Wronskian Addition Formula and Darboux-Pöschl-Teller Potentials

Received 30 November 2012; Accepted 25 February 2013 Academic Editor: Alberto Enciso Copyright © 2013 P. Gaillard and V. Matveev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For the famous Darboux-Pöschl-Teller equation, we present new wronskian representation both for the potential and the related eigenfunctions. The simplest application of this new formula is the explicit description of dynamics of the DPT potentials and the action of the KdV hierarchy. The key point of the proof is some evaluation formulas for special wronskian determinant.


Introduction
In 1882 Darboux [1] proved the integrability of the Sturm-Liouville equation (Darboux also considered and solved an equation which is a generalisation in elliptic functions of (1): see [2].A hundred years later the integrability of the same elliptic model was obtained by Treibich and Verdier using completely different methods: see [3,4]): Later in 1933, in the frame of the study of the quantum theory of two atomic molecules, Pöschl and Teller [5] (creator of the hydrogenous bomb) rediscover the hyperbolic version of (1) and proved independently of Darboux the integrability of the equation obtained from (1) by the transformation  → .A lot of authors studied the Schrödinger equation with the same potentials (mainly called Pöschl-Teller potentials in the literature of quantum mechanics, sometimes called the Nantanzon potentials or Morse-Rosen potentials).In the following, these potentials will be called the Darboux-Pöschl-Teller potentials or most simply the DPT potentials.Darboux gave two methods to solve (1) (he considered the case  = 1).
First in exploring the fact the functions defined by  , () = cos +1 () sin +1 () , ỹ, () = cos − () sin − () (3) represent particular solutions of (1) with, respectively,  =  2 ( +  + 2) 2 and  =  2 ( + ) 2 , precisely using them as generator functions  of the transformation defined by that we call today Darboux transformation; he obtained the general solution of (1) in terms of elementary functions (for  solution of (2)): with Then taking sin 2  as a new independent variable, he reduced (1) to a Gauss hypergeometric equation and expressed explicitly the solutions in terms of Gauss hypergeometric functions:  = sin  () cos  ()  (sin 2 ()) , (7) where  is a solution of Gauss hypergeometric equation: In particular for 2  1 defined by with ) is a solution of (1).We present here new representations for the Darboux-Pöschl-Teller potentials and for eigenfunctions of the Schrödinger operators corresponding in terms of wronskians.To explain their nature, we must recall results obtained by Crum [6] in 1955.As usual the wronskian of  functions is defined by the formula We denoted by   ,  some independent solutions of (2) associated, respectively, with   and ; that is, We define then the new potential ũ and the function Ψ by the formulas Then we have the following result (Crum's Th. [6]).
Theorem 1. Ψ represents a general solution of (2): with ũ defined by (13), if  is a solution of (2) associated with  and .
Then we can ask the following natural question: how to choose the functions   solution of (2) in the case  = 0 so that the potential ũ defined by ( 13) is equal to DPT potential The choice of the introduction of parameter  makes it possible to change from trigonometric model to hyperbolic model and vice versa without any difficulty: it is sufficient for this to change  in .
We first compute a certain type of wronskians which gives a sort of addition formula.
Then we get a representation of the solutions of (2) in terms of functions   if we replace in the general solution of ( 2) with Darboux's formula defined by ( 5) represents then a direct consequence of ( 13) and ( 14) and the formalism of Darboux's transformations methods developed by Matveev [7,8] (see also [9]).One of the advantages of this potential representation with wronskians (14) is that it makes it possible to describe the action of the flow of the KdV equation and its hierarchy on these DPT potentials taken as initial data; by replacing the arguments of functions   () by convenient arguments   ( + 4), we get the solutions of the hierarchy of the KdV equations with DPT initial data without difficulty as in [7,8].
Initial motivation of this paper was the following question.How to select the solutions   of the free Shrödinger equation −   =     in order to get the DPT potential from Crum formula (63) with V = 0.The choice of   was clear from [10] but at every point   we have the generic solution   sin(√   +   ), where   plays no role in computing  2  log ( 1 , . . .,   ).In general, there is no compact expression for the wronskian provided that the phases are chosen arbitrarily.It was also clear that some selection of phases   leading to DPT potential via (63) should exist but the fact that trivial phases provide the right answer a priori was not obvious and it was necessary to prove (24) (at least modulo constant normalization factor   ) in order to confirm it.
This result first appeared in [11], and its abbreviated version makes a part of [12].Here we give a more detailed and partially new proof of these results.Another different proof was very recently obtained in [13].
We can compare this result with these obtained recently for the discrete version of the Schrodinger equation by the authors [14][15][16], respectively, called DDPT-I and DDPT-II models.Like in the continuous version where the solutions can be expressed with wronskian determinant, the solutions of these DDPT models can be expressed with the Casorati determinants [17,18].

Wronskian Addition Formula for Sine Functions
Let  and  be some nonnegative integers,  =  + ,  and  0 be some real parameters, and   are the functions defined as follows: We use below the standard notation for wronskian determinant of any  functions and the short notation   =   (, ,  0 ) = ( 1 , . . .,   ), where   is defined in (19).
With these notations we have the following formulas (product of factorials in this formulas can be also written as a product of powers): Theorem 2. Consider the following: We note Ŵ to be the determinant obtained from   replacing sine functions   by the sinh functions   of the same argument: Then the following hyperbolic versions of ( 23) and (24) hold.
where   is the same as before.
Replacing (19) by the system of functions, we get for their wronskian the following analogue of ( 23) and (24).
Theorem 4. Consider the following: where   has the same meaning as before.
In the following, we use the notation  =   .We define the functions   by the formula Then it is easy to see that We define the sequence of wronskians  −, of order  −  by the formulas It is obvious that   () =  0 (2).We consider the functions equal to   defined in (19).
The structure of the proof of the first result (( 24), ( 23)) is the following.
We start with the proof of ( 23).Next we establish the recursion relation for each  such that 0 ≤  ≤  −  − 1: For brevity, we replace  −  0 by  in the following.
For this, we use the expansion Therefore   can be reduced to by replacing each term in the expansion of sin 2, formula (38) by their first terms, that is, sin 2(2 cos 2) −1 , 2 ≤  ≤ .
first proved by Frobenius [19].The second one is a well-known formula of change of independent variable in a wronskian.
Proposition 7.For  = () and f defined by   () = f (): Then we can factorize (sin (2))  in the preceding determinant   to get Here, we make the change of variable defined by  = () = cos 2.This leads us to appearance of the extra factor (−2 sin 2) (−1)/2 .Therefore the wronskian   can be written as × (sin (2))   (1, , . . .,  −1 ) . (44) The wronskian (1, , . . .,  −1 ) has the upper triangular form and thus is equal to the product of its diagonal elements; in other words, So   can be written as Expanding sin 2, we get finally In a second step, we establish a recursion relation between the wronskians   and  −1, (here  > ).

Journal of Mathematics
In the case where  = 0,  0 can be computed with the obvious relation  0 (2) =   ().So we have Thus, one gets the results of Theorem 2, and consequently those of Theorems 3, 4 and 5.
Formulas ( 24) represent a very strong reduction with respect to combinatorial definition of the related determinant representing a trigonometric polynomial containing ! terms.It is reduced by (24) to a single monomial term.
Proposition 9.The solution  of the Schrödinger equation is given by the formula In the case  = 1, this proposition was found and proved by Darboux [10], and in general case by Crum [6].Remarkably, their result can be extended to the partial differential equations and their nonabelian and lattice versions, [7] which finds many applications in modern theory of solitons [8,9].Below we need this extension only for the particular case of the scalar evolution PDE of the following form: Let  1 ,  2 , . . .,   be any  linearly independent solutions of (65) different from .Then the following statement holds.
The RHS of (64) satisfy the following PDE: where the coefficients   (, ) can be explicitly written in terms of V  and   (, ).In particular, assuming that V  = const and V −1 = 0 we obtain the following formulas for   ,  −1 ,  −2 : The RHS of (64 Assume now that  = (1, 2, ..., ), V = 0, and   =   , where   are defined in (19).Combining formula (24) with Proposition 9 we get the following result.The function  corresponding to the case   =   , V = 0 solves the Schrödinger equation with singular periodic potential  , : where  , = − 2 The solution of (71) found by Darboux [10] corresponds to particular factorization (70) with  = (1, 2, . . ., ); that is, the related function  can be written as The related functions   now can be computed combining (24) and (70), where, of course, the constant factor in the RHS of ( 24) can be ignored: Original derivation [10] by Darboux of (77) was based on the observation that the functions sin  ( −  0 ) cos  ( −  0 ) represent the particular solutions of the Schrödinger equation (71) with potential  −1,−1 corresponding to  =  2 ( + ) 2 and was not using general Crum's formulas as we did.
The advantage of the Crum representation for the DPT potentials is that it allows to describe immediately the action of KdV and higher flows of the KdV hierarchy on those potentials taken as initial data by using Proposition 10.The jth equation of the KdV hierarchy can be obtained as a compatibility condition of (62) and the following evolution equation: In particular for  = 1,   = −4, and V 0 = 3V  compatibility of (62) and (78) implies that V satisfies the KdV equation In general the choice of   is arbitrary: it fixes normalization of the KdV hierarchy.Now according to Proposition 10 for any given solution V(,   ) of th KdV equation the sequence  1 , . . .,   of the common solutions of (62) and ( 78 where the functions   (, ) are defined as follows: (, ) := cosh   ( −  0 − 4 =  cot  ( −  0 ) ,  ≤  − ,  −+ = ( −  + )  cot  ( −  0 ) −  tan  ( −  0 ) ,  = 1, . . ., .