Building extended resolvent of heat operator via twisting transformations

Building extended resolv en t of heat op erator v ia t wisting trans formati ons M. Boiti ∗ , F. P empinelli ∗ , A. K. P ogrebk o v † , and B. Prinari ∗ ∗ Dipartimen to di Fisica, Univ ersit` a del Salen to and Sezione INFN, Lecce, Italy † Steklo v Mathematical Institute, Mosco w, R ussia Abstract Twisting transformations for the heat operator are introduced. They are used, at th e same time, to sup erimp ose ` a la Darb oux N solitons to a generic smooth, decaying at infinity , p otential and to generate t h e corre- sp on d ing Jost solutions. These tw isting op erators are als o used to study the existence of the related exten ded resolven t. Existence and u niqueness of the extended resolven t in the case of N solitons with N “ingoing” ra ys and one “outgoing” ra y is studied in details. 1 In tro duction The Kadomtsev–Petviash vili equation in its version called KPI I ( u t − 6 uu x 1 + u x 1 x 1 x 1 ) x 1 = − 3 u x 2 x 2 , (1.1) is a (2+1)-dimensiona l generalization of the celebrated Korteweg–de V ries (KdV) equation. As a conse q uence, the KP I I e q uation admits solutions that b ehave at space infinit y like the solutio ns o f the KdV equation. F or ins tance, if u 1 ( t, x 1 ) ob eys KdV, then u ( t, x 1 , x 2 ) = u 1 ( t, x 1 + µx 2 − 3 µ 2 t ) solves KPI I for an arbi- trary c onstant µ ∈ R . Thus, it is impor tant to consider s olutions of (1.1) that decay at spa ce infinit y in all dir ections with exception o f a finite num ber of 1-dimensional rays with b ehavior of the type of u 1 . Even though KPI I has b een known to b e in tegrable for more than three decades [1, 2], its genera l theory (in volving such non- decaying solutions ) is far from being complete. Thus, the Cauch y pro blem fo r KPI I with rapidly decaying initial data was so lved in [3 , 4] by using the In v erse Sca tter ing T ra nsform (IST) method, bas e d on the sp ectral analysis of the heat op era tor L ( x, ∂ x ) = − ∂ x 2 + ∂ 2 x 1 − u ( x ) , x = ( x 1 , x 2 ) , (1.2) 1 that gives the a sso ciated linea r problem for the KPI I equation. The standar d approach to the sp e ctral theory of the op er ator (1.2) is bas e d on integral equa- tions for the Jost solution Φ( x, k ), where k ∈ C denotes the sp ectra l par ameter, or for Ψ( x, k ), Jo st s olution of the dual op er ator L d . How ev er, it is known that in the case of p otentials with one-dimensio nal asymptotic b ehavior these int egral equations ar e ill-defined. In order to overcome these difficulties, a r e solven t appro ach w as developed in [5–14]. In this approach a spa ce o f op er ators A ( q ) with kernels A ( x, x ′ ; q ) belo nging to the space of temp ered distributions o f v ariables x, x ′ , q ∈ R 2 was int ro duced. Ordinary differential o pe r ators L ( x, ∂ x ) a re imbedded in this spac e as op erator s with kernels L ( x, x ′ ; q ) ≡ L ( x, ∂ x + q ) δ ( x − x ′ ) . (1.3) Due to the additional dep endence o n the para meter q these o p er ators are called extended op er a tors. In this space a genera lization of the resolvent of a differen- tial opera tor, called extended resolven t, was in tro duced, enabling consideration of the spe ctral theory of op er a tors with nontrivial asympto tic b ehavior at space infinit y . In [11–14] we considere d the nonstationary Schr¨ odinger and heat op e rators with potentials with only one direc tio n of no ndec aying b e havior. The firs t step in s olving the problem was the embedding of a pure one-dimens io nal s p ec tral theory in the t wo-dimensional one, building the t wo-dimensional extended re - solven t for an op erator with p otential u ( x ) ≡ u 1 ( x 1 ). The second step was the dressing o f this resolven t b y an a r bitrary bidimensional perturba tion of the po tential u 1 . Finally , a ll mathematical en tities app earing in the IST theo ry were o btained from this dr essed resolven t by a reduction pro cedure, allowing the formulation of the direc t and inverse pro blems. Let us a lso mention that the standard sp ectra l theory for the hea t o pe rator in the ca s e o f p otentials no n- decaying in one spa ce dire ction was developed in [15 ], but under some sp ecial and inexplicit conditions on the p o tential. Here we consider the substantially more complicated problem: the case of a p otential u non decaying a long multiple non parallel r ays. Th us now there is no ana logy with the one- dimensional case a nd the whole theo r y has to be constructed directly , without embedding o ne-dimensional entities in tw o di- mensions. Therefore, we are oblige d to consider directly true bidimensional po tentials as it was alrea dy done in [16, 17] for the nonstationary Sc hr¨ odinger op erator . In fact, we apply the same pro c edure that w as successfully applied in that case. P r ecisely , in order to get the p otential corr esp onding to N solitons “sup erimp osed” to a generic smo oth decaying po tential and the rela ted J ost so- lutions we construct these entit ies directly by means of t wisting tra nsformations instead of using recursively the (binary ) Darb oux trans fo rmation. In this w ay we r ecov er p o tent ials and Jos t s o lutions tha t were obtained in [11] by using a recursive pro cedure and, also , so me alternative representations of the pure soli- tonic p otentials obtained in [18–22 ] by using the τ functions. It is w orth noting that the hea t ope r ator is no t self-dua l and thus sing ula r b ehaviors of the left 2 and r ight t wisting op erator s a re not oblig ed to b e corr elated. This explains why the structure of the N soliton solution of KPI I is muc h richer than for KP I. In particular, the N soliton solution can hav e a differen t n um be r of “ingoing” a nd “outgoing” (in the sense of x 2 → −∞ , or + ∞ ) rays. The article is org anized as follows. In Sec. 2 w e briefly review some asp ects and basic ideas of the extended resolven t appr oach, referring to the exa mple of an op erator (1.2) with a smo oth, r apidly decaying p otential u ( x ). F or fur- ther details, we refer the interested readers to [5 –9]. In Sec. 3 we in troduce the t wisting op erato rs, i.e., op erator s that “ twist” the o p erator L , extension in the sense of (1.3) of the o p erator L in (1 .2 ), to an op era tor L ′ of the same kind with a p otential u ′ describing N so litons “sup erimp osed” ` a la Dar b oux to the background po tential u . In Sec. 4, by using these twisting op erato rs, under the assumption of the existence o f the resolven t M ′ ( q ) w e derive an explicit expres- sion for its kernel M ′ ( x, x ′ ; q ). So the pr oblem of the e xistence of the resolven t M ′ ( q ) is reduced to the pro blem of finding the re gion in the q - plane, wher e this kernel is a temp ered distribution. This problem resulted to b e more ar duous than in the cas e o f the nonstationary Schr¨ odinger op era tor and, therefo r e, in Sec. 5 , we first consider the pure N -so lito n p otentials a nd, afterwards, in Sec. 6, the s p ec ial simple sub clas s of N -soliton p otentials with N “ingoing” r ays a nd only o ne “o utg oing” ray . W e prove that for N > 1 the resolven t M ′ ( q ) exists only in the reg ion of the q -plane exter ior to a p olyg on with N + 1 sides. More sp ecifically , we prov e that there exists a v a lue k 0 of the s p ectral par a meter k , for which the dual Jost solution Ψ ′ ( x, k ) is such that Ψ ′ ( x, k 0 ) exp( q 1 x 1 + q 2 x 2 ) is exp onentially decaying on the x -plane for any v alue of q b elonging to this po lygon and, conse quently , we deduce that inside the p olygon the t wisted op- erator L ′ ( q ) has a left annihilator and cannot hav e r ight inv erse. Nonetheless, the Gr een’s function of L ′ exists and can be uniquely der ived via a reduction from M ′ ( q ). In a forthcoming publica tio n we plan to consider the gener ic case of an arbitrar y num ber of inco ming and outgo ing rays and to elucidate the role of these annihilators in the sp ectr a l theory of such potentials. 2 Bac kground theory Let us intro duce the s pace of extended op er ators A ( q ), i.e., oper ators with kernel A ( x, x ′ ; q ) b elong ing to the space S ′ of temp ered distributio ns o f the six real v ariables x = ( x 1 , x 2 ), x ′ = ( x ′ 1 , x ′ 2 ), and q = ( q 1 , q 2 ). F or t wo extended op erator s A ( q ) and B ( q ) with kernels A ( x, x ′ ; q ) and B ( x, x ′ ; q ) we int ro duce the comp osition law ( AB )( x, x ′ ; q ) = Z dx ′′ A ( x, x ′′ ; q ) B ( x ′′ , x ′ ; q ) , (2.1) provided the in tegral exists in terms of distributions. An oper ator A can have an inv erse, A − 1 , in the sense of this co mp o sition: AA − 1 = I and A − 1 A = I , wher e I is the unity op er ator, i.e., the op era tor with kernel I ( x, x ′ ; q ) = δ ( x 1 − x ′ 1 ) δ ( x 2 − x ′ 2 ). A sp ecific s ubcla ss o f extended op e rators is given by the extensions L ( q ) 3 of differential op er a tors a s defined in (1.3), where L ( x, ∂ x ) denotes a differential op erator whose co efficients are smo oth functions of x . Let us a sso ciate to any op erator A ( q ) with kernel A ( x, x ′ ; q ) its “hat”- kernel b A ( x, x ′ ; q ) = e q ( x − x ′ ) A ( x, x ′ ; q ) , (2.2) where q x = q 1 x 1 + q 2 x 2 . F or a differential op e rator L ( q ) this pro cedure is the inv erse of the extensio n introduced in (1.3), i.e., b L ( x, x ′ ; q ) = L ( x, x ′ ). Notice, how ev er, that, since the kernels of the op erator s A are o nly s ub jected to the requirement o f belong ing to the space o f temp ered distributions, in general, the hat-kernel b A still dep ends on q a nd is not necessar ily b ounded. The co njugate A ∗ ( q ) of an op er ator A ( q ) is defined by A ∗ ( x, x ′ ; q ) = A ( x, x ′ ; q ) . (2.3) It is co nv enien t to intro duce the r epresentation of the op era tor A ( q ) in the p -space which is defined b y the F o urier transfo r m A ( p ; q ) = 1 (2 π ) 2 Z dx Z dx ′ e i ( p + q ℜ ) x − i q ℜ x ′ A ( x, x ′ ; q ℑ ) , (2.4) where p = ( p 1 , p 2 ) ∈ R 2 and we intro duced a t wo dimens io nal complex vector q = q ℜ + i q ℑ , q ≡ q ℑ , q ℜ , q ℑ ∈ R 2 . (2.5) The compo sition (2.1) in the p -space beco mes a sort of shifted convolution ( AB )( p ; q ) = Z dp ′ A ( p − p ′ ; q + p ′ ) B ( p ′ ; q ) . (2.6) Then, with these nota tions, the extension of the heat op er ator (1.2) reads L = L 0 − u, u ( x, x ′ ; q ) = u ( x ) δ ( x − x ′ ) , (2.7) where L 0 is the extension in the sense of (1.3) of the differen tial part L 0 of the heat op erator (1.2) and has , in the p -space, thanks to (2.4), kernel given by L 0 ( p ; q ) = ( i q 2 − q 2 1 ) δ ( p ) . (2.8) The main ob ject of our appro ach is the e x tended reso lven t (or reso lven t, for short) M ( q ) of the op erator L ( q ), which is the inv erse of the op era tor L in the sense of comp osition (2.1) (or (2 .6)), i.e., LM = M L = I , (2.9) and can be defined as the solution of the in tegral equations M = M 0 + M 0 uM , M = M 0 + uM M 0 , (2.10) where M 0 is the reso lven t of the zero potential (bar e) o p erator L 0 . Thanks to (2.8) we have M 0 ( p ; q ) = δ ( p ) / ( i q 2 − q 2 1 ) for the kernel of this op era tor in 4 the p -space. In the cas e of a r apidly decaying p otential u ( x ), the existence and uniqueness of the s o lution of the equa tions in (2 .1 0) can b e proved in analo gy with [4]. Notice that for a real p otential u ( x ) bo th L and M a re self-conjugate op erator s in the sense of definition (2.3). Thanks to (2.6) and the explicit form of M 0 ( p ; q ) g iven a b ov e the integral equations (2.10) written in the p -space show that the kernel M ( p ; q ) is singular for q 2 = − i q 2 1 and for q 2 + p 2 = − i ( q 1 + p 1 ) 2 . Therefore, it is natura l to int ro duce the following truncated and reduced v alues of the resolven t ν ( p ; q ) = ( M L 0 )( p ; q )    q 2 = − i q 2 1 , ω ( p ; q ) = ( L 0 M )( p ; q )    q 2 = − i ( q 1 + p 1 ) 2 − p 2 . (2.11) One can s how that the op erato r L ( q ) and its res olven t M ( q ) admit the following bilinear represe nt ations in terms of ν and ω L = ν L 0 ω , M = ν M 0 ω . (2.12) Corresp o ndingly , we call the op era tors ν and ω dressing op er a tors since they “dress” the bare oper ators L 0 and M 0 . Notice that ν ( p ; q ) and ω ( p ; q ) satisfy asymptotics lim q 1 →∞ ν ( p ; q ) = δ ( p ) a nd lim q 1 →∞ ω ( p ; q ) = δ ( p ) and do not depe nd on q 2 , which, if necessary , we make clear by wr iting ν ( p ; q 1 ) and ω ( p ; q 1 ). The dressing op era to rs ν and ω satisfy the eq ua tions Lν = ν L 0 , ω L = L 0 ω , (2.13) and are mutually inv erse ω ν = I , ν ω = I . (2.14) In order to define the Jost solutions by mea ns of the dressing op erator s we int ro duce the following F ourier tra nsforms: χ ( x, q 1 ) = Z dp e − ipx ν ( p ; q 1 ) , ξ ( x, q 1 ) = Z dp e − ipx ω ( p ; q 1 − p 1 ) . (2.15) Then the Jost and the dual Jo s t solutions can b e defined, resp ectively , as Φ( x, k ) = e − i k x 1 − k 2 x 2 χ ( x, k ) , Ψ( x, k ) = e i k x 1 + k 2 x 2 ξ ( x, k ) , (2.16) where w e denoted q 1 as k in order to meet the standard nota tio ns for the sp ectral pa r ameter . Thanks to (2.13), they ob ey the heat equa tion a nd its dual ( − ∂ x 2 + ∂ 2 x 1 − u ( x ))Φ( x, k ) = 0 , ( ∂ x 2 + ∂ 2 x 1 − u ( x ))Ψ( x, k ) = 0 . (2.17) 3 Darb oux transformations via t wist ing op era- tors ζ and η In order to build a tw o-dimensional p o tential des cribing N so litons sup erim- po sed to a generic smoo th bac kgro und, we bypass the recursive pro cedure used 5 in [1 1] and, by using the o p er ator for mulation introduce d in the pre v ious s ec- tion, w e construct directly the final Darboux tra nsformation. The main to ol in accomplishing this result is what we call the twisting op er ators. Let us consider a transfor mation from the op er ator L in (2.7 ) to a new op erator of the same form L ′ = L 0 − u ′ , u ′ ( x, x ′ ; q ) = u ′ ( x ) δ ( x − x ′ ) , (3.1) given by means of an op erator pair ζ , η acco rding to the formulas L ′ ζ = ζ L , η L ′ = L η , (3.2) “twisting” L to L ′ . W e consider the p otential u ( x ) in L to b e a real, smo oth and rapidly decaying function of x and we loo k for self-co njug a te ζ a nd η such that the new po tent ial u ′ ( x ) is also real and smo oth. In a dditio n, we require η to be the left inv erse of ζ , i.e., to ob ey the condition η ζ = I , (3.3) so that L = η L ′ ζ . (3.4) In order to get a new p otential u ′ ( x ) not decaying along some directions of the plane, the tw o o p erator L and L ′ m ust b e related by a transformation more general than a similarity one. Therefor e, w e search for twisting oper ators ha ving a pro duct ζ η , which is not equal to I , but which is given by ζ η = I − P , (3.5) where P is an o rthogona l s elf-conjugate pr o jector , since as a consequence of (3.3) P 2 = P and, thanks to the self-co njuga te pro p erty of ζ and η , P ∗ = P . The t wisting op erato r s ζ and η genera te a new potential u ′ via (3.2), but also the new dressing op erator s ν ′ and ω ′ . In fact, taking into account (2 .13), w e get by (3.2) L ′ ζ ν = ζ L ν = ζ ν L 0 , and by (2.13) ω η L ′ = ω Lη = L 0 ω η . Therefore, the op erato r s ν ′ and ω ′ defined by ν ′ = ζ ν, ω ′ = ω η , (3.6) ob ey equations L ′ ν ′ = ν ′ L 0 , ω ′ L ′ = L 0 ω ′ , (3.7) analogo us to (2.13) sa tisfied by the dressing op erator s ν a nd ω . Note that b eca use o f (3.3) the scalar pro duct o f these dressing op erato rs is equal to I like for the origina l o ne in (2.14): ω ′ ν ′ = I . (3.8) How ev er, in cont rast to ν and ω , these op er a tors do not sa tisfy a completeness relation, since, thanks to (2.1 4) and unlike it, one has by (3.5) ν ′ ω ′ + P = I . (3.9) 6 In or der to obtain a Darb o ux transforma tion w e must sp ecify the a nalyticity prop erties of the kernels ζ ( p ; q ) a nd η ( p ; q ) of the twisting op erato rs with resp ect to the v ariables q . Since, thanks to (2.14) and (3.6), w e hav e ζ = ν ′ ω and η = ν ω ′ , the singularities of these kernels are related to the prop erties o f ν ′ ( p ; q ) and ω ′ ( p ; q ). W e re quire that 1. the kernels ν ′ ( p ; q ) and ω ′ ( p ; q ) ar e indep endent of q 2 and have as ymptotic behavior lim q 1 →∞ ν ′ ( p ; q 1 ) = δ ( p ) a nd lim q 1 →∞ ω ′ ( p ; q 1 ) = δ ( p ), 2. the k ernels ν ′ ( p ; q 1 ) and ω ′ ( p ; q 1 ) hav e (corr esp ondingly , rig ht and left) simple p o les with resp ect to the v ar ia ble q 1 , i.e., ther e exist non trivial limits ν ′ b l ( p ) = lim q 1 → ib l ( q 1 − ib l ) ν ′ ( p ; q 1 ) , (3.10) ω ′ a j ( p ) = lim q 1 →− p 1 + ia j ( q 1 + p 1 − ia j ) ω ′ ( p ; q 1 ) , (3.11) where a 1 , . . . , a N a , b 1 , . . . , b N b are N a + N b parameters , which w e c ho ose to be a ll different, and rea l in order to guar antee reality of the tra ns formed po tential, and, moreov er, that 3. the kernels ζ ( p ; q ) and η ( p ; q ) b esides the discontin uities a t q 1 ℑ = b l and q 1 ℑ = a j , which follow fro m requirement 2, ha ve no additional departures from analy ticit y with resp ect to q 1 . Then, the kernels of the ope r ators ζ , η are given by the following e quations ζ ( p ; q 1 ) = δ ( p ) + N b X l =1 Z dp ′ ν ′ b l ( p − p ′ ) ω ( p ′ ; ib l − p ′ 1 ) q 1 + p ′ 1 − ib l , (3.12) η ( p ; q 1 ) = δ ( p ) + N a X j =1 Z dp ′ ν ( p − p ′ ; ia j ) ω ′ a j ( p ′ ) q 1 + p ′ 1 − ia j , (3.13) where notations (3.1 0) a nd (3.1 1) were used. Let now χ ′ ( x, k ) and ξ ′ ( p ; k ) de- note the functions defined in ter ms of the dressing o p er ators ν ′ and ω ′ in analogy with (2.15) and let χ ′ b l ( x ) and ξ ′ a j ( x ) b e their r esidua (cf. (3.10) and (3.1 1)). Then, it is eas y to show that co ndition (3 .3) is equiv alen t to the set o f equa tions χ ′ b l ( x ) = − i N a X j =1 χ ( x, ia j ) m j l ( x ) , l = 1 , . . . , N b , (3.14) ξ ′ a j ( x ) = i N b X l =1 m j l ( x ) ξ ( x, ib l ) , j = 1 , . . . , N a . (3.15) 7 where m ( x ) is the N a × N b -matrix with elements m j l ( x ) = ( a j − b l ) ∞ Z x 1 dy 1 e ( a j − b l )( x 1 − y 1 ) ξ ′ a j ( y ) χ ′ b l ( y )    y 2 = x 2 , (3.16) which is well defined for bo unded χ ′ b l and ξ ′ a j and ob eys ∂ x 1 m j l ( x ) = ( a j − b l ) m j l ( x ) − ξ ′ a j ( x ) χ ′ b l ( x ) . (3.17) In addition, from (3.1) and (3.2), we get for the transfor med p otential u ′ ( x ) u ′ ( x ) = u ( x ) − 2 ∂ x 1 N a X j =1 N b X l =1 ξ ( x, ib l ) χ ( x, ia j ) m j l ( x ) . (3.18) Then, due to (3.12) and (3.13), in order to define ζ and η we must spe c ify ν ′ b l and ω ′ a j , i.e., χ ′ b l ( x ) and ξ ′ a j ( x ). Thanks to (3.14) and (3.1 5), this means that we have to define the matrix m j l ( x ). Ins erting eq uations (3 .14) a nd (3 .15) int o (3.17) and using (2.16), w e write the fo llowing equation for the matrix b m j l ( x ) = e i ( ℓ ( ia j ) − ℓ ( ib l )) x m j l ( x ) ∂ x 1 b m j l ( x ) = − N a X j ′ =1 N b X l ′ =1 b m j l ′ ( x )Ψ( x, ib l ′ )Φ( x, ia j ′ ) b m j ′ l ( x ) . (3.19 ) Omitting details, we pr esent, here, directly its so lution in the tw o equiv alent forms b m ( x ) = ( E N a + c F ( x )) − 1 c = c ( E N b + F ( x ) c ) − 1 , (3.20) where c is an a rbitrar y real constant N a × N b matrix, E N a and E N b are the unit y N a × N a and N b × N b matrices, co rresp onding ly , and F ( x ) is a N b × N a matrix with elements F lj ( x ) = F ( x, ib l , ia j ), where F ( x, k , k ′ ) = x 1 Z ( k ℑ − k ′ ℑ ) ∞ dx ′ 1 Ψ( x ′ , k )Φ( x ′ , k ′ )    x ′ 2 = x 2 . (3.21) Now the p otential (3.18) also can b e written in the tw o fo rms u ′ ( x ) = u ( x ) − 2 ∂ 2 x 1 ln det( E N b + F c ) = u ( x ) − 2 ∂ 2 x 1 ln det( E N a + c F ) . (3.22) Inserting the ma trix m found ab ov e int o (3.1 4), (3.15) and, then, the o btained equations into (3.1 2), (3.13), w e derive the ex plicit formulae for the dressing op erator s ζ and η . Finally , fro m (3.6 ) using equation (2.16) and its analog for the transformed (primed) Jost and dual Jost solutions we g et for them Φ ′ ( x, k ) = Φ( x, k ) − Φ( x, ia )( E N a + c F ( x )) − 1 c F ( x, ib, k ) = = Φ( x, k ) − Φ( x, ia ) c ( E N b + F ( x ) c ) − 1 F ( x, ib, k ) , (3.23) 8 Ψ ′ ( x, k ) = Ψ( x, k ) − F ( x, k , ia )( E N a + c F ( x )) − 1 c Ψ( x, ib ) = = Ψ( x, k ) − F ( x, k , ia ) c ( E N b + F ( x ) c ) − 1 Ψ( x, ib ) , (3.24) where j = 1 , . . . , N a , l = 1 , . . . , N b , and Φ( x, ia ) = diag { Φ( x, ia j ) } , Ψ( x, ib ) = diag { Ψ( x, ib l ) } , F ( x, k , ia ) = dia g {F ( x, k , ia j ) } , F ( x, ib , k ) = diag {F ( x, ib l , k ) } , and where any of the tw o forms can b e used. It is easy to see that b o th Φ ′ ( x, k ) and Ψ ′ ( x, k ) have p oles at k = ib l and k = ia j , corr esp ondingly , and thanks to (3.2 3) and (3.24) we get that the res idua of these functions are g iven in terms of their v alues in the dual points by means of the relations Φ ′ b l ( x ) = − i N a X j =1 Φ ′ ( x, ia j ) c j l , Ψ ′ a j ( x ) = i N b X l =1 c j l Ψ ′ ( ib l ) , (3.25) as exp ected. One ca n show tha t the potential u ′ ( x ) in (3.22) and the Jost solutions from (3.2 3) and (3.24) coinc ide with those obta ine d in [11], including the cas e N a 6 = N b considered here, which is rec overed by cho osing s o me zero rows or columns in the constant matr ix C introduced in [11]. W e p o stp one to Sec. 5 the discussion on the reg ularity conditions for the po tential given by (3.22). 4 Resolv en t Once the transfor med op era tor L ′ has b een obtained, the op era tors ζ and η can be used to inv estigate its sp ectra l prop er ties and existence and uniqueness of the cor resp onding re s olven t M ′ . Multiplying the first and the second equation in (3.2 ), r esp ectively , from the r ight b y η and fro m the left by ζ , and recalling the definition (3.5) of P , w e get the intert wining relation L ′ = ζ Lη + L ∆ , (4.1) where we intro duce d L ∆ = L ′ P = P L ′ . (4.2) Thu s we lo o k for a resolven t in the form M ′ = ζ M η + M ∆ , (4.3) where ζ M η is determined by the ab ove c o nstruction. Indeed, due to (2.12) and (2.14) we deduce that ζ M η = ν ′ M 0 ω ′ , (4.4) and, then, we get for its kernel in the x -space the following bilinear e xpression in terms of the transfor med Jo st solutions ( ζ M η )( x, x ′ ; q ) = − sgn( x 2 − x ′ 2 ) e − q ( x − x ′ ) 2 π Z dp 1 θ  ( q 2 + p 2 1 − q 2 1 )( x 2 − x ′ 2 )  × 9 × Φ ′ ( x ; p 1 + iq 1 )Ψ ′ ( x ′ ; p 1 + iq 1 ) . (4.5) The second term in (4.3), M ∆ , is to be determined. By using (3.2), (2.9 ), and (3.5), w e deduce that M ′ in (4.3) is the right or left inv erse of L ′ iff M ∆ is a solution, resp ectively , of the first o r the second of the op era tor equations L ′ M ∆ = P , M ∆ L ′ = P . (4.6) Below w e consider the so lv a bility of these equations in the ca se of pure soliton po tentials. In order to co mplete the dis cussion o f the gener ic case, we men tion that an explicit form of the oper ator P defined in (3.5) can be der ived by inserting there ζ and η given in (3.12) and (3.13). Thanks ag ain to (2.16), we get for the hat-kernel o f this op era tor the expression b P ( x, x ′ ; q ) = i δ ( x 2 − x ′ 2 ) N X n =1 θ ( q 1 − α n ) res k = iα n Φ ′ ( x, k )Ψ ′ ( x ′ , k ) . (4.7) Here we introduced the set of parameter s { α 1 , . . . , α N } = { a 1 , . . . , a N a , b 1 . . . , b N b } , N = N a + N b , (4.8) in order to mak e explicit the symmetry pr op erties with resp ect to the pa rameters a j , b l used a bove. This kernel is different from zero only in the interv al where q 1 is b e t ween the lowest and highest v alues of the α m ’s. Indeed, while for q 1 below this in terv al this is obvious b y (4.7), for the v alues of q 1 ab ov e the interv al this follows from the equality N X n =1 res k = iα n Φ ′ ( x, k )Ψ ′ ( x ′ , k ) = 0 , (4.9) that in its turn is a consequence of (3 .25). Now, a ccording to the dis cussion ab ov e, we exp ect that c M ∆ has a structure similar to b P ( x, x ′ ; q ). It is clear that the kernel c M ∆ ( x, x ′ ; q ) = ∓ iθ  ± ( x 2 − x ′ 2 )  N X n =1 θ ( q 1 − α n ) res k = iα n Φ ′ ( x, k )Ψ ′ ( x ′ , k ) , (4.10) ob eys equations L ′ x c M ∆ ( x, x ′ ; q ) = P ( x, x ′ ; q ), L ′ d x ′ c M ∆ ( x, x ′ ; q ) = P ( x, x ′ ; q ) for any sign. B ut it is easy to see that, in general, the kernel M ∆ ( x, x ′ ; q ) con- structed by means of (2.2) is growing at space infinit y and cannot b e the kernel of an extended op erator as defined in Sec. 2. Below we in vestigate this pro blem in detail by means of a s p ecific example. 5 Pure soliton p oten tial and Jost solutions In the ca se u ( x ) ≡ 0 o ne gets the g eneral N -soliton solutio n, where N = max { N a , N b } . The tr ansformed p o tent ial u ′ ( x ) a nd the cor resp onding Jost so - lutions can b e eas ily obta ine d from the general ex pression der ived in Sec. (3). 10 Thu s, the transformed potential is g iven by any of the formulae in (3.22) where now the matr ix F ( x ) has elements F lj ( x ) = e ( a j − b l )( x 1 +( a j + b l ) x 2 ) a j − b l . (5.1) By expanding the determina nts on the r.h.s. of (3.2 2) and by using the Binet– Cauch y formula o ne gets that, in order to hav e a regular potential, it is s uffi- cient that the real matrix c satisfies the following characterization r equirements (equiv a lent to thos e in [20]) Λ  l 1 , l 2 , . . . , l n j 1 , j 2 , . . . , j n  c  j 1 , j 2 , . . . , j n l 1 , l 2 , . . . , l n  ≥ 0 , Λ lj = 1 a j − b l (5.2) for any 1 ≤ n ≤ min { N a , N b } and all minor s, i.e., any choice of 1 ≤ l 1 < l 2 < · · · < l n ≤ N b and 1 ≤ j 1 < j 2 < . . . j n ≤ N a . Here, for the minors we used the standard notation A  j 1 , j 2 , . . . , j n l 1 , l 2 , . . . , l n  = det                 a j 1 l 1 a j 1 l 2 · · · a j 1 l n a j 2 l 1 a j 2 l 2 · · · a j 2 l n · · · · · · · · · · · · a j n l 1 a j n l 2 · · · a j n l n                 . (5.3) F or mulae (3.23) and (3.24), a fter r ather cumbers ome calcula tions that we skip here, can b e mor e simply expres s ed as ratios of deter minants. As in (3.22) o ne can alterna tively , but eq uiv a lently , use determinants of ( N b × N b )- o r ( N a × N a )- matrices. W e us e b e low the last case and we use the symmetric notation (4.8). In thes e terms for the functions χ ′ and ξ ′ related the Jost so lutions by (2 .16) we get χ ′ ( x, k ) = N b Y l =1 ( b l + i k ) − 1 ! τ χ ( x, k ) τ ( x ) , (5.4) ξ ′ ( x ′ , k ) = N b Y l =1 ( b l + i k ) ! τ ξ ( x ′ , k ) τ ( x ′ ) , (5.5) where the τ functions are determinants de fined b y τ χ ( x, k ) = det  A e A ( x ) ( α + i k ) D  , (5 .6) τ ξ ( x ′ , k ) = det  A e A ( x ′ ) ( α + i k ) − 1 D  , (5.7) τ ( x ) = det  A e A ( x ) D  , (5.8) i.e., deter minants of ( N b × N b )-matrices obtained as pro ducts o f square and rectangular matrices. P recisely , A ln = α N b − l n , α + i k = diag { α n + i k } (5.9) e A ( x ) = dia g { e A n ( x ) } , A n ( x ) = α n x 1 + α 2 n x 2 (5.10) 11 D =  d E N b  , d j l = c j l Q N b l ′ =1 , l ′ 6 = l ( b l − b l ′ ) Q N b l ′ =1 ( a j − b l ′ ) , (5 .11) where l = 1 , 2 , . . . , N b , j = 1 , 2 , . . . , N a , n = 1 , 2 , . . . , N . As far as the potential is concer ned, it can b e expres s ed as u ′ ( x ) = − 2 ∂ 2 x 1 log τ ( x ) and, taking in to account (5.8), one recov ers the expres sion obtained in [19 , 20, 22 ] using the tau functions. 6 N -soli ton solutions in the case N b = 1 Now, we restrict o urselves to the sp ecial case of the N -soliton p otential with N b = 1 and N a = N arbitr a ry . In addition, for simplicit y , without los s of generality , w e c ho ose α 1 < α 2 < · · · < α N . In this situatio n the potential u ′ ( x ) has on the x -plane N a ingoing r ays and one o utgoing r ay , as s chematically shown in Fig. 1 . In this ca se τ - functions (5.6)–(5.8) ta ke the simple form τ χ ( x, k ) = N X m =1 f m e A ( x ) ( α m + i k ) , τ ξ ( x, k ) = N X m =1 f m e A ( x ) α m + i k , τ ( x ) = N X m =1 f m e A ( x ) , (6.1) where the f m ’s are o btained from the elements d m 1 ’s of ma tr ix d in (5.11) by a per mutation whic h ta ke into acc o unt the c hosen o r dering of the α m ’s. Condition (5.2) here means that a ll f m > 0. x x 1 2 α 1 α 2 + α 2 α 3 + α 3 α 4 + α 4 α 1 + α α α α 1 2 3 4 q q 1 2 Figure 1: Rays and Polygon for N = 4 W e wan t to show that the extended oper ator L ′ ( q ) cor resp onding to this po tential, as announced in the introductio n, ha s a left self-co njugate annihilato r K ( q ) for q b elo ng ing to a domain of the q -plane. Pr ecisely , let us consider the domain inside the p oly gon inscrib ed in the para b ola q 2 = q 2 1 of the q - plane and with vertices at po ints q 1 = α m , m = 1 , . . . , N , whose characteristic function is given by κ ( q ) = N − 1 X m =1 [ θ ( q 1 − α m +1 ) − θ ( q 1 − α m )] × 12 × [ θ ( q 2 − ( α m + α n ) q 1 + α m α n ) − θ ( q 2 − ( α 1 + α N ) q 1 + α 1 α N ) . (6.2) Notice that this poly gon can b e considered as dua l to the r ay structure (see Fig. 1 for the case N = 4). The main remark needed in order to get the annihilator K ( q ) is that the function ψ ( x ; q ) = κ ( q ) e qx τ ( x ) (6.3) is bo unded in the x -plane. More pre c is ely , a s one can prove after a detailed study , it is exp onentially decaying for x going to infinit y in a ny direction on the plane, for q inside the p o lygon, and, for q on the b or der s of the polyg on, it has directions of nondecaying (but b ounded) b ehavior. Then, since thanks to (2.16), (5.4) a nd (6.1) 1 /τ is pr o p ortional to the v a lue of the dual Jost solutio n Ψ ′ ( x, k ) at k = ib 1 , w e hav e L ′ x ′ d b ψ ( x ′ ; q ) = 0 , (6.4) and, consequently , K L ′ = 0 (6.5) for any o pe rator K with kernel K ( x, x ′ ; q ) = ϕ ( x ; q ) ψ ( x ′ ; q ) (6.6) where ϕ ( x ; q ) is a ny a rbitrar y self-conjugate function, b ounded in x and iden ti- cally zero outside the p olyg on in tro duced ab ove. One c a n verify directly that K P = K . If, in addition, we choose ϕ ( x ; q ) of the form ϕ ( x ; q ) = ( P γ )( x ; q ) with γ ( x ; q ) a self-conjugate function, bo unded in x and iden tically zer o outside the po ly gon, and such that Z dx γ ( x ; q ) ψ ( x, q ) = κ ( q ) , (6.7) one also get P ϕ = ϕ, Z dx ϕ ( x ; q ) ψ ( x, q ) = κ ( q ) . (6.8) Then, it is easy to chec k that K is a self-conjugate pro jector commuting with the pro jector P , i.e., we hav e K ∗ = K , K 2 = K , P K = K P = K . (6.9) The existence of this annihilator pr ov es that the oper ator L ′ cannot hav e right inv erse for q b elonging to the p olyg on defined b y the c haracteris tic function (6.2). In order to pr ov e that, on the co ntrary , the resolven t M ′ ( q ) exists for q o utside the poly gon we go back to the hat-kernel c M ∆ ( x, x ′ ; q ) defined in (4.10) and we study the b oundedness prop erties of M ∆ ( x, x ′ ; q ) = e − q ( x − x ′ ) c M ∆ ( x, x ′ ; q ). Since, thanks to (4.9), this function is iden tically equal to zero outside the strip 13 α 1 < q 1 < α N +1 on the q -plane, we need to investigate its prop erties o nly in this strip. Using (2.16), (5.4), (5.5), and (6.1) we rewrite it explicitly as M ∆ ( x, x ′ ; q ) = ± e − q ( x − x ′ ) θ  ± ( x 2 − x ′ 2 )  τ ( x ) τ ( x ′ ) × × N X m,n =1 f m f n θ ( q 1 − α m )( α m − α n ) e A m ( x )+ A n ( x ) . (6.10) Then, one c a n pr ov e that, choosing the upp er (bo ttom) sign for q ab ov e (b elow) the p olyg on in the strip α 1 < q 1 < α N +1 , the kernel M ∆ ( x, x ′ ; q ) is a bo unded function of x and q and it defines the kernel o f an extended op erato r M ∆ ( q ) according to the definition in Sec. 2. W e conclude that, for q outside the p olyg on, M ∆ ( q ) sa tis fy eq uations (4.6) and, consequently , the r esolven t M ′ ( q ) exists a nd is given by (4.3). Notice that the total Gre en’s function G ( x, x ′ , k ) of the op erator L ′ can be defined (see [11, 1 2]) as the v a lue o f the kernel M ′ ( x, x ′ ; q ) at q 1 = k ℑ , q 2 = k 2 ℑ − k 2 ℜ for a complex s pe ctral par ameter k = k ℜ + i k ℑ . Thes e v alues of q lie outside the parab ola q 2 = q 2 1 , th us o utside the p oly gon and touch it at the vertices only . Then the Green’s function exists for a ny k but it is singular at the p o ints k = iα m corres p o nding to the vertices of the polyg on. These are the only singularities of the Gr een’s function, since the dis c ontin uities of the fir st term in (4.3) at q 1 = α m (see (4.5)) are comp ensated (outside the p olygon) by the discontin uities of the second term, as follows from (4.10). Finally , it is worth noting that the case N a = 1 and N b arbitrar y can b e handled in a analog o us wa y , having the op er a tor L ′ ( q ) a rig ht instead of a left annihilator for q inside the p o lygon, and that in the ca s e N a = N b = 1 the po lygon re duces to the s egment o f the line q 2 = ( α 1 + α 2 ) q 1 − α 1 α 2 with end po ints q 1 = α 1 and q 1 = α 2 . In this case one recov ers the results obtained in [11]. Ac kno wledgmen ts This work is suppo rted in par t by the gr ant RFBR– CE # 06-01 -9205 7, g rant NWO–RFBR # 047.011 .2 004.0 59, gra nt RFBR # 08-01 -0050 1, Scien tific Sc hoo ls 795.2 008.1 , by the P rogr a m of RAS ”Ma the- matical Methods o f the Nonlinear Dynamics”, by INFN and by Conso rtium E.I.N.S.T.E.IN. AKP thanks Department of P hysics of the University of Salento (Lecce) for kind hospitality . References [1] V. S. Dryuma, Sov. JETP Lett. 19 387– 388 (19 74). [2] V. E. Zak harov and A. B. Shaba t, F unc. Anal. Appl. 8 226– 235 (19 74). [3] M. J. 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