 10 ( v 1 ) 2 + 4 w 10 v 1 v ′ 1 + 8 w 10 v 1 v 0 + 4 w 00 ( v 1 ) 2 − 6( w 10 ) 2 ( v 1 ) 2  + C 4 ( v 1 ) 4 + 16 C 10 = 0 , (33) C 0 i  v ′′′ 1 − 2 w ′′ 10 v 1 − 3 w ′ 10 v ′ 1 − 2 w 10 v ′′ 1 + 4 w ′ 10 v 0 + 2 w 10 v ′ 0 + 2 w ′ 00 v 1 + 4 w 00 v ′ 1 − 2 w 10 w ′ 10 v 1 − ( w 10 ) 2 v ′ 1 − 2( w 10 ) 2 v 0 − 4 w 10 w 00 v 1 + 2( w 10 ) 3 v 1 + C 2 ( v 1 ) 2 ( − v ′ 1 − 2 v 0 + 2 w 10 v 1 )  + 8 C 1 i = 0 . (34) It is eviden t that the equations (3 2 ) and (34) hav e the trivial solutions C 1 i = C 0 i = 0. On the other hand, for non- trivial solutions the com bination C 1 i /C 0 i := ˜ C ( i = 1 , 2 , 3) should not dep end on the index i . Let us first consider t he set of conditions (29) and (3 0 ). W e find that the tw o com binatio ns w 10 × (29)+ C 2 v 1 × (30) and v 1 × (29)+ w 10 × (30) are total differen tials and th us a re inte gr ated resp ectiv ely as 2 w 10 w ′′ 10 − ( w ′ 10 ) 2 − 2( w 10 ) 2 w ′ 10 + 4( w 10 ) 2 w 00 − ( w 10 ) 4 + C 2  2 v 1 v ′′ 1 − ( v ′ 1 ) 2 − 2 w ′ 10 ( v 1 ) 2 − 4 w 10 v 1 v ′ 1 + 8 w 10 v 1 v 0 + 4 w 00 ( v 1 ) 2 − 6( w 10 ) 2 ( v 1 ) 2  − C 4 ( v 1 ) 4 = D 1 , (35) 6 and w ′′ 10 v 1 − w ′ 10 v ′ 1 + w 10 v ′′ 1 − 2 w 10 w ′ 10 v 1 − ( w 10 ) 2 v ′ 1 + 2( w 10 ) 2 v 0 + 4 w 10 w 00 v 1 − 2( w 10 ) 3 v 1 − C 2 ( v 1 ) 2 ( v ′ 1 − 2 v 0 + 2 w 10 v 1 ) = D 2 , (36) where D 1 and D 2 are inte gr al constants . It is easily c hec ked tha t (35) and (36) a re compatible with all t he remaining conditions (31)–(34) if and only if D 1 = 16 C 10 , D 2 = 8 C 1 i /C 0 i = 8 ˜ C . (37) Hence, t he only remaining problem is to analyze (35) and (36). They can b e regarded as sim ultaneous linear equations for w 00 and v 0 . F or the non-degenerate case v 1 6 = C − 1 w 10 , they are uniquely solved as 4  ( w 10 ) 2 − C 2 ( v 1 ) 2  2 w 00 = ( w 10 ) 2  − 2 w 10 w ′′ 10 + ( w ′ 10 ) 2 + 2( w 10 ) 2 w ′ 10 + ( w 10 ) 4 + 16 C 10  + C 2  2 w 10 w ′′ 10 ( v 1 ) 2 + ( w ′ 10 ) 2 ( v 1 ) 2 − 4 w 10 w ′ 10 v 1 v ′ 1 + 2( w 10 ) 2 v 1 v ′′ 1 + ( w 10 ) 2 ( v ′ 1 ) 2 − 4( w 10 ) 2 w ′ 10 ( v 1 ) 2 − ( w 10 ) 4 ( v 1 ) 2 − 32 ˜ C w 10 v 1 + 16 C 10 ( v 1 ) 2  − C 4 ( v 1 ) 2  2 v 1 v ′′ 1 − ( v ′ 1 ) 2 − 2 w ′ 10 ( v 1 ) 2 + ( w 10 ) 2 ( v 1 ) 2  + C 6 ( v 1 ) 6 , (38) and 2  ( w 10 ) 2 − C 2 ( v 1 ) 2  2 v 0 = w 10  w 10 w ′′ 10 v 1 − ( w ′ 10 ) 2 v 1 + w 10 w ′ 10 v ′ 1 − ( w 10 ) 2 v ′′ 1 + ( w 10 ) 3 v ′ 1 + ( w 10 ) 4 v 1 + 8 ˜ C w 10 − 16 C 10 v 1  + C 2 v 1  − w ′′ 10 ( v 1 ) 2 + w ′ 10 v 1 v ′ 1 + 2 w 10 v 1 v ′′ 1 − w 10 ( v ′ 1 ) 2 − 2( w 10 ) 2 v 1 v ′ 1 − 2( w 10 ) 3 ( v 1 ) 2 + 8 ˜ C v 1  + C 4 ( v 1 ) 4 ( v ′ 1 + w 10 v 1 ) . (39) Finally , the general form of a 2 × 2 Hermitian mat r ix 2-fold SUSY system for the non- degenerate case is give n b y H + = − 1 2 d 2 d q 2 + 1 4  3 w ′ 10 − 2 w 00 + ( w 10 ) 2 + C 2 ( v 1 ) 2  + 1 4 (3 v ′ 1 − 2 v 0 + 2 w 10 v 1 ) 3 X i =1 C 0 i σ i − C 0 , (40) H − = − 1 2 d 2 d q 2 + 1 4  − w ′ 10 − 2 w 00 + ( w 10 ) 2 + C 2 ( v 1 ) 2  + 1 4 ( − v ′ 1 − 2 v 0 + 2 w 10 v 1 ) 3 X i =1 C 0 i σ i − C 0 , (41) P − 2 = d 2 d q 2 + w 10 + v 1 3 X i =1 C 0 i σ i ! d d q + w 00 + v 0 3 X i =1 C 0 i σ i . (42) The tw o functions w 00 and v 0 in the ab ov e can b e eliminated b y using (38) and ( 3 9). Hence, the system can b e expressed solely in terms of the t wo functions w 10 and v 1 . F or the degenerate case v 1 = C − 1 w 10 , the t wo equations (35) and (36 ) are not indep enden t and are equiv alent with the following single equation 4( w 10 ) 2 ( w 00 + C v 0 ) = − 2 w 10 w ′′ 10 + ( w ′ 10 ) 2 + 4( w 10 ) 2 w ′ 10 + 4( w 10 ) 4 + 8 C 10 , (43) 7 with C 10 = ˜ C C . Hence, w e can again eliminate t wo of the four functions, e.g., v 1 and v 0 . The general f o rm of a 2 × 2 Hermitian matrix 2-fold SUSY system for the degenerate case is given by H + = − 1 2 d 2 d q 2 + 1 4  3 w ′ 10 − 2 w 00 + 2( w 10 ) 2  + 1 4 C " w ′ 10 + 2 w 00 + w ′′ 10 w 10 − ( w ′ 10 ) 2 2( w 10 ) 2 − 4 ˜ C C ( w 10 ) 2 # 3 X i =1 C 0 i σ i − C 0 , (44) H − = − 1 2 d 2 d q 2 + 1 4  − w ′ 10 − 2 w 00 + 2( w 10 ) 2  + 1 4 C " − 3 w ′ 10 + 2 w 00 + w ′′ 10 w 10 − ( w ′ 10 ) 2 2( w 10 ) 2 − 4 ˜ C C ( w 10 ) 2 # 3 X i =1 C 0 i σ i − C 0 , (45) P − 2 = d 2 d q 2 + w 10 1 + C − 1 3 X i =1 C 0 i σ i ! d d q + w 00 + 1 C " w ′ 10 − w 00 + ( w 10 ) 2 − w ′′ 10 2 w 10 + ( w ′ 10 ) 2 4( w 10 ) 2 + 2 ˜ C C ( w 10 ) 2 # 3 X i =1 C 0 i σ i . (46) It is in teresting to note that in the limit C 0 i → 0 ( i = 1 , 2 , 3), t he non-degenerate system (40)–(42) r educes to the most general 2 -fold SUSY scalar system [16, 19, 20] while t he degenerate system (4 4 )–(46) do es not . IV. DISCUSSION AND SUMMAR Y In this a rticle, w e hav e for t he first time fo r mulated generically N -fo ld SUSY in quan tum mec hanical matrix mo dels and constructed the general 2 × 2 Hermitian matrix 2- fold SUSY systems without recourse to an y a ssumption or ansatz. In a ddition to the detailed studies for larger n × n matrices ( n > 2) and N > 2 cases, there are many f uture issues to b e follo wed after this w ork as the follo wing: 1. First of all, it is imp ortan t to clarif y g eneral asp ects of N - fold SUSY in matrix systems, as we re done in [16, 17] for scalar systems. In the scalar case, there are t w o significan t features, namely , the equiv alence b et we en N -fo ld SUSY and w eak quasi-solv ability and the equiv alence b et wee n the conditions (3 a) a nd (3b). In the case of 2 × 2 Hermitian matrix systems , ho we ve r, it do es not seem that the conditions (10) and (13 ) coming from the former are equiv alent with the conditions (16), (19 ), (22) , a nd (25) coming from the latter although the other conditions are certainly equiv alent. That is exactly the reason why we considered the b oth to deriv e the formula (27). W e exp ect that the general approac h [21] for the scalar case recen tly prop osed b y us would b e also efficien t fo r the matrix case. 2. In t he scalar case, the systematic alg orithm fo r constructing an N -fold SUSY system [22] based on quasi-solv ability has show n to b e quite effectiv e. Hence, it s generalization to the mat r ix case is desirable. It w ould also enable us to connect directly the p ossible t yp es of matrix N -fold SUSY systems with the p ossible linear spaces of m ulti-comp onen t functions preserv ed by a second-order matrix linear differen tial op erator. F or example, it is 8 in teresting to see the connection with the quasi-solv able matrix op erato r s constructed fro m the generato r s of sl (2) in Ref. [23]. 3. Shap e in v ariance is a w ell-known sufficien t condition for s olvability [24]. It means in par- ticular that it alw ay s implies N - fold SUSY in the scalar case. In fact, some shap e-in v ariant scalar p oten tials w ere systematically constructed as particular cases of N -fold SUSY with in termediate Hamiltonians [25, 26]. Recen tly , sev eral shap e-inv ariant matrix p oten tials w ere constructed in Refs. [2, 6, 1 2 –14], and we exp ect that our formu la t ion of N - fold SUSY w ould b e also quite efficien t in constructing shap e-inv ar ia n t matrix mo dels. 4. Extension to m or e general second-order matrix linear differen tial op era t o rs w ould b e p ossible, e.g., b y admitting a non- diagonal second-order op era t o r and b y adding a matrix-v alued first-order op erator . 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