# Wronskian Addition Formula and Darboux-Pöschl-Teller Potentials

Authors: Pierre Gaillard; Vladimir Matveev
Journal: Journal of Mathematics
 (2013)

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## 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.


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## Body

## 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]):
(1)-y′′+(m(m+1)α2sin2αx+n(n+1)α2cos2αx)y=λy.
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 α→iα.  A lot of authors studied the Schrödinger equation
(2)-∂x2f+u(x)f=λf,
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 α=1).First in exploring the fact the functions defined by(3)ym,n(x)=cosn+1(αx)sinm+1(αx),y~m,n(x)=cos-n(αx)sin-m(αx)
represent particular solutions of (1) with, respectively, λ=α2(m+n+2)2 and λ=α2(m+n)2, precisely using them as generator functions φ of the transformation defined by
(4)∂x-φxφI,
that we call today Darboux transformation; he obtained the general solution of (1) in terms of elementary functions (for f solution of (2)):
(5)y=Dm·Dm-1⋯D1f,Dk=∂x-σkI,
with
(6)σj=jαcotα(x-x0),j≤m-n,σm-n+k=(m-n+k)αcotα(x-x0)-kαtanα(x-x0),k=1,…,n.
Then taking sin2αx as a new independent variable, he reduced (1) to a Gauss hypergeometric equation and expressed explicitly the solutions in terms of Gauss hypergeometric functions:
(7)y=sinm(αx)cosn(αx)g(sin2(αx)),
where g is a solution of Gauss hypergeometric equation:
(8)x(1-x)y′′+(γ-(δ+β+1)x)y′-δβy=0.
In particular for 2F1 defined by
(9)2F1(δ,β,γ,z)=∑k=0+∞‍(δ)k(β)k(γ)k(1)kzk,
with
(10)(a)0=1,(a)k=a(a+1)⋯(a+k-1),k≠0,γ=m+12,δ=m+n2+12αλ,β=m+n2-12αλ,y=sinm(δx)cosn(αx)2F1(α,β,γ,sin2αx) 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 m functions is defined by the formula
(11)W(f1,f2,…,fm):=det∥∂xj-1fk(x)∥;j,k=1,…,m.
We denoted by φj, f some independent solutions of (2) associated, respectively, with λj and λ; that is,
(12)-φj′′+uφj=λjφj,-f′′+uf=λf.
We define then the new potential u~ and the function Ψ by the formulas
(13)u~=u-2∂x2[lnW(φ1,…,φm)],(14)Ψ=W(φ1,…,φm,f)W(φ1,…,φm).
Then we have the following result (Crum's Th. [6]).Theorem 1.
Ψ represents a general solution of (2):
(15)-∂x2Ψ+u~Ψ=λΨ,
with u~ defined by (13), if f is a solution of (2) associated with u and λ.Then we can ask the following natural question: how to choose the functionsφj solution of (2) in the case u=0 so that the potential u~ defined by (13) is equal to DPT potential
(16)(m(m+1)α2sin2αx+n(n+1)α2cos2αx).
The choice of the introduction of parameter α makes it possible to change from trigonometric model to hyperbolic model and vice versa without any difficulty: it is sufficient for this to change α in iα.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 φi if we replace in the general solution
(17)y~m,n=W(φ1,…,φm,f)W(φ1,…,φm)
of (2) with
(18)u=-2∂x2[lnW(φ1,…,φm)],φi′′=λiφi.
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 φp(x) by convenient arguments φp(x+4λt), 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 solutionsfj of the free Shrödinger equation -fj′′=Ejfj in order to get the DPT potential from Crum formula (63) with v=0. The choice of Ej was clear from [10] but at every point Ej we have the generic solution Ajsin(Ejx+xj), where Aj plays no role in computing ∂x2logW(f1,…,fm). 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 xj 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 cmn) 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

Letn and d be some nonnegative integers, m=n+d, α and x0 be some real parameters, and φp are the functions defined as follows:
(19)φp(x)=sinαap(x-x0),ap=p,ifd≠0,1≤p≤m-n,(20)aj=aj(m,n)=n-m+2j,ifn≠0,m-n+1≤j≤m.
We use below the standard notation
(21)W(f1,f2,…,fm):=det∥∂xj-1fk(x)∥;j,k=1,…,m,
for wronskian determinant of any m functions and the short notation Wmn=Wmn(α,x,x0)=W(φ1,…,φm), where φp 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):(22)1!·2!⋯k!=1k·2k-1·3k-2⋯k.Theorem 2.
Consider the following:(23)Wm0=(-2α)m(m-1)/2∏k=0m-1‍k!·sinm(m+1)/2α(x-x0),(24)Wmn=cmn·sinm(m+1)/2α(x-x0)cosn(n+1)/2α(x-x0),(25)cmn=(-2α)m(m-1)/2·2n2∏p=0d-1‍p!∏k=1n‍(d+2k-1)!(2k-1)!(k-1)!,ifm≠n,m≥2,cmm=(-2α)m(m-1)/2·2m2∏k=1m-1‍k!,m≥2.
We note W^mn to be the determinant obtained from Wmn replacing sine functions φj by the sinh functions gj of the same argument:
(26)W^mn=W(g1,…,gm),gp(x)=sinhαap(x-x0),ap=p,ifd≠0,1≤p≤m-n;am-n+j=m-n+2j,ifn≠0,1≤j≤n.
Then the following hyperbolic versions of (23) and (24) hold.Theorem 3.
Consider the following:(27)W^m0=(2α)m(m-1)/2∏k=0m-1‍k!·sinhm(m+1)/2α(x-x0),W^mn=|cmn|·sinhm(m+1)/2α(x-x0)×coshn(n+1)/2α(x-x0),m≥2,
wherecmn is the same as before.Replacing (19) by the system of functions,
(28)hp(x)=cosαap(x-x0),ap=p,ifpisodd,hp(x)=sinαap(x-x0),ap=p,ifpiseven,ifd≠0,1≤p≤m-n,am-n+j=m-n+2j,ifn≠0,1≤j≤n,hm-n+j(x)=cosαam-n+j(x-x0),ifm-nisodd,hm-n+j(x)=sinαam-n+j(x-x0),ifm-niseven;
we get for their wronskian
(29)wmn=W(h1,…,hm)
the following analogue of (23) and (24).Theorem 4.
Consider the following:(30)wm0=(-1)(m-1)m2/2(2α)m(m-1)/2∏k=0m-1‍k!·cosm(m+1)/2α(x-x0),wmn=(-1)(m-1)((m-n)2+n)/2|cmn|·cosm(m+1)/2α(x-x0)×sinn(n+1)/2α(x-x0),m≥2,
where cmn has the same meaning as before.Finally, replacing in the latter formulassin and cos functions hj by sinh and cosh functions lj of the same arguments and using the notation w^mn for their wronskian we get the formulas
(31)w^mn=W(l1…,lm),lp(x)=coshαap(x-x0),ap=p,ifpisodd,lp(x)=sinhαap(x-x0),ap=p,ifpiseven,ifd≠0,1≤p≤m-n,am-n+j=m-n+2j,ifn≠0,1≤j≤n,lm-n+j(x)=coshαam-n+j(x-x0),ifm-nisodd,lm-n+j(x)=sinhαam-n+j(x-x0),ifm-niseven.Theorem 5.
Consider the following:(32)w^m0=(2α)m(m-1)/2∏k=0m-1‍k!·coshm(m+1)/2α(x-x0),(33)w^mn=|cmn|·coshm(m+1)/2α(x-x0)×sinhn(n+1)/2α(x-x0),m≥2.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 (x-x0)→i(x-x0), (30) from ((24), (23)) by the change (x-x0)→(π/2α)+(x-x0), and ((33), (32)) from ((24), (23)) by the change (x-x0)→(π/2α)+i(x-x0). It is sufficient to prove the first result ((24), (23)).
In the following, we use the notationq=eiα. We define the functions fj by the formula
(34)fj=sin(αjx),withj∈ℤ.
Then it is easy to see that
(35)Wmn=W(f1,f2,…,fm-n,fm-n+2,fm-n+4,…,fm+n).
We define the sequence of wronskians Wm-k,n of order m-k by the formulas
(36)Wm-1,n=W(f1,f2,…,fm-n-1,fm-n+1,fm-n+3,…,fm+n-1),Wm-k,n=W(f1,f2,…,fm-n-k,fm-n-k+2,fm-n-k+4,…,fm-k+n),Wn,n=W(f2,f4,…,f2n).
It is obvious that Wnn(α)=Wn0(2α). We consider the functions equal to φj 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 k such that 0≤k≤m-n-1:
(37)Wm-k,n=f1m-k(-2α)m-k-1(m-k-1-n)!×∏j=1n‍(m-k-1-n+2j)·Wm-k-1,n.
For brevity, we replace x-x0 by x in the following.
We calculate the wronskianWnn=W(f2,…,f2n).
For this, we use the expansion(38)sinpx=sinx∑k=0[(p-1)/2]‍(-1)kCp-k-1k·(2cosx)p-1-2k=sinx[(2cosx)p-1-(p-2)(2cosx)p-3+⋯].
Therefore Wnn can be reduced to
(39)Wnn=W(sin2αx,2sin2αxcos2αx,…,sin2αx(cos2αx)n-1),
by replacing each term in the expansion of sin2pαx, formula (38) by their first terms, that is, sin2αx(2cos2αx)p-1, 2≤p≤n.
We can also factorize in each columnp the term 2p-1 for 2≤p≤n, so we can drop out the factor 2n(n-1)/2; Wnn can be written as
(40)Wnn=2n(n-1)/2W×(sin2αx,sin2αxcos2αx,…,sin2αx(cos2αx)n-1).
Then we use the two following statements.
Proposition 6.For any smooth functionsf,φ1,…,φn,  the wronskianW(f·φ1,…,f·φn)verify(41)W(f·φ1,…,f·φn)=(f(x))nW(φ1,…,φn),first proved by Frobenius [19]. The second one is a well-known formula of change of independent variable in a wronskian.
Proposition 7.Fort=φ(x)andf~jdefined byfj(x)=f~j(t):
(42)W(f1,…,fn)=(φ′(x))n(n-1)/2·W(f~1,f~2,…,f~n).
Then we can factorize (sin(2αx))n in the preceding determinant Wn to get
(43)Wnn=2n(n-1)/2(sin(2αx))nW(1,cos2αx,…,(cos2αx)n-1).
Here, we make the change of variable defined by t=φ(x)=cos2αx. This leads us to appearance of the extra factor (-2αsin2αx)n(n-1)/2. Therefore the wronskian Wn can be written as
(44)Wnn=2n(n-1)/2(-2αsin2αx)n(n-1)/2×(sin(2αx))nW(1,t,…,tn-1).
The wronskian W(1,t,…,tn-1) has the upper triangular form and thus is equal to the product of its diagonal elements; in other words,
(45)W(1,t,…,tn-1)=∏j=1n-1‍j!.
So Wn can be written as
(46)Wnn=2n(n-1)(-α)n(n-1)/2∏j=1n-1‍j!·(sin2αx)n(n+1)/2.
Expanding sin2αx, we get finally
(47)Wnn=2n(n-1)+(n(n+1)/2)(-α)n(n-1)/2×∏j=1n-1‍j!·(sinαx)n(n+1)/2(cosαx)n(n+1)/2.
In a second step, we establish a recursion relation between the wronskians Wmn and Wm-1,n (here m>n).
We can factorizeφ1m in Wmn to get
(48)Wmn=φ1m·|1φ2φ1⋯φmφ1φ1(1)φ1φ2(1)φ1⋯φm(1)φ1⋮⋮⋮⋮φ1(m-1)φ1φ2(m-1)φ1⋯φm(m-1)φ1|.
It can be written as
(49)Wmn=φ1m·|(φ2φ1)(1)(φ3φ1)(1)⋯(φmφ1)(1)⋮⋮⋮⋮(φ2φ1)(m-1)(φ3φ1)(m-1)⋯(φmφ1)(m-1)|.
In a shortened way we get the recursion relation
(50)Wmn=φ1m·W((φ2φ1)(1),…,(φmφ1)(1)).
We can evaluate the quotient φp/φ1 in function of q; namely,
(51)φpφ1=qap(x)-q-ap(x)q(x)-q-(x)=q(ap-1)(x)+q(ap-3)(x)+⋯+q-(ap-1)(x).
So the jth row of the column p of the determinant W((φ2/φ1)(1),…,(φm/φ1)(1)) can be reduced to (q(ap-1)(x)+q-(ap-1)(x))(j+1) because other terms are combinations of the preceding columns, except for p=1 or p=2; in this case the constant terms are eliminated by the derivations (e.g., for a3=3, (φ3/φ1)=q2x+1+q-2x, and the derivations eliminate the constant term).
Then each term(φp/φ1)(1) in the determinant W((φ2/φ1)(1),…,(φm/φ1)(1)) can be replaced by (q(ap-1)(x)+q-(ap-1)(x))(1), which can be rewritten as αi(ap-1)(q(ap-1)(x)-q-(ap-1)(x))=-2α(ap-1)sin((ap-1)x) for 2≤p≤m.
LetΦp be φp/φ1. It is obvious that ap(m,n)-1=ap-1(m-1,n), for 2≤p≤m. So the sequence (Φ2,…,Φm-n,Φm-n+1,…,Φm) can be replaced by (f1,…fm-n-1,fm-n+1,…,fm+n-1). We get the equality
(52)Wmn=f1mW×(-2αf1,…,-2α(m-n-1)fm-n-1,-2α(m-n+1)fm-n+1,…,-2α(m+n-1)fm+n-1).
We can drop out the constants appearing in each column to get
(53)Wmn=f1m(-2α)m-1(m-1-n)!×∏j=1n‍(m-1-n+2j)·Wm-1,n.
We can make these transformations for each integer k such that 0≤k≤m-n-1, so we get the recursion relation
(54)Wm-k,n=f1m-k(-2α)m-k-1(m-k-1-n)!×∏j=1n‍(m-k-1-n+2j)·Wm-k-1,n.
We apply the recursion relation (54) (m-n) times and we get the expression
(55)Wmn=sinα(x)(m-n)(m+n+1)/2(-2α)(m-n)(m+n-1)/2·∏j=0m-n-1‍j!∏k=1m-n‍∏j=1n‍(k+2j-1)·Wn,n.
To compute Wmn it is now sufficient to replace Wnn given by (47) in Wm,n defined by (55). So we get
(56)Wmn=(-1)m(m-1)/22((m(m-1)/2)+(n(n+1)/2)+(n(n-1)/2))αm(m-1)/2·∏j=1n-1‍j!∏j=1m-n-1‍j!∏k=1m-n‍∏j=1n‍(k+2j-1)!·(sinαx)m+⋯+n+1+(n(n+1)/2)(cosαx)n(n+1)/2.
We can rearrange these terms and get
(57)Wmn=(-2α)m(m-1)/22n2·∏j=1n-1‍j!∏j=1m-n-1‍j!∏k=1m-n‍∏j=1n‍(k+2j-1)!·(sinαx)m(m+1)/2(cosαx)n(n+1)/2.
It is easy to verify the identity
(58)∏k=1m-n‍∏j=1n‍(k+2j-1)!=∏j=1n‍(m-n+2j-1)!(2j-1)!.
Finally, we get
(59)Wmn=(-2α)m(m-1)/22n2·∏j=1m-n-1‍j!∏j=1n‍(m-n+2j-1)!(2j-1)!×(j-1)!·(sinαx)m(m+1)/2(cosαx)n(n+1)/2.
Taking into account the notation d=m-n, we recover exactly the formula (24).
In the case wheren=0, Wm0 can be computed with the obvious relation Wn0(2α)=Wnn(α). So we have
(60)Wm0=(-2α)m(m-1)/2∏j=1m-1‍j!(sinαx)m(m+1)/2.
Thus, one gets the results of Theorem 2, and consequently those of Theorems 3, 4 and 5.Example 8.
The following example of wronskianWmn illustrates well the behavior of the coefficients cmn:
(61)W(sinx,sin3x,sin5x,sin7x,sin9x)=235·9cos10xsin15x=309237645312cos10xsin15x.
Formulas (24) represent a very strong reduction with respect to combinatorial definition of the related determinant representing a trigonometric polynomial containing m! 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.Letv(x) be any potential and f1(x,E1),f2(x,E2),…,fm(x,Em) be any solutions of the 1D Schrödinger (Sturm-Liouville) equation
(62)-f′′+v(x)f=Ef,
corresponding to E=E1,E2,…,Em.Proposition 9.
The solutionψ of the Schrödinger equation
(63)-ψ′′+uψ=Eψ,u=v-2∂x2logW(f1,f2,…,fm)
is given by the formula
(64)ψ(x,E)=W(f1,f2,…,fm,f(x))W(f1,f2,…,fm).In the casem=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:
(65)∂tf=∑j=0N‍vm(x,t)∂xjf(x,t).
Let f1,f2,…,fm be any m linearly independent solutions of (65) different from f. Then the following statement holds.Proposition 10 (see [8, 9] for the proof and further details).
The RHS of (64) satisfy the following PDE:
(66)∂tψ=∑j=0N‍um(x,t)∂xjψ(x,t),
where the coefficients um(x,t) can be explicitly written in terms of vj and fk(x,t). In particular, assuming that vN=const and vN-1=0 we obtain the following formulas for uN,uN-1,uN-2:
(67)uN=vN,uN-1=vN-1=0,uN-2=vN-2+NvN∂x2logW(f1,f2,…,fm).
The RHS of (64) represents the action of the differential operator of order m on the function f:
(68)ψ=D(m)f:=(∂xm+∑k=0m-1‍gk(x,t)∂xk)f.D(m) can be also represented in a form of a product of the first-order differential operators in many different ways. Let J=(j1,j2,…,jm) be any permutation of the numbers (1,2,…,m). To each permutation corresponds the following factorization of D(m):
(69)D(m)=Djm·Djm-1⋯Dj1,Djk=∂x-σjk,(70)σjk:=∂xlogW(fj1,fj2,…,fjk)W(fj1,fj2,…,fjk-1).
Assume now that J=(1,2,...,m), v=0, and fj=φj, where φj are defined in (19). Combining formula (24) with Proposition 9 we get the following result. The function ψ corresponding to the case fj=φj,v=0 solves the Schrödinger equation with singular periodic potential Um,n:
(71)-ψ′′+Um,n(x)ψ=Eψ,
where
(72)Um,n=-2∂x2logWmn=α2m(m+1)sin2α(x)+α2n(n+1)cos2α(x),m>n.
Replacing Wmn by W^mn we obtain the potential U~m,n exponentially decreasing at infinity:
(73)U~m,n=-2∂x2logW^mn=α2m(m+1)sinh2α(x)-α2n(n+1)cosh2α(x),m>n.
General solution of the related Shrödinger equation (71) is obtained from (64) by choosing fj=gj. Quite similarly we have
(74)umn≔-2∂x2logwmn=α2m(m+1)cos2α(x-x0)+α2n(n+1)sin2α(x-x0),m>n,
where wmn was defined in (2) and the solution ψ(x,E) of the related Schrödinger equation can be obtained by choosing fj=hj in (64).It is clear as well that(75)u^mn≔-2∂x2logw^mn=-α2m(m+1)cosh2α(x-x0)+α2n(n+1)sinh2α(x-x0),m>n,
and the solutions of the related Schrödinger equation are obtained from (64) by choosing fj=lj. In the case n=0, u^mn is a nonsingular reflectionless potential and the discrete negative eigenvalues of the related Schrödinger operator on the real line are Ej=-α2j2.The integrability of (71) with potential (72) with α=1,x0=0 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 J=(1,2,…,m); that is, the related function ψ can be written as
(76)ψ=Dm·Dm-1⋯D1f,Dk=∂x-σk.
The related functions σk now can be computed combining (24) and (70), where, of course, the constant factor in the RHS of (24) can be ignored:
(77)σj=αjcotα(x-x0),j≤m-n,σm-n+k=(m-n+k)αcotα(x-x0)-kαtanα(x-x0),k=1,…,n.
Original derivation [10] by Darboux of (77) was based on the observation that the functions sinm(x-x0)cosnα(x-x0) represent the particular solutions of the Schrödinger equation (71) with potential Um-1,n-1 corresponding to E=α2(m+n)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 Proposition10. The jth equation of the KdV hierarchy can be obtained as a compatibility condition of (62) and the following evolution equation:
(78)∂tjf=cj∂x2j+1f-(2j+1)cj2v(x,tj)∂x2j-1f+∑p=02j-2‍vp(x,tj)∂xpf.
In particular for j=1, cj=-4, and v0=3vx compatibility of (62) and (78) implies that v satisfies the KdV equation
(79)vt=6vvx-vxxx.
In general the choice of cj is arbitrary: it fixes normalization of the KdV hierarchy. Now according to Proposition 10 for any given solution v(x,tj) of jth KdV equation the sequence f1,…,fm of the common solutions of (62) and (78) generates the new solution u(x,tj) of the same equation:
(80)u(x,tj)=v(x,tj)-2∂x2logW(f1,f2,…,fm).
In particular, in the case
(81)v=0,fp(x,t)=sin[αp(x-x0)+(-1)jcjα2j+1p2j+1tj],p=1,…,d,d=m-n,fd+l(x,t)=sin[α(d+2l)(x-x0)+(-1)jcjα2j+1(d+2l)2j+1tj],l=1,…,n,m≥n;
the RHS of (80) is the solution of the jth 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 v=0 and
(82)fp(x,t)=sin(αp(x-x0)+4α3p3t),p=1,…,d,d=m-n,fd+l(x,t)=sin[α(d+2l)(x-x0)+4α3(d+2l)3t],l=1,…,n,m≥n.
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 v=0, is given by the formula
(83)v(x,t)=-2∂x2logW(L1,…,Lm),
where the functions Lj(x,t) are defined as follows:
(84)Lp(x,t)≔coshαap(x-x0-4α2ap2t),ap=p,ifpisodd,Lp(x,t)≔sinhαap(x-x0-4α2ap2t),ap=p,ifpiseven,ifd≠01≤p≤m-n,am-n+j=m-n+2j,ifn≠0,j=1,…,n,Lm-n+j(x,t)≔coshαam-n+j(x-x0-4α2am-n+j2t),ifm-nisodd,Lm-n+j(x)≔sinhαam-n+j(x-x0-4α2am-n+j2t),ifm-niseven.


### 3.2. Differential Identities

Formulas (23) and (24) lead to some nontrivial identities. Let D^ be a differential operator defined by the formula
(85)(D^f)(x)=(∏l=2m‍∂xll-1)f(x1,x2,…,xm)|x1=x2=⋯=xm=x.
Let Dm be the following determinant: Dm=detaj,k, aj,k=sinjxk, with 1≤j,k≤m.Clearly, we have(86)D^Dm=det∂xj-1sinkx=W(sinx,sin2x,…,sinmx)=Wm0.
The determinant Dm 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 m-dimensional abelian varieties over finite fields Fα when α→∞):
(87)Dm=2m(m-1)/2∏k=1m‍sinxk∏m≥k>j≥1‍(cosxk-cosxj).
Now, substituting this expression with Dm in (86) and taking into account the formula (24) for Wm0, we get the following identity:
(88)(∏l=2m‍∂xll-1)[(∏m≥k>j≥1‍(cosxj-cosxk))(∏k=1m‍sinxk)(∏l=2m‍∂xll-1)·(∏m≥k>j≥1‍(cosxj-cosxk))]|x1=x2=⋯=xm=x=∏k=1m-1‍k!·sinm(m+1)/2x,m≥2.
It is obvious that the wronskian
(89)W~mn≔W(1,cosx,cos2x,…,cosdx,cos(d+2)x,cos(d+4)x,…,cos(d+2n)x)
is proportional to Wmn:
(90)W~mn=(-1)m(d-1)!∏k=0n‍(d+2k)·Wmn,W~m0=(-1)mm!·Wm0.
This follows immediately from the expansion of W~ by the elements of its first column. Assuming that n=0, and introducing the determinant,
(91)Gm=detB,Bjk=cosjxk,j,k=0,1,…m,withx0=x;
we obviously have the following formula:
(92)W~m0=(∏j=1m‍∂xjj)Gm|x0=x1=x2=⋯=xm=x.
From the other side, Gm can be calculated explicitly and the reduction to a Vandermonde determinant is obvious:
(93)Gm=2m(m-1)/2∏m≥j>k≥0‍(cosxj-cosxk).
Comparing these two formulas with explicit expression for W~m0, following immediately from (90), we obtain the following identity:
(94)(∏l=1m‍∂xll)·(∏m≥j>k≥0‍(cosxk-cosxj))|x0=x1=x2=⋯=xm=x=(∏k=1m‍k!)·sinm(m+1)/2x,m≥2.
Similar but more complicated identities might be obtained along the same lines, comparing (24) with explicit expression for the determinant detB,Bpj=φp(xj),j,p=1,…,m, where the functions φp(x) were defined in (19). For instance for n=1,α=1,  and  x0=0 this leads to the following identity:
(95)(∏l=2m‍∂xll-1)[(∏k=1m‍sinxk)·(∑i=1m‍cosxi)(∏l=2m‍∂xll-1)·(∏m≥k>j≥1‍(cosxj-cosxk))]|x1=x2=⋯=xm=x=m!(∏k=0m-2‍k!)cosx·sinm(m+1)/2x.
In a more general formulation, using the same arguments as before, we get the identity
(96)(∏l=2m‍∂xll-1)[2m(m-1)/2∏k=1m‍sinxk·|1cosx1⋯cosm-n-1x1cosm-n+1x1⋯cosm+n-1x1⋮⋮⋮⋮⋮⋮⋮1cosxm⋯cosm-n-1xmcosm-n+1xm⋯cosm+n-1xm|]|x1=x2=⋯=xm=x=cmn·sinm(m+1)/2x·cosn(n+1)/2x,
with cmn defined by (25).


## 3.1. Wronskian Representation of the DPT Potentials and KdV Equation

Below we need the following two propositions.Letv(x) be any potential and f1(x,E1),f2(x,E2),…,fm(x,Em) be any solutions of the 1D Schrödinger (Sturm-Liouville) equation
(62)-f′′+v(x)f=Ef,
corresponding to E=E1,E2,…,Em.Proposition 9.
The solutionψ of the Schrödinger equation
(63)-ψ′′+uψ=Eψ,u=v-2∂x2logW(f1,f2,…,fm)
is given by the formula
(64)ψ(x,E)=W(f1,f2,…,fm,f(x))W(f1,f2,…,fm).In the casem=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:
(65)∂tf=∑j=0N‍vm(x,t)∂xjf(x,t).
Let f1,f2,…,fm be any m linearly independent solutions of (65) different from f. Then the following statement holds.Proposition 10 (see [8, 9] for the proof and further details).
The RHS of (64) satisfy the following PDE:
(66)∂tψ=∑j=0N‍um(x,t)∂xjψ(x,t),
where the coefficients um(x,t) can be explicitly written in terms of vj and fk(x,t). In particular, assuming that vN=const and vN-1=0 we obtain the following formulas for uN,uN-1,uN-2:
(67)uN=vN,uN-1=vN-1=0,uN-2=vN-2+NvN∂x2logW(f1,f2,…,fm).
The RHS of (64) represents the action of the differential operator of order m on the function f:
(68)ψ=D(m)f:=(∂xm+∑k=0m-1‍gk(x,t)∂xk)f.D(m) can be also represented in a form of a product of the first-order differential operators in many different ways. Let J=(j1,j2,…,jm) be any permutation of the numbers (1,2,…,m). To each permutation corresponds the following factorization of D(m):
(69)D(m)=Djm·Djm-1⋯Dj1,Djk=∂x-σjk,(70)σjk:=∂xlogW(fj1,fj2,…,fjk)W(fj1,fj2,…,fjk-1).
Assume now that J=(1,2,...,m), v=0, and fj=φj, where φj are defined in (19). Combining formula (24) with Proposition 9 we get the following result. The function ψ corresponding to the case fj=φj,v=0 solves the Schrödinger equation with singular periodic potential Um,n:
(71)-ψ′′+Um,n(x)ψ=Eψ,
where
(72)Um,n=-2∂x2logWmn=α2m(m+1)sin2α(x)+α2n(n+1)cos2α(x),m>n.
Replacing Wmn by W^mn we obtain the potential U~m,n exponentially decreasing at infinity:
(73)U~m,n=-2∂x2logW^mn=α2m(m+1)sinh2α(x)-α2n(n+1)cosh2α(x),m>n.
General solution of the related Shrödinger equation (71) is obtained from (64) by choosing fj=gj. Quite similarly we have
(74)umn≔-2∂x2logwmn=α2m(m+1)cos2α(x-x0)+α2n(n+1)sin2α(x-x0),m>n,
where wmn was defined in (2) and the solution ψ(x,E) of the related Schrödinger equation can be obtained by choosing fj=hj in (64).It is clear as well that(75)u^mn≔-2∂x2logw^mn=-α2m(m+1)cosh2α(x-x0)+α2n(n+1)sinh2α(x-x0),m>n,
and the solutions of the related Schrödinger equation are obtained from (64) by choosing fj=lj. In the case n=0, u^mn is a nonsingular reflectionless potential and the discrete negative eigenvalues of the related Schrödinger operator on the real line are Ej=-α2j2.The integrability of (71) with potential (72) with α=1,x0=0 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 J=(1,2,…,m); that is, the related function ψ can be written as
(76)ψ=Dm·Dm-1⋯D1f,Dk=∂x-σk.
The related functions σk now can be computed combining (24) and (70), where, of course, the constant factor in the RHS of (24) can be ignored:
(77)σj=αjcotα(x-x0),j≤m-n,σm-n+k=(m-n+k)αcotα(x-x0)-kαtanα(x-x0),k=1,…,n.
Original derivation [10] by Darboux of (77) was based on the observation that the functions sinm(x-x0)cosnα(x-x0) represent the particular solutions of the Schrödinger equation (71) with potential Um-1,n-1 corresponding to E=α2(m+n)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 Proposition10. The jth equation of the KdV hierarchy can be obtained as a compatibility condition of (62) and the following evolution equation:
(78)∂tjf=cj∂x2j+1f-(2j+1)cj2v(x,tj)∂x2j-1f+∑p=02j-2‍vp(x,tj)∂xpf.
In particular for j=1, cj=-4, and v0=3vx compatibility of (62) and (78) implies that v satisfies the KdV equation
(79)vt=6vvx-vxxx.
In general the choice of cj is arbitrary: it fixes normalization of the KdV hierarchy. Now according to Proposition 10 for any given solution v(x,tj) of jth KdV equation the sequence f1,…,fm of the common solutions of (62) and (78) generates the new solution u(x,tj) of the same equation:
(80)u(x,tj)=v(x,tj)-2∂x2logW(f1,f2,…,fm).
In particular, in the case
(81)v=0,fp(x,t)=sin[αp(x-x0)+(-1)jcjα2j+1p2j+1tj],p=1,…,d,d=m-n,fd+l(x,t)=sin[α(d+2l)(x-x0)+(-1)jcjα2j+1(d+2l)2j+1tj],l=1,…,n,m≥n;
the RHS of (80) is the solution of the jth 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 v=0 and
(82)fp(x,t)=sin(αp(x-x0)+4α3p3t),p=1,…,d,d=m-n,fd+l(x,t)=sin[α(d+2l)(x-x0)+4α3(d+2l)3t],l=1,…,n,m≥n.
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 v=0, is given by the formula
(83)v(x,t)=-2∂x2logW(L1,…,Lm),
where the functions Lj(x,t) are defined as follows:
(84)Lp(x,t)≔coshαap(x-x0-4α2ap2t),ap=p,ifpisodd,Lp(x,t)≔sinhαap(x-x0-4α2ap2t),ap=p,ifpiseven,ifd≠01≤p≤m-n,am-n+j=m-n+2j,ifn≠0,j=1,…,n,Lm-n+j(x,t)≔coshαam-n+j(x-x0-4α2am-n+j2t),ifm-nisodd,Lm-n+j(x)≔sinhαam-n+j(x-x0-4α2am-n+j2t),ifm-niseven.


## 3.2. Differential Identities

Formulas (23) and (24) lead to some nontrivial identities. Let D^ be a differential operator defined by the formula
(85)(D^f)(x)=(∏l=2m‍∂xll-1)f(x1,x2,…,xm)|x1=x2=⋯=xm=x.
Let Dm be the following determinant: Dm=detaj,k, aj,k=sinjxk, with 1≤j,k≤m.Clearly, we have(86)D^Dm=det∂xj-1sinkx=W(sinx,sin2x,…,sinmx)=Wm0.
The determinant Dm 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 m-dimensional abelian varieties over finite fields Fα when α→∞):
(87)Dm=2m(m-1)/2∏k=1m‍sinxk∏m≥k>j≥1‍(cosxk-cosxj).
Now, substituting this expression with Dm in (86) and taking into account the formula (24) for Wm0, we get the following identity:
(88)(∏l=2m‍∂xll-1)[(∏m≥k>j≥1‍(cosxj-cosxk))(∏k=1m‍sinxk)(∏l=2m‍∂xll-1)·(∏m≥k>j≥1‍(cosxj-cosxk))]|x1=x2=⋯=xm=x=∏k=1m-1‍k!·sinm(m+1)/2x,m≥2.
It is obvious that the wronskian
(89)W~mn≔W(1,cosx,cos2x,…,cosdx,cos(d+2)x,cos(d+4)x,…,cos(d+2n)x)
is proportional to Wmn:
(90)W~mn=(-1)m(d-1)!∏k=0n‍(d+2k)·Wmn,W~m0=(-1)mm!·Wm0.
This follows immediately from the expansion of W~ by the elements of its first column. Assuming that n=0, and introducing the determinant,
(91)Gm=detB,Bjk=cosjxk,j,k=0,1,…m,withx0=x;
we obviously have the following formula:
(92)W~m0=(∏j=1m‍∂xjj)Gm|x0=x1=x2=⋯=xm=x.
From the other side, Gm can be calculated explicitly and the reduction to a Vandermonde determinant is obvious:
(93)Gm=2m(m-1)/2∏m≥j>k≥0‍(cosxj-cosxk).
Comparing these two formulas with explicit expression for W~m0, following immediately from (90), we obtain the following identity:
(94)(∏l=1m‍∂xll)·(∏m≥j>k≥0‍(cosxk-cosxj))|x0=x1=x2=⋯=xm=x=(∏k=1m‍k!)·sinm(m+1)/2x,m≥2.
Similar but more complicated identities might be obtained along the same lines, comparing (24) with explicit expression for the determinant detB,Bpj=φp(xj),j,p=1,…,m, where the functions φp(x) were defined in (19). For instance for n=1,α=1,  and  x0=0 this leads to the following identity:
(95)(∏l=2m‍∂xll-1)[(∏k=1m‍sinxk)·(∑i=1m‍cosxi)(∏l=2m‍∂xll-1)·(∏m≥k>j≥1‍(cosxj-cosxk))]|x1=x2=⋯=xm=x=m!(∏k=0m-2‍k!)cosx·sinm(m+1)/2x.
In a more general formulation, using the same arguments as before, we get the identity
(96)(∏l=2m‍∂xll-1)[2m(m-1)/2∏k=1m‍sinxk·|1cosx1⋯cosm-n-1x1cosm-n+1x1⋯cosm+n-1x1⋮⋮⋮⋮⋮⋮⋮1cosxm⋯cosm-n-1xmcosm-n+1xm⋯cosm+n-1xm|]|x1=x2=⋯=xm=x=cmn·sinm(m+1)/2x·cosn(n+1)/2x,
with cmn defined by (25).


## 4. Additional Comments

(i)
There is some similarity of (24), especially of the case n=0 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 W(σ(x),σ(2x),…,σ(nx)), where σ(x) is a Weierstrass sigma function.
(ii)
Takingf=eikx,k2=E in the formula (64) for the solutions of (71) we see that the related function is ψ(x,E)=P(x,k)·eikx where P(x,k) is a polynomial of k. 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 α,(2α)2,…,(dα)2,(d+2)2α2,…(d+2n)2α2.
(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:
(97)u(x)=m(m-1)k2cn2xdn2x+α(α-1)dn2xcn2x+p(p-1)sn2x+n(n-1)k2sn2x.
In (97) sn,cn, and dn are usual Jacobi elliptic functions of modulus k and m,α,p,n are arbitrary  nonnegative integers. In the case m=α=n=0, 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 Section1 with [29–31] shows that the DPT trigonometric potentials produce the subfamily of two dimensional Huygens potentials V(r,φ) via a formula
(98)V(r,φ)=1r2[m(m+1)sin2φ+n(n+1)cos2φ],
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 m+1 cosine functions):
(99)V(r,φ)=-2r2∂2∂φ2lnW(χ1,…,χm),χj=sin(kjφ+δj),j=1,…,m;kj∈ℤ,δj∈ℝ.Therefore, takingkj=aj, where aj are defined in (19) and taking δj=0, we get the aforementioned link between the DPT potentials and Huygens potentials.



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*Source: 645752-2013-04-28.xml*