Journal of Mathematics

Volume 2013 (2013), Article ID 645752, 10 pages

http://dx.doi.org/10.1155/2013/645752

Research Article

Wronskian Addition Formula and Darboux-Pöschl-Teller Potentials

Pierre Gaillard and Vladimir Matveev

Institut de Mathématiques de Bourgogne, UMR 5584 CNRS, Université de Bourgogne, Faculté des Sciences Mirande, 9 Avenue Alain Savary, BP 47870, 21078 Dijon Cedex, France

Received 30 November 2012; Accepted 25 February 2013

Academic Editor: Alberto Enciso

Copyright © 2013 Pierre Gaillard and Vladimir 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.

Abstract.

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.

1. 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 ).

First in exploring the fact the functions defined by

represent particular solutions of (1) with, respectively, and , 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 as a new independent variable, he reduced (1) to a Gauss hypergeometric equation and expressed explicitly the solutions in terms of Gauss hypergeometric functions:

where is a solution of Gauss hypergeometric equation:

In particular for 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 and the function by the formulas

Then we have the following result (Crum's Th. [6]).

Theorem 1. represents a general solution of (2):

with 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 so that the potential 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 , 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 . The choice of was clear from [10] but at every point we have the generic solution , where plays no role in computing ). 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 [14–16], 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].

2. Wronskian Addition Formula for Sine Functions

Let and be some nonnegative integers, , and 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 , 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 to be the determinant obtained from replacing functions by the functions of the same argument:

Then the following hyperbolic versions of (23) and (24) hold.

Theorem 3. Consider the following:

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.

Finally, replacing in the latter formulas and functions by and functions of the same arguments and using the notation for their wronskian we get the formulas

Theorem 5. Consider the following:

Proof. All the results given in (24), (23), (27), (30), (33), and (32) can be proved in the same way; we present, for example, the case ((24), (23)) where the functions are defined by (19). Then all the others results can be obtained by changes of variables; precisely, wet get (27) from ((24), (23)) by the change , (30) from ((24), (23)) by the change , and ((33), (32)) from ((24), (23)) by the change . It is sufficient to prove the first result ((24), (23)).

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 . 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 :

For brevity, we replace by in the following.

We calculate the wronskian .

For this, we use the expansion

Therefore can be reduced to

by replacing each term in the expansion of , formula (38) by their first terms, that is, , .

We can also factorize in each column the term for , so we can drop out the factor ; can be written as

Then we use the two following statements.

Proposition 6. For any smooth functions , the wronskian verify

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 defined by :

Then we can factorize in the preceding determinant to get

Here, we make the change of variable defined by . This leads us to appearance of the extra factor . Therefore the wronskian can be written as

The wronskian 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 , we get finally

In a second step, we establish a recursion relation between the wronskians and (here ).

We can factorize in to get

It can be written as

In a shortened way we get the recursion relation

We can evaluate the quotient in function of ; namely,

So the th row of the column of the determinant can be reduced to because other terms are combinations of the preceding columns, except for or ; in this case the constant terms are eliminated by the derivations (e.g., for , , and the derivations eliminate the constant term).

Then each term in the determinant can be replaced by , which can be rewritten as for .

Let be . It is obvious that , for . So the sequence can be replaced by . We get the equality

We can drop out the constants appearing in each column to get

We can make these transformations for each integer such that , so we get the recursion relation

We apply the recursion relation (54) times and we get the expression

To compute it is now sufficient to replace given by (47) in defined by (55). So we get

We can rearrange these terms and get

It is easy to verify the identity

Finally, we get

Taking into account the notation , we recover exactly the formula (24).

In the case where , can be computed with the obvious relation . So we have

Thus, one gets the results of Theorem 2, and consequently those of Theorems 3, 4 and 5.

Example 8. The following example of wronskian illustrates well the behavior of the coefficients :

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.

3. Applications of the Addition Formula

3.1. Wronskian Representation of the DPT Potentials and KdV Equation

Below we need the following two propositions.

Let be any potential and be any solutions of the 1D Schrödinger (Sturm-Liouville) equation

corresponding to .

Proposition 9. The solution of the Schrödinger equation

is given by the formula

In the case , 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 be any linearly independent solutions of (65) different from . Then the following statement holds.

Proposition 10 (see [8, 9] for the proof and further details). The RHS of (64) satisfy the following PDE:

where the coefficients can be explicitly written in terms of and . In particular, assuming that const and we obtain the following formulas for :

The RHS of (64) represents the action of the differential operator of order on the function :

can be also represented in a form of a product of the first-order differential operators in many different ways. Let be any permutation of the numbers . To each permutation corresponds the following factorization of :

Assume now that , , and , where are defined in (19). Combining formula (24) with Proposition 9 we get the following result. The function corresponding to the case solves the Schrödinger equation with singular periodic potential :

where

Replacing by we obtain the potential exponentially decreasing at infinity:

General solution of the related Shrödinger equation (71) is obtained from (64) by choosing . Quite similarly we have

where was defined in (2) and the solution of the related Schrödinger equation can be obtained by choosing in (64).

It is clear as well that

and the solutions of the related Schrödinger equation are obtained from (64) by choosing . In the case , is a nonsingular reflectionless potential and the discrete negative eigenvalues of the related Schrödinger operator on the real line are .

The integrability of (71) with potential (72) with was the discovered first time by Darboux in 1882 and rediscovered for its hyperbolic version (73) by Pöschl and Teller in 1933. Therefore, it is reasonable to call (72), (73), (74), and (75) the Darboux-Pöschl-Teller or shortly DPT potentials.

The solution of (71) found by Darboux [10] corresponds to particular factorization (70) with ; 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 represent the particular solutions of the Schrödinger equation (71) with potential corresponding to 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 th equation of the KdV hierarchy can be obtained as a compatibility condition of (62) and the following evolution equation:

In particular for , , and compatibility of (62) and (78) implies that 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 of th KdV equation the sequence of the common solutions of (62) and (78) generates the new solution of the same equation:

In particular, in the case

the RHS of (80) is the solution of the th KdV equation with the initial data equal to the DPT potential (72). Therefore, we obtain a very simple description of the isospectral deformations of the DPT potentials generated by the KdV hierarchy. In particular the solution of (79) with the initial condition (72) is given by the formula (80) with and

Similar formulas for the action of KdV and higher KdV flows on the potentials (74), (73), and (75) are also obvious from the previous discussion. For instance, the solutions of the KdV equation (79) with the initial condition (75), with , is given by the formula

where the functions are defined as follows:

3.2. Differential Identities

Formulas (23) and (24) lead to some nontrivial identities. Let be a differential operator defined by the formula

Let be the following determinant: , , with .

Clearly, we have

The determinant can be easily computed and further obvious reduction to a Vandermonde determinant (the same expression (up to a different constant normalization factor) as the RHS of (33) recently appeared [20, (Th. A, Th. 5.1)] as a density of probability measure describing the asymptotic distribution of Frobenius roots on the -dimensional abelian varieties over finite fields when :

Now, substituting this expression with in (86) and taking into account the formula (24) for , we get the following identity:

It is obvious that the wronskian

is proportional to :

This follows immediately from the expansion of by the elements of its first column. Assuming that , and introducing the determinant,

we obviously have the following formula:

From the other side, can be calculated explicitly and the reduction to a Vandermonde determinant is obvious:

Comparing these two formulas with explicit expression for , following immediately from (90), we obtain the following identity:

Similar but more complicated identities might be obtained along the same lines, comparing (24) with explicit expression for the determinant , where the functions were defined in (19). For instance for ,  and   this leads to the following identity:

In a more general formulation, using the same arguments as before, we get the identity

with defined by (25).

4. Additional Comments

(i)There is some similarity of (24), especially of the case with some addition formulas for elliptic functions obtained by Kiepert, Brioschi, Frobenius, and Stickelberger [21, 22] although we were not able to find a direct connection with their results. In particular, we do not know the compact expression for , where is a Weierstrass sigma function. (ii)Taking in the formula (64) for the solutions of (71) we see that the related function is where is a polynomial of . In general the potentials allowing the solutions of this form are called Bargmann potentials. All Bargmann potentials might be considered as a degenerate case of the hyperelliptic finite gap periodic potentials [23]. In particular the DPT potentials (72) might be obtained from the general m-gap hyper elliptic potentials by closing the gaps to the points . (iii) Darboux [2] also found the integrable elliptic generalization of the DPT potentials also representing the 4-parametric generalization of the Lamé equation. This elliptic Darboux potential has the following form:

 In (97) , and are usual Jacobi elliptic functions of modulus and are arbitrary nonnegative integers. In the case , this equation reduces to Lamé equation first integrated by Hermite. The potentials of the form (97) were rediscovered recently by Trebich and Verdier [24] as a special reduction of the general formula [23, 25, 26] describing all finite-gap periodic or quasi periodic potentials together with explicit solutions of the related Schrödinger equation (Baker-Akhiezer functions). More general connections between the finite gap potentials and Fuchsian equations were recently discussed by Smirnov [27]. In the limit when one of the periods of the elliptic functions tends to infinity the Darboux-Verdier-Trebich potential reduces to the Darboux-Pöschl-Teller potential. (iv) The DPT potentials and the Darboux-Verdier-Trebich potentials recently found some applications to the algebraic Bethe Ansatz construction for trigonometric case of the quantum Calogero-Moser multi particle system [28]. (v) Comparison of the results of Section 1 with [29–31] shows that the DPT trigonometric potentials produce the subfamily of two dimensional Huygens potentials via a formula

  since the generic 2D Huygens potentials in polar coordinates are described by the formula (in [29–31] the formula below was written in a slightly different but strictly equivalent form, using the wronskian of cosine functions):

Therefore, taking , where are defined in (19) and taking , we get the aforementioned link between the DPT potentials and Huygens potentials.

Acknowledgment

The authors wish to thank Dr. Y. Grandati for sending his preprint [13].

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