Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials

Exactl y Solv able Quan tum Mechanics and Infini te F amil ies of Mult i-indexed Orthog onal Polynomia ls Satoru Odake a, ∗ , Ryu Sasak i b a Dep artment of Physics, Shinshu Unive rsity, Matsumoto 390-86 21, Jap an b Y ukawa Institute for The or etic al Physics, Kyoto Universit y, Kyoto 606-85 02, Jap an Abstract Infinite families of multi-indexe d orthogo nal po lynomials a re discovered as the s olutions of e x actly solv able one- dimensional quantum mec hanical systems. The simplest examples, the one- index ed orthogona l po lynomials, a re the infinite families of the exc eptional Lag uerre and Jacobi p oly nomials of type I and II constructed by the present authors. The totality of the integer indices of the new p olynomia ls ar e finite and they co rresp ond to the degrees of the ‘ vi rtual state wav efunctions ’ which are ‘deleted’ by the generalis a tion of Crum-Adler theor em. Each p olynomial has another int eger n whic h co un ts the no des. Keywor ds: shap e inv ariance, orthog o nal p olynomia ls P A CS: 03.65.-w, 03.65.Ca, 03.65.Fd, 03.65 .Ge, 0 2 .30.Ik, 02.30 .Gp 1. Introduction W e will rep ort on the discov ery of infinite families of multi-indexe d orthog onal p olyno mials, which form the main parts of the eigenfunctions of exactly solv able one- dimensional q ua nt um mechanical systems. This will a dd a huge num ber of new member s to the in v entory of ex- actly solv able quantum mechanics [1, 2], whic h had seen rapid incr e a se due to the recent disco very of v ar ious kinds of infinite families of exc eptional o r thogonal p olynomials [3]–[8]. The metho d for deriving the m ulti-indexe d or- thogonal poly no mials is simple generalisa tion of the well- known metho d of Cr um [9] and its mo dification by K rein- Adler [10]. Starting from a n exactly so lv able Hamiltonia n system, Crum’s metho d provides another exactly solv able system by successively deleting the low es t lying eigen- state. The mo dification due to Krein-Adler uses dele- tion of a finite num b er of higher eigenstates sa tis fying certain conditions. The present method is based on the same s pirit of ‘deleting’ a finite num ber of virtual states instead of the eig enstates. Here the virtual sta te wa v e- functions a re the solutions of the Schr¨ odinger equa tio n of the original system and they ha ve n e gative ener gies , with resp ect to the groundstate eig env alue. The virtual state wa vefunctions hav e no no de in the interior and they ar e squar e-non- inte gr able . Their rec ipro cals are squar e-non- inte gr able , too . W e apply this metho d to the Hamiltonian systems of the radial oscillato r potential and the Darbo ux- P¨ o schl-T eller p otential, which are the b e s t known exa m- ples of exactly solv able quantum mechanics [1, 2]. Their ∗ Corresp onding author. Email addr e ss: odake@azusa .shinshu- u.ac.jp (Satoru Odake) eigenfunctions consist of the Laguerr e and the Jaco bi p o ly- nomials, resp ectively . When only one virtual sta te of deg ree ℓ ( ≥ 1) is deleted, the re s ulting system is the exce ptio nal X ℓ Laguerr e or Ja - cobi p olynomials of type I or I I [3, 5]. In gener al M dis- tinct vir tual states of degree v j ( ≥ 1) are de le ted and the resulting eig enfunctions co nsist of orthogo na l p olynomials indexed b y the integers (v 1 , . . . , v M ). The multi-indexed orthogo nal p olynomials are na tur al g e ne r alisation o f the exceptional or tho gonal po lynomials. They start a t degree ℓ ≥ 1 instea d of a degree 0 constant term, thus a voiding the constraints by Bo chner’s theore m [11, 12]. As s olutions of exactly solv able quantum mechanical systems they consti- tute a complete system o f orthogona l functions. The first examples of the exceptional Laguerre and Jacobi p olyno - mials a re constructed by G´ omez-Ullate et al. [12] in the framework of Stur m- Liouville theor y and by Quesne [13] within the shap e inv ariant [14] and exa ctly s olv able quan- tum mechanics. They are equiv alen t to the low es t mem- ber s ( ℓ = 1) of the X ℓ Laguerr e and Jacobi po lynomials derived in [3] by the pr esent authors. The present metho d could b e consider ed as the genera lisation of the Darb oux- Crum transfor mations [15] applied for the deriv a tion of the exceptional or tho gonal p oly no mials [16]–[20]. It is exp ected that these new p olyno mials would play impo rtant roles in ph ysics, mathematics and related dis- ciplines. The m ulti-indexed Jacobi polyno mials provide infinitely many global solutions o f second o r der F uc h- sian differential equations with 3 + ℓ regular singulari- ties [21]. Other theoretical developmen ts are: non-linear ident ities under lying the shap e inv ariance [22] are clari- fied, the structur e of the ex tr a zer os of the new o rthog- onal p olynomia ls [23] are exemplified, tw o-step Darb oux Pr eprint submitted to Physic s L etters B Octob er 24, 2018 transformatio ns [24] ar e explicitly co nstructed for the t ype II exceptional Laguerre p olynomia ls, v a rious use of non- ph ysical states [25, 26] for genera ting the exceptio nal p oly- nomials are sug g ested and several applications [2 7, 28] are rep orted. In this connection, let us men tion references of multi-step Darb oux- Crum tra nsformations, higher o r- der s usy transfor mations, N -fold susy transformatio ns , etc [29]. Some o f them utilise no n-physical solutions . This Letter is org a nised as follows. The general scheme of v irtual states deletion is pre s ent ed in section tw o. The explicit forms of the new p olynomials are derived in section three, together with the for ward-bac kward shift op e rators and the second o r der differen tial op erator s. The final sec - tion is f or a summar y and comments. 2. Adle r-Crum Sc heme for Virtual States Deleti on Here we present the metho d of v irtual states deletion in its generic and formal form. F or rigorous tr e atment s of po ssible sing ularities at the b oundar ies, we need to sp ecify the details of the Hamiltonia n systems, which will b e done in the next sectio n. Throughout this paper we choose all the solutions of the Schr¨ odinger eq ua tions to be real. W e start with an exactly solv able o ne dimensional qua ntum mechanical sy stem defined in an in ter v al x 1 < x < x 2 , H φ n ( x ) = E n φ n ( x ) ( n ∈ Z ≥ 0 ) , (1) ( φ m , φ n ) def = Z x 2 x 1 dx φ m ( x ) ∗ φ n ( x ) = h n δ m n . (2) F or simplicit y , we as s ume that it has discrete eigenv alues only with v anis hing g roundstate energy: 0 = E 0 < E 1 < E 2 < · · · , which allows us to expr ess the p otential in ter ms of the groundsta te wa v efunction φ 0 ( x ) which has no no de: H = − d 2 dx 2 + U ( x ) , U ( x ) = ∂ 2 x φ 0 ( x ) φ 0 ( x ) . (3) Let us suppose that the Schr¨ odinge r equation has a finite or infinite num b er of virtual state so lutions with ne gative ener gies : H ˜ φ v ( x ) = ˜ E v ˜ φ v ( x ) , ˜ E v < 0 (v ∈ V ) , (4) which hav e no no de in the in terior x 1 < x < x 2 . Her e V denotes the index s et of the virtua l s ta tes. W e als o require the infinite norm conditions, ( ˜ φ v , ˜ φ v ) = (1 / ˜ φ v , 1 / ˜ φ v ) = ∞ and this means that the v irtual s tate wa vefunctions v anish at one b oundar y and diverge at the other: lim x → x j ˜ φ v ( x ) = lim x → x j ∂ x ˜ φ v ( x ) = 0 , lim x → x k ˜ φ v ( x ) = lim x → x k ∂ x ˜ φ v ( x ) = ∞ , (5) where j 6 = k ∈ { 1 , 2 } and ∀ v ∈ V . The set of eigenfunctions { φ n ( x ) } forms a complete set in an a ppropriate Hilb ert space H , for example: H def =  ψ ( x ) ∈ L 2 [ x 1 , x 2 ]   lim x ↓ x 1 ( x − x 1 ) 1 2 ∂ x ψ ( x ) = 0 , lim x ↑ x 2 ( x 2 − x ) 1 2 ∂ x ψ ( x ) = 0  . (F or x 1 = −∞ a nd x 2 = ∞ , replace ( x − x 1 ) 1 2 → ( − x ) 1 2 and ( x 2 − x ) 1 2 → x 1 2 .) That is an y function in H which is orthogo nal to a ll the eigenfunctions has a z e ro norm: (Ψ , φ n ) = 0 ( ∀ n ∈ Z ≥ 0 ) ⇒ (Ψ , Ψ) = 0 . After M -steps of deleting the low est lying eigensta tes by Crum’s metho d [9], the resulting Hamiltonian system is iso- sp e c tral to the origina l one g iven by H [ M ] φ [ M ] n ( x ) = E n φ [ M ] n ( x ) ( n = M , M + 1 , . . . ) , φ [ M ] n ( x ) def = W[ φ 0 , φ 1 , . . . , φ M − 1 , φ n ]( x ) W[ φ 0 , φ 1 , . . . , φ M − 1 ]( x ) , ( φ [ M ] m , φ [ M ] n ) = M − 1 Y j =0 ( E n − E j ) · h n δ m n , U [ M ] ( x ) def = U ( x ) − 2 ∂ 2 x log   (W[ φ 0 , φ 1 , . . . , φ M − 1 ]( x )   , in which W[ f 1 , · · · , f n ]( x ) def = det  ∂ j − 1 x f k ( x )  1 ≤ j,k ≤ n is the W ronskian. The s e formulas hav e b een rep eatedly re dis cov- ered under v ario us names, higher deriv ative susy , N -fold susy , etc [29]. The corre s po nding Adler-Crum formulas for deleting the set o f M distinct eigenstates sp ecified b y D def = { d 1 , d 2 , . . . , d M } , d j ≥ 0, a r e H [ M ] φ [ M ] n ( x ) = E n φ [ M ] n ( x ) ( n ∈ Z ≥ 0 \D ) , φ [ M ] n ( x ) def = W[ φ d 1 , φ d 2 , . . . , φ d M , φ n ]( x ) W[ φ d 1 , φ d 2 , . . . , φ d M ]( x ) , ( φ [ M ] m , φ [ M ] n ) = M Y j =1 ( E n − E d j ) · h n δ m n , U [ M ] ( x ) def = U ( x ) − 2 ∂ 2 x log   W[ φ d 1 , φ d 2 , . . . , φ d M ]( x )   . The set D must sa tisfy the conditions Q M j =1 ( m − d j ) ≥ 0, ∀ m ∈ Z ≥ 0 [10, 3 0]. The ex a ctly solv able Hamiltonian system obtained b y dele ting M distinct virtual states D = { d 1 , . . . , d M } , d j ∈ V lo o ks almost the same: H [ M ] φ [ M ] n ( x ) = E n φ [ M ] n ( x ) ( n ∈ Z ≥ 0 ) , (6) H [ M ] ˜ φ [ M ] v ( x ) = ˜ E v ˜ φ [ M ] v ( x ) (v ∈ V \D ) , (7) φ [ M ] n ( x ) def = W[ ˜ φ d 1 , ˜ φ d 2 , . . . , ˜ φ d M , φ n ]( x ) W[ ˜ φ d 1 , ˜ φ d 2 , . . . , ˜ φ d M ]( x ) , (8) ( φ [ M ] m , φ [ M ] n ) = M Y j =1 ( E n − ˜ E d j ) · h n δ m n , (9) ˜ φ [ M ] v ( x ) def = W[ ˜ φ d 1 , ˜ φ d 2 , . . . , ˜ φ d M , ˜ φ v ]( x ) W[ ˜ φ d 1 , ˜ φ d 2 , . . . , ˜ φ d M ]( x ) , (10) U [ M ] ( x ) def = U ( x ) − 2 ∂ 2 x log   W[ ˜ φ d 1 , ˜ φ d 2 , . . . , ˜ φ d M ]( x )   . (11) It should be stresse d that the final r esults after M deletions are independent of the orders of deletions. The algebr aic 2 side of the der iv ation of (6)–(1 1) is basically the same as that for the Adler-Crum cas e. The essential difference is the verification of the inher ita nce at e ach step of the pr op- erties of the virtual states; no-no deness and the prop er bo undary b ehaviours. This is gua ranteed by choo sing the parameters of the theory g and h la rge e nough, se e (23) and (24). Here we show the pro cess H → H [1] . It is elementary to show H = ˆ A † d ˆ A d + ˜ E d , ˆ A d def = d dx − ∂ x ˜ φ d ( x ) ˜ φ d ( x ) , ∀ d ∈ V . (12) Since ˜ φ d ( x ) ha s no no de, the op erator ˆ A d and ˆ A † d are r eg- ular in the in ter ior. Next w e define H [1] by interchanging the order of these t wo op erators : H [1] def = ˆ A d ˆ A † d + ˜ E d , U [1] ( x ) = U ( x ) − 2 ∂ 2 x log   ˜ φ d ( x )   . (13) It is stra ightf orward to show that H a nd H [1] are iso - sp ectral H [1] φ [1] n ( x ) = E n φ [1] n ( x ) , φ [1] n ( x ) def = ˆ A d φ n ( x ) = W[ ˜ φ d , φ n ]( x ) ˜ φ d ( x ) ( n ∈ Z ≥ 0 ) , H [1] ˜ φ [1] v ( x ) = ˜ E v ˜ φ [1] v ( x ) , ˜ φ [1] v ( x ) def = ˆ A d ˜ φ v ( x ) = W[ ˜ φ d , ˜ φ v ]( x ) ˜ φ d ( x ) (v ∈ V \ { d } ) , φ n ( x ) = ˆ A † d E n − ˜ E d φ [1] n ( x ) , ( φ [1] m , φ [1] n ) = ( E n − ˜ E d )( φ m , φ n ) = ( E n − ˜ E d ) h n δ m n . Obviously the state d is delete d from the virtual s tate sp ec- trum of H [1] , for ˆ A d ˜ φ d ( x ) = 0. Since the zero-mo des of ˆ A d and ˆ A † d are ˜ φ d ( x ) and 1 / ˜ φ d ( x ), which hav e infinite norms, the s ets of eigenfunctions { φ n ( x ) } a nd { φ [1] n ( x ) } are iso- sp ectral a nd in one to one corresp ondence. Thus the ne w set { φ [1] n ( x ) } is c omplete . If not, there must exist a finite norm vector Ψ in the Hilb ert space which is o r thogonal to all the basis: (Ψ , φ [1] n ) = 0, ∀ n ∈ Z ≥ 0 . This w ould lead to a contradiction ( ˆ A † d Ψ , φ n ) = 0 , ∀ n ∈ Z ≥ 0 , since ˆ A † d Ψ 6 = 0. Next we show the logic for demonstrating that the virtua l state solutions { ˜ φ [1] v ( x ) } ha v e no node in th e in terior. By using the Sc hr ¨ odinge r equations for t hem we obtain ∂ x W[ ˜ φ d , ˜ φ v ]( x ) =  ˜ E d − ˜ E v  ˜ φ d ( x ) ˜ φ v ( x ) , (14) which has no no de. Since w e can verify that W[ ˜ φ d , ˜ φ v ]( x ) v anishes at one b o unda ry , no-no deness of W[ ˜ φ d , ˜ φ v ]( x ) in the interior follows. When the vir tual s tates d and v b elong to different types (I a nd I I) the W ronskians might not v anish at b oth bo undaries. In that case we can show that they hav e the same sign a t b oth bo und- aries W [ ˜ φ d , ˜ φ v ]( x 1 ) W [ ˜ φ d , ˜ φ v ]( x 2 ) > 0. Likewise w e can show, for a ll the explicit examples in the ne x t section that ˜ φ [1] v and 1 / ˜ φ [1] v hav e infinite norms. The steps go ing from H [1] → H [2] and further ar e e s sentially the same. 3. Mul ti-indexed Laguerre and Jacobi Polynomials Here we will a pply the method of virtual states deletion to t wo well-kno wn sys tems o f exactly s olv able quantum mechanics with the radial oscillator and Darb oux-P¨ oschl- T eller p otentials: U ( x ) =                x 2 + g ( g − 1) x 2 − (1 + 2 g ) , x 1 = 0 , x 2 = ∞ , g > 1 2 , : L g ( g − 1) sin 2 x + h ( h − 1) cos 2 x − ( g + h ) 2 , x 1 = 0 , x 2 = π 2 , g , h > 1 2 , : J , (15) in which L a nd J stand for the names of their eigenfunc- tions, the La guerre and J acobi p olynomials. F or the a c - tual pa rameter ra ng es consistent with sp ecific dele tio ns, see (23)-(24). The eigenfunctions ar e factorised into the groundstate eig enfunction and the po lynomial in the sinu- soidal c o or dinate η = η ( x ) [31]: φ n ( x ; λ ) = φ 0 ( x ; λ ) P n ( η ( x ); λ ) , (16) in which λ stands for the parameters , g fo r L and ( g , h ) for J. Their explicit forms are L : φ 0 ( x ; g ) def = e − 1 2 x 2 x g , η ( x ) def = x 2 , P n ( η ; g ) def = L ( g − 1 2 ) n ( η ) , E n ( g ) def = 4 n, h n ( g ) def = 1 2 n ! Γ( n + g + 1 2 ) , (17) J : φ 0 ( x ; g , h ) def = (sin x ) g (cos x ) h , η ( x ) def = cos 2 x, P n ( η ; g , h ) def = P ( g − 1 2 ,h − 1 2 ) n ( η ) , E n ( g , h ) def = 4 n ( n + g + h ) , h n ( g , h ) def = Γ( n + g + 1 2 )Γ( n + h + 1 2 ) 2 n !(2 n + g + h )Γ( n + g + h ) . (18) They hav e t wo t yp e s of virtual states, which are again p olynomial solutions . The total n um ber is finite dep ending on the parameter v alues g and h , except for L1 which has infinite. The vir tua l states wa vefunctions for L are: L1 : ˜ φ I v ( x ) def = e 1 2 x 2 x g ξ I v ( η ( x ); g ) , ξ I v ( η ; g ) def = P v ( − η ; g ) , v ∈ Z ≥ 0 , ˜ E I v def = − 4( g + v + 1 2 ) , ˜ δ I def = − 1 , (19) L2 : ˜ φ II v ( x ) def = e − 1 2 x 2 x 1 − g ξ II v ( η ( x ); g ) , ξ II v ( η ; g ) def = P v ( η ; 1 − g ) , v = 0 , 1 , . . . , [ g − 1 2 ] ′ , ˜ E II v def = − 4( g − v − 1 2 ) , ˜ δ II def = 1 , (20) in which [ a ] ′ denotes the greates t integer less than a and ˜ δ I , I I will be use d later. F or no-no denes s of ξ I , I I v , see (2 .39) of [22]. The vir tual states wav e functions for J a re: J1 : ˜ φ I v ( x ) def = (sin x ) g (cos x ) 1 − h ξ I v ( η ( x ); g , h ) , 3 ξ I v ( η ; g , h ) def = P v ( η ; g , 1 − h ) , v = 0 , 1 , . . . , [ h − 1 2 ] ′ , ˜ E I v def = − 4( g + v + 1 2 )( h − v − 1 2 ) , ˜ δ I def = ( − 1 , 1) , (21) J2 : ˜ φ II v ( x ) def = (sin x ) 1 − g (cos x ) h ξ II v ( η ( x ); g , h ) , ξ II v ( η ; g , h ) def = P v ( η ; 1 − g , h ) , v = 0 , 1 , . . . , [ g − 1 2 ] ′ , ˜ E II v def = − 4( g − v − 1 2 )( h + v + 1 2 ) , ˜ δ II def = (1 , − 1) . (22) The larg er the parameter g and/o r h b ecome, the more the virtual s ta tes a re ‘crea ted’. The L1 sy s tem is ob- tained from the J1 in the limit h → ∞ [5]. This ex- plains the infinitely many virtual states of L1. Let us denote by V I , II the index sets of the virtua l sta tes of t yp e I and I I for L and J. Due to the par it y pro p er t y of the Jacobi p olyno mial P ( α,β ) n ( − x ) = ( − 1 ) n P ( β ,α ) n ( x ), the tw o vir tual state polyno mials ξ I v and ξ II v for J are re- lated b y ξ II v ( − η ; g , h ) = ( − 1) v ξ I v ( η ; h, g ). F or no-no deness of ξ I , I I v , s ee (2.40) of [22] a nd (3.2) of [6]. It is also obvi- ous that the vir tual state w av efunctions ˜ φ v and their re- cipro cals 1 / ˜ φ v are not squar e integrable in all four types. There is a third type of nega tive energy p olyno mial so- lutions for b oth L and J. F or J, the degree n so lution is (sin x ) 1 − g (cos x ) 1 − h P ( − g + 1 2 , − h + 1 2 ) n ( η ). Even when they hav e no no de in the interior, their r ecipro cals a r e square int egrable and they cannot b e used for the virtual states deletion. The lab el 0 is sp ecial in that the w avefunctions satisfy ˜ φ I 0 ( x ; λ ) ˜ φ II 0 ( x ; λ ) = c − 1 F dη ( x ) dx since ξ I , I I 0 = 1. Here the co nstant c F = 2 for L and c F = − 4 for J. W e will not use the la bel 0 s tates for deletion. ˆ A d 1 ˆ A † d 1 ˆ A d 1 d 2 ˆ A † d 1 d 2 0 1 2 3 3 d 1 d 2 . . . H H [1] H [2] H [ M + N ] original system d 1 deleted system d 1 , d 2 deleted system M + N virtual states deleted system Figure 1: Schematic picture of virtual sta tes de le tion. The black circ les denote eige ns tates. The down and up triangles denote vir tual states of type I and I I. The deleted virtual sta tes are denoted b y white triangles. Since ther e are t wo types of virtual states av ailable, the general deletion is sp ecified by the set of M + N p ositive int egers D def = { d I 1 , . . . , d I M , d II 1 , . . . , d II N } , d I , I I j ≥ 1, which are the degrees of the deleted virtual state wa vefunctions. Note that the sub cases of e ither M = 0 or N = 0 a re meaningful. In or der to acco mmo date all these virtual states, the para meters g a nd h must b e la rger than cer ta in bo unds: L : g > max { N + 3 2 , d II j + 1 2 } , (23) J : g > max { N + 2 , d II j + 1 2 } , h > max { M + 2 , d I j + 1 2 } . (24) See Figure 1 for the schematic structure of virtua l states deletion. Like the original eige nfunctions (16) the n -th eigenfunc- tion φ D ,n ( x ; λ ) ≡ φ [ M ,N ] n ( x ) after the deletion (9) can b e clearly factorised in to a n x -dep endent part, the co mmon denominator p olynomial Ξ D in η and the multi-indexe d p olynomial P D ,n in η : φ [ M ,N ] n ( x ) ≡ φ D ,n ( x ; λ ) = c M + N F ψ D ( x ; λ ) P D ,n ( η ( x ); λ ) , ψ D ( x ; λ ) def = φ 0 ( x ; λ [ M ,N ] ) Ξ D ( η ( x ); λ ) , (25) in whic h φ 0 ( x ; λ ) is the groundstate w av efunction (17)- (18). Here the shifted parameters λ [ M ,N ] after the [ M , N ] deletion are λ [ M ,N ] def = λ − M ˜ δ I − N ˜ δ II , explicitly they ar e λ [ M ,N ] =  g + M − N : L ( g + M − N , h − M + N ) : J . (26) Needless to say , the denominator p olyno mial Ξ D has no no de in the interior. The p olynomials P D ,n and Ξ D are expressed in terms of W ronskians of the v ariable η : P D ,n ( η ; λ ) def = W[ µ 1 , . . . , µ M , ν 1 , . . . , ν N , P n ]( η ) × ( e − M η η ( M + g + 1 2 ) N : L  1 − η 2  ( M + g + 1 2 ) N  1+ η 2  ( N + h + 1 2 ) M : J , (27) Ξ D ( η ; λ ) def = W[ µ 1 , . . . , µ M , ν 1 , . . . , ν N ]( η ) × ( e − M η η ( M + g − 1 2 ) N : L  1 − η 2  ( M + g − 1 2 ) N  1+ η 2  ( N + h − 1 2 ) M : J , (28) µ j =    e η ξ I d I j ( η ; g ) : L  1+ η 2  1 2 − h ξ I d I j ( η ; g , h ) : J , ν j =    η 1 2 − g ξ II d II j ( η ; g ) : L  1 − η 2  1 2 − g ξ II d II j ( η ; g , h ) : J , (29 ) in whic h P n in (27) de no tes the o riginal polynomial, P n ( η ; g ) for L and P n ( η ; g , h ) for J. These p olyno mials de- pend on the order of deletions o nly through the sign of per mut ation. The multi-indexed poly nomial P D ,n is of de- gree ℓ + n a nd the denominato r p o lynomial Ξ D is of degr ee ℓ in η , in which ℓ is given b y ℓ def = M X j =1 d I j + N X j =1 d II j − 1 2 M ( M − 1) − 1 2 N ( N − 1) + M N ≥ 1 . (30) 4 Here the lab el n sp ecifies the e ner gy eigenv alue E n of φ D ,n . Hence it also counts the no des due to the oscillation the- orem. Although they do not satisfy the three term re- currence relations, the m ulti-indexed po lynomials { P D ,n } form a complete set of orthogonal po ly nomials with the orthogo nality r elations: Z dη W ( η ; λ [ M ,N ] ) Ξ D ( η ; λ ) 2 P D ,m ( η ; λ ) P D ,n ( η ; λ ) = h n ( λ ) δ nm ×      Q M j =1 ( n + g + d I j + 1 2 ) · Q N j =1 ( n + g − d II j − 1 2 ) : L 4 − M − N Q M j =1 ( n + g + d I j + 1 2 )( n + h − d I j − 1 2 ) × Q N j =1 ( n + g − d II j − 1 2 )( n + h + d II j + 1 2 ) : J , (31) where the weigh t function of the o riginal p olynomials W ( η ; λ ) dη = φ 0 ( x ; λ ) 2 dx rea ds ex plicitly W ( η ; λ ) def =  1 2 e − η η g − 1 2 : L 1 2 g + h +1 (1 − η ) g − 1 2 (1 + η ) h − 1 2 : J . (32) This is obtained by rewr iting the or thogonality relations of the eig enfunctions (9) after the [ M , N ] deletion. In the res t of this section we explo re v arious prop er- ties o f the new m ulti-indexed p olyno mials { P D ,n } . As for the exce ptio nal ortho g onal p oly no mials [3, 4, 5, 21, 20, 7], the low e st degree p oly nomial P D , 0 ( η ; λ ) is r elated to the denominator p olyno mia l Ξ D ( η ; λ ) by the parameter shift λ → λ + δ ( δ = 1 for L a nd δ = (1 , 1) for J ): P D , 0 ( η ; λ ) = Ξ D ( η ; λ + δ ) ×        ( − 1) M Q N j =1 ( g − d II j − 1 2 ) : L 2 − M Q M j =1 ( h − d I j − 1 2 ) × ( − 2) − N Q N j =1 ( g − d II j − 1 2 ) : J . (33) The virtual state w av efunction with the index v ∈ D lead- ing to the final [ M , N ] deletion is ˜ φ [ M ′ ,N ′ ] v (10) with the index set D ′ def = D \{ v } , in which [ M ′ , N ′ ] = [ M − 1 , N ] if v ∈ V I , and [ M , N − 1 ] if v ∈ V II . It has the following form: ˜ φ [ M ′ ,N ′ ] v ( x ; λ ) = c M + N − 1 F Ξ D ( η ( x ); λ ) Ξ D ′ ( η ( x ); λ ) ×            ( − 1) N φ 0 ( ix ; λ [ M − 1 ,N ] ) i − ( g + M − N − 1) , v ∈ V I : L φ 0 ( x ; t II ( λ [ M ,N − 1] )) , v ∈ V II : L ( − 1) N φ 0 ( x ; t I ( λ [ M − 1 ,N ] )) , v ∈ V I : J φ 0 ( x ; t II ( λ [ M ,N − 1] )) , v ∈ V II : J . (34) Here the t wist operato r t acting on the parameter s is de- fined a s t II ( g ) = 1 − g for L, t I ( g , h ) = ( g , 1 − h ) and t II ( g , h ) = (1 − g , h ) for J. Note that the vir tua l states (20)– (22) hav e the form ˜ φ v ( x ; λ ) = φ v ( x ; t ( λ )). These rela tions are e s sential for the determination of the low er b ounds of the para meters g iven in (23)–(24), which guarantee that the virtual state pr o pe rties are cor rectly inherited a t each step of deletion. The Hamiltonian H D ( λ ) ≡ H [ M ,N ] of the [ M , N ] deleted system can b e express ed in terms of its g roundstate eigen- function φ D , 0 ( x ; λ ) ≡ φ [ M ,N ] 0 ( x ; λ ) with the help of (33 ): H D ( λ ) = A D ( λ ) † A D ( λ ) , (35) A D ( λ ) def = d dx − ∂ x φ D , 0 ( x ; λ ) φ D , 0 ( x ; λ ) , φ D , 0 ( x ; λ ) ∝ φ 0 ( x ; λ [ M ,N ] ) Ξ D ( η ( x ); λ + δ ) Ξ D ( η ( x ); λ ) . (36) This has the s ame form as the v arious Hamiltonians of the exceptional or thogonal p olyno mia ls including those of the discr ete q uantum mec hanics [3, 4 , 5, 7]. Reflecting the construction [19, 6 ] it is sha pe in v ariant [14, 3, 2 2] A D ( λ ) A D ( λ ) † = A D ( λ + δ ) † A D ( λ + δ ) + E 1 ( λ ) . (3 7) This means that the operato r s A D ( λ ) a nd A D ( λ ) † relate the eigenfunctions of neigh bo uring degr ees and par ame- ters: A D ( λ ) φ D ,n ( x ; λ ) = f n ( λ ) φ D ,n − 1 ( x ; λ + δ ) , (3 8) A D ( λ ) † φ D ,n − 1 ( x ; λ + δ ) = b n − 1 ( λ ) φ D ,n ( x ; λ ) , (39) in which the constants f n ( λ ) a nd b n − 1 ( λ ) are the factors of the eig env alue f n ( λ ) b n − 1 ( λ ) = E n ( λ ): f n ( λ ) =  − 2 : L − 2( n + g + h ) : J , b n − 1 ( λ ) = − 2 n : L, J . The forward and backw a rd shift op erators are defined by F D ( λ ) def = ψ D ( x ; λ + δ ) − 1 ◦ A D ( λ ) ◦ ψ D ( x ; λ ) (40) = c F Ξ D ( η ; λ + δ ) Ξ D ( η ; λ )  d dη − ∂ η Ξ D ( η ; λ + δ ) Ξ D ( η ; λ + δ )  , (41) B D ( λ ) def = ψ D ( x ; λ ) − 1 ◦ A D ( λ ) † ◦ ψ D ( x ; λ + δ ) (42) = − 4 c − 1 F c 2 ( η ) Ξ D ( η ; λ ) Ξ D ( η ; λ + δ ) ×  d dη + c 1 ( η , λ [ M ,N ] ) c 2 ( η ) − ∂ η Ξ D ( η ; λ ) Ξ D ( η ; λ )  , (43 ) in which the functions c 1 ( η ; λ ) and c 2 ( η ) a re those a p- pea ring in the (co nfluen t) hyper geometric equatio ns for the Laguer re and Jacobi po lynomials L : c 1 ( η , λ ) def = g + 1 2 − η , c 2 ( η ) def = η , J : c 1 ( η , λ ) def = h − g − ( g + h + 1) η , c 2 ( η ) def = 1 − η 2 . Their action on the multi-indexed p o lynomials P D ,n ( η ; λ ) is F D ( λ ) P D ,n ( η ; λ ) = f n ( λ ) P D ,n − 1 ( η ; λ + δ ) , (44) B D ( λ ) P D ,n − 1 ( η ; λ + δ ) = b n − 1 ( λ ) P D ,n ( η ; λ ) . (45) 5 The s e c ond o rder differential op era to r e H D ( λ ) g overning the multi-indexed po ly nomials is: e H D ( λ ) def = ψ D ( x ; λ ) − 1 ◦ H D ( λ ) ◦ ψ D ( x ; λ ) = B D ( λ ) F D ( λ ) = − 4  c 2 ( η ) d 2 dη 2 +  c 1 ( η , λ [ M ,N ] ) − 2 c 2 ( η ) ∂ η Ξ D ( η ; λ ) Ξ D ( η ; λ )  d dη + c 2 ( η ) ∂ 2 η Ξ D ( η ; λ ) Ξ D ( η ; λ ) − c 1 ( η , λ [ M ,N ] − δ ) ∂ η Ξ D ( η ; λ ) Ξ D ( η ; λ )  , (46) e H D ( λ ) P D ,n ( η ; λ ) = E n ( λ ) P D ,n ( η ; λ ) . (47) Since all the ze r os of Ξ D ( η ; λ ) are simple, (47) is a F u chsian differential equation for the J ca se. The characteristic ex- po nent s at the zeros of Ξ D ( η ; λ ) a re the same everywhere, 0 and 3. The multi-indexed po ly nomials { P D ,n ( η ; λ ) } pr o - vide infinitely ma n y global solutions of the a bove F uc hsian equation (47) with 3 + ℓ regula r singularities for the J case [21]. The L case is obtained in a confluent limit. These sit- uations a re basic ally the same as those o f the exceptional po lynomials. Although we have r estricted d I , I I j ≥ 1 , there is no ob- struction for deletion of d I , I I j = 0. In terms of the multi- indexed po lynomial (27), the level 0 dele tio ns imply the following: P D ,n ( η ; λ )    d I M =0 = P D ′ ,n ( η ; λ − ˜ δ I ) × A, D ′ = { d I 1 − 1 , . . . , d I M − 1 − 1 , d II 1 + 1 , . . . , d II N + 1 } , (4 8) P D ,n ( η ; λ )    d II N =0 = P D ′ ,n ( η ; λ − ˜ δ II ) × B , D ′ = { d I 1 + 1 , . . . , d I M + 1 , d II 1 − 1 , . . . , d II N − 1 − 1 } , (4 9) where the multiplicativ e factors A a nd B are A =        ( − 1) M Q N j =1 ( d II j + 1 ) : L − ( − 2) − M Q M − 1 j =1 ( g − h + d I j + 1 ) × ( − 2) − N Q N j =1 ( d II j + 1 ) · ( n + h − 1 2 ) : J , B =        ( − 1) M Q M j =1 ( d I j + 1 ) · ( n + g − 1 2 ) : L 2 − M Q M j =1 ( d I j + 1 ) · ( − 2) − N × Q N − 1 j =1 ( h − g + d II j + 1 ) · ( n + g − 1 2 ) : J . F rom (33), Ξ D behaves simila rly . Ther efore inc luding the level 0 deletion corre spo nds to M + N − 1 virtual states deletions. This is why w e hav e re stricted d I , I I j ≥ 1. These re la tions (48)–(49) can b e used for studying the equiv alence of H D . It should be stressed that the sa me po lynomials can hav e differen t but equiv alent s ets of multi- indices D , which defines the denomina tor p olynomial Ξ D . Since the overall sca le of Ξ D is irrelev a nt (see (36)) for the m ulti-indexed p olynomia l { P D ,n } , we consider the equiv- alence class of Ξ D . F or example, it is straightforward to verify the f ollowing equiv alence and their duals I ↔ I I: Ξ { 1 I ,...,k I } ( η ; λ − ˜ δ II ) ∼ Ξ { k II } ( η ; λ − k ˜ δ I ) ( k ≥ 1 ) , (50) Ξ { m I ,..., ( k + m ) I } ( η ; λ − m ˜ δ II ) ∼ Ξ { ( k +1) II ,..., ( k + m ) II } ( η ; λ − ( k + 1) ˜ δ I ) ( k ≥ 1) . (51) The simplest, k = 1, of (50) corre s po nds to the well known fact that the X 1 t yp e I and I I p olynomia ls a r e ident ical for L and J [5]. Deleting the fir st k consecutive excited s ta tes, like the l.h.s. of (50) has led to many interesting phe- nomena in the ordinar y and discrete quantum mec hanics [30, 3 3, 32]. Classifica tion of the equiv alent classes leading to the same polynomials is a challenging future problem. The exceptiona l X ℓ orthogo nal p olynomials o f type I and I I, [12, 13, 3, 5 , 21, 19] cor resp ond to the simplest cases of one virtual state deletion of that type, D = { ℓ I } or { ℓ II } , ℓ ≥ 1 : ξ ℓ ( η ; λ ) = Ξ D ( η ; λ + ℓ δ + ˜ δ ) , (52) P ℓ,n ( η ; λ ) = P D ,n ( η ; λ + ℓ δ + ˜ δ ) × A, (53) where ˜ δ = ˜ δ I , I I and the multiplicative facto r A is A = − 1 for XL1, ( n + g + 1 2 ) − 1 for XL2, 2 ( n + h + 1 2 ) − 1 for XJ1 and − 2( n + g + 1 2 ) − 1 for XJ2. Most formulas betw een (35) and (47) lo ok almost the same as thos e app earing in the theory of the exceptional orthogona l p olynomials [3, 4, 5, 2 1, 19, 20, 6, 7]. In a recent pap er [24] G´ omez-Ullate et al discussed “tw o- step Darb oux transfor mations,” which corresp onds to the examples o f D = { m II 1 , m II 2 } for L in our scheme. This pap er stimulated the present work. 4. Summ ary and Com men ts This is a first shor t r epo rt on the discovery of infinite families of multi-indexed ortho gonal p olynomials . They are obtained as the main part of exa ctly solv able qua n- tum mec hanical sys tems, which ar e deformatio ns of the radial osc illator and the Darb oux-P ¨ oschl-T eller p o tent ials. Although these new p olynomials sta rt at degree ℓ ( ≥ 1), they fo r m a complete set of o r thogonal p olynomials. The exactly s olv able modified Hamiltonian and the eigenfunc- tions are obtained by a pply ing the modified Crum’s theo- rem due to Adler to the or iginal system to delete a finitely many virtual state solutions of type I and II. The sim- plest c a se of one virtual state deletion repr o duces the ex- ceptional or thogonal p olynomials [12 , 13, 3, 5, 2 1]. F or the Jacobi case, the new po lynomials pr ovide infinitely many examples of global s olutions o f second order F uc h- sian differen tial equations with 3 + ℓ regular singularities [21]. Like the undeformed theor y , the Lag uerre results c an be o btained from the Jacobi results in certa in confluence limits. But we prese n ted all r esults in parallel, for b etter understanding of the structure. The parameter rang e s (23) and (24) are conserv a tive s ufficien t conditions with which the prop er b o undary b ehaviours and the completeness ar e guaranteed. As a first rep or t, o nly the basic r esults are pr esented. Some imp ortant issues ca nnot be included due to space 6 restrictions, for example, an a lternative pr o of o f shap e in- v aria nce, the action of v arious intertwining op era tors, etc. Many imp ortant pro blems are remaining to b e clarified. F or exa mple, the bisp ectr a lit y [19], generating functions [21], prop erties of the zeros of the denominato r p olynomial Ξ D and the extra zero s o f the m ulti-indexed p olynomials { P D ,n } [23], class ific a tion of the equiv alent cla sses lea ding to the s a me new p olynomials . The exceptional Wilso n, Askey-Wilson, Ra cah and q - Racah po lynomials were constructed in the framework o f discrete qua nt um mechanics [4, 20, 7, 8], and the Crum- Adler theorem for disc rete quantum mechanics w ere also presented [33, 30, 32]. The method of vir tual states dele- tion presented in this Letter is applicable to dis crete quan- tum mechanics and it is easy to wr ite down gener al for- m ulas like as (6)–(11). This gives mult i-indexed orthogo- nal polynomials corresp onding to the deformations of the Wilson, Askey-Wilson, Ra cah and q -Racah p olynomials. In co ncrete examples, more tha n tw o types of virtual state solutions are av ailable and muc h richer structures are ex- pec ted. W e will re p or t on these topics els e where. Before c losing this Letter, let us make tw o small c om- men ts. The o riginal Schr¨ odinger equation (1) has a g e neral solution for an arbitrary energ y E , H ψ ( x ) = E ψ ( x ). If w e define the co rresp onding [ M , N ] deletion s olution ψ [ M ,N ] ( x ) def = W[ ˜ φ I d 1 , . . . , ˜ φ I d M , ˜ φ II d 1 , . . . , ˜ φ II d N , ψ ]( x ) W[ ˜ φ I d 1 , . . . , ˜ φ I d M , ˜ φ II d 1 , . . . , ˜ φ II d N ]( x ) , it so lves the deformed Schr¨ odinger equation with the same energy , too H [ M ,N ] ψ [ M ,N ] ( x ) = E ψ [ M ,N ] ( x ) . 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