Wronskian Addition Formula and Darboux-Pöschl-Teller Potentials

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]): −𝑦^′′+𝑚(𝑚+1)𝛼²sin²+𝑛()𝛼𝛼𝑥𝑛+1²cos²𝛼𝑥𝑦=𝜆𝑦. (1) 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_𝑥𝑓+𝑢(𝑥)𝑓=𝜆𝑓, (2) 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 𝑦_𝑚,𝑛(𝑥)=cos^𝑛+1(𝛼𝑥)sin^𝑚+1(𝛼𝑥_∼),𝑦_𝑚,𝑛(𝑥)=cos^−𝑛(𝛼𝑥)sin^−𝑚(𝛼𝑥) (3) 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 𝜕_𝑥−𝜑_𝑥𝜑𝐼, (4) that we call today Darboux transformation; he obtained the general solution of (1) in terms of elementary functions (for 𝑓 solution of (2)): 𝑦=𝐷_𝑚⋅𝐷_𝑚−1⋯𝐷₁𝑓,𝐷_𝑘=𝜕_𝑥−𝜎_𝑘𝐼, (5) with 𝜎_𝑗=𝑗𝛼cot𝛼𝑥−𝑥₀𝜎,𝑗≤𝑚−𝑛,_{𝑚−𝑛+𝑘}=(𝑚−𝑛+𝑘)𝛼cot𝛼𝑥−𝑥₀−𝑘𝛼tan𝛼𝑥−𝑥₀,𝑘=1,…,𝑛. (6) 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: 𝑦=sin^𝑚(𝛼𝑥)cos^𝑛(𝛼𝑥)𝑔sin²(𝛼𝑥), (7) where 𝑔 is a solution of Gauss hypergeometric equation: 𝑥(1−𝑥)𝑦^′′+𝛾−𝛿+𝛽+1𝑥𝑦^′−𝛿𝛽𝑦=0. (8) In particular for ₂𝐹₁ defined by ₂𝐹₁𝛿,𝛽,𝛾,𝑧=^+∞_𝑘=0(𝛿)_𝑘𝛽_𝑘𝛾_𝑘(1)_𝑘𝑧^𝑘, (9) with (𝑎)₀=1,(𝑎)_𝑘()⋯()1=𝑎𝑎+1𝑎+𝑘−1,𝑘≠0,𝛾=𝑚+2,𝛿=𝑚+𝑛2+1√2𝛼𝜆,𝛽=𝑚+𝑛2−1√2𝛼𝜆, (10)𝑦=sin^𝑚(𝛿𝑥)cos^𝑛(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 𝑊𝑓₁,𝑓₂,…,𝑓_𝑚∶=det𝜕_𝑥^𝑗−1𝑓_𝑘(𝑥);𝑗,𝑘=1,…,𝑚. (11) We denoted by 𝜑_𝑗, 𝑓 some independent solutions of (2) associated, respectively, with 𝜆_𝑗 and 𝜆; that is, −𝜑_𝑗^′′+𝑢𝜑_𝑗=𝜆_𝑗𝜑_𝑗,−𝑓^′′+𝑢𝑓=𝜆𝑓. (12) We define then the new potential 𝑢^∼ and the function Ψ by the formulas 𝑢^∼=𝑢−2𝜕²_𝑥ln𝑊𝜑₁,…,𝜑_𝑚,𝑊𝜑Ψ=¹,…,𝜑_𝑚,𝑓𝑊𝜑₁,…,𝜑_𝑚. (13)(14) Then we have the following result (Crum's Th. [6]).

Theorem 1. Ψ represents a general solution of (2): −𝜕²_𝑥Ψ+𝑢^∼Ψ=𝜆Ψ, (15) 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 functions 𝜑_𝑗 solution of (2) in the case 𝑢=0 so that the potential 𝑢^∼ defined by (13) is equal to DPT potential 𝑚(𝑚+1)𝛼²sin²+𝑛()𝛼𝛼𝑥𝑛+1²cos².𝛼𝑥 (16) 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 𝜑_𝑖 if we replace in the general solution 𝑦^∼_𝑚,𝑛=𝑊𝜑₁,…,𝜑_𝑚,𝑓𝑊𝜑₁,…,𝜑_𝑚 (17) of (2) with 𝑢=−2𝜕²_𝑥ln𝑊𝜑₁,…,𝜑_𝑚,𝜑_𝑖^′′=𝜆_𝑖𝜑_𝑖. (18) 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 𝜑_𝑝(𝑥) by convenient arguments 𝜑_𝑝(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 solutions 𝑓_𝑗 of the free Shrödinger equation −𝑓_𝑗^′′=𝐸_𝑗𝑓_𝑗 in order to get the DPT potential from Crum formula (63) with 𝑣=0. The choice of 𝐸_𝑗 was clear from [10] but at every point 𝐸_𝑗 we have the generic solution 𝐴_𝑗sin(𝐸_𝑗𝑥+𝑥_𝑗), where 𝐴_𝑗 plays no role in computing 𝜕²_𝑥log𝑊(𝑓₁,…,𝑓_𝑚). 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 𝑥_𝑗 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 𝑐_𝑚𝑛) 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 𝑛 and 𝑑 be some nonnegative integers, 𝑚=𝑛+𝑑, 𝛼 and 𝑥₀ be some real parameters, and 𝜑_𝑝 are the functions defined as follows: 𝜑_𝑝(𝑥)=sin𝛼𝑎_𝑝𝑥−𝑥₀,𝑎_𝑝𝑎=𝑝,if𝑑≠0,1≤𝑝≤𝑚−𝑛,_𝑗=𝑎_𝑗(𝑚,𝑛)=𝑛−𝑚+2𝑗,if𝑛≠0,𝑚−𝑛+1≤𝑗≤𝑚. (19)(20) We use below the standard notation 𝑊𝑓₁,𝑓₂,…,𝑓_𝑚∶=det𝜕_𝑥^𝑗−1𝑓_𝑘(𝑥);𝑗,𝑘=1,…,𝑚, (21) for wronskian determinant of any 𝑚 functions and the short notation 𝑊_𝑚𝑛=𝑊_𝑚𝑛(𝛼,𝑥,𝑥₀)=𝑊(𝜑₁,…,𝜑_𝑚), where 𝜑_𝑝 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): 1!⋅2!⋯𝑘!=1^𝑘⋅2^𝑘−1⋅3^𝑘−2⋯𝑘. (22)

Theorem 2. Consider the following: 𝑊_𝑚0=(−2𝛼)^{𝑚(𝑚−1)/2𝑚−1}_𝑘=0𝑘!⋅sin^{𝑚(𝑚+1)/2}𝛼𝑥−𝑥₀,𝑊_𝑚𝑛=𝑐_𝑚𝑛⋅sin^{𝑚(𝑚+1)/2}𝛼𝑥−𝑥₀cos^{𝑛(𝑛+1)/2}𝛼𝑥−𝑥₀,𝑐_𝑚𝑛=(−2𝛼)^{𝑚(𝑚−1)/2}⋅2^{𝑛2𝑑−1}_𝑝=0𝑝!^𝑛_𝑘=1(𝑑+2𝑘−1)!()!()𝑐2𝑘−1𝑘−1!,if𝑚≠𝑛,𝑚≥2,_𝑚𝑚=(−2𝛼)^{𝑚(𝑚−1)/2}⋅2^{𝑚2𝑚−1}_𝑘=1𝑘!,𝑚≥2. (23)(24)(25) We note 𝑊_𝑚𝑛 to be the determinant obtained from 𝑊_𝑚𝑛 replacing sine functions 𝜑_𝑗 by the sinh functions 𝑔_𝑗 of the same argument: 𝑊_𝑚𝑛=𝑊𝑔₁,…,𝑔_𝑚,𝑔_𝑝(𝑥)=sinh𝛼𝑎_𝑝𝑥−𝑥₀,𝑎_𝑝𝑎=𝑝,if𝑑≠0,1≤𝑝≤𝑚−𝑛;_{𝑚−𝑛+𝑗}=𝑚−𝑛+2𝑗,if𝑛≠0,1≤𝑗≤𝑛. (26) Then the following hyperbolic versions of (23) and (24) hold.

Theorem 3. Consider the following: 𝑊_𝑚0=(2𝛼)^{𝑚(𝑚−1)/2𝑚−1}_𝑘=0𝑘!⋅sinh^{𝑚(𝑚+1)/2}𝛼𝑥−𝑥₀,𝑊_𝑚𝑛=𝑐_𝑚𝑛⋅sinh^{𝑚(𝑚+1)/2}𝛼𝑥−𝑥₀×cosh^{𝑛(𝑛+1)/2}𝛼𝑥−𝑥₀,𝑚≥2, (27)
where 𝑐_𝑚𝑛 is the same as before.

Replacing (19) by the system of functions, ℎ_𝑝(𝑥)=cos𝛼𝑎_𝑝𝑥−𝑥₀,𝑎_𝑝ℎ=𝑝,if𝑝isodd,_𝑝(𝑥)=sin𝛼𝑎_𝑝𝑥−𝑥₀,𝑎_𝑝𝑎=𝑝,if𝑝iseven,if𝑑≠0,1≤𝑝≤𝑚−𝑛,_{𝑚−𝑛+𝑗}ℎ=𝑚−𝑛+2𝑗,if𝑛≠0,1≤𝑗≤𝑛,_{𝑚−𝑛+𝑗}(𝑥)=cos𝛼𝑎_{𝑚−𝑛+𝑗}𝑥−𝑥₀ℎ,if𝑚−𝑛isodd,_{𝑚−𝑛+𝑗}(𝑥)=sin𝛼𝑎_{𝑚−𝑛+𝑗}𝑥−𝑥₀,if𝑚−𝑛iseven; (28) we get for their wronskian 𝑤_𝑚𝑛=𝑊ℎ₁,…,ℎ_𝑚 (29) the following analogue of (23) and (24).

Theorem 4. Consider the following: 𝑤_𝑚0=(−1)^{(𝑚−1)𝑚2/2}(2𝛼)^{𝑚(𝑚−1)/2𝑚−1}_𝑘=0𝑘!⋅cos^{𝑚(𝑚+1)/2}𝛼𝑥−𝑥₀,𝑤_𝑚𝑛=(−1)^{(𝑚−1)((𝑚−𝑛)2+𝑛)/2}𝑐_𝑚𝑛⋅cos^{𝑚(𝑚+1)/2}𝛼𝑥−𝑥₀×sin^{𝑛(𝑛+1)/2}𝛼𝑥−𝑥₀,𝑚≥2, (30) where 𝑐_𝑚𝑛 has the same meaning as before.

Finally, replacing in the latter formulas sin and cos functions ℎ_𝑗 by sinh and cosh functions 𝑙_𝑗 of the same arguments and using the notation 𝑤_𝑚𝑛 for their wronskian we get the formulas 𝑤_𝑚𝑛=𝑊𝑙₁…,𝑙_𝑚,𝑙_𝑝(𝑥)=cosh𝛼𝑎_𝑝𝑥−𝑥₀,𝑎_𝑝𝑙=𝑝,if𝑝isodd,_𝑝(𝑥)=sinh𝛼𝑎_𝑝𝑥−𝑥₀,𝑎_𝑝𝑎=𝑝,if𝑝iseven,if𝑑≠0,1≤𝑝≤𝑚−𝑛,_{𝑚−𝑛+𝑗}𝑙=𝑚−𝑛+2𝑗,if𝑛≠0,1≤𝑗≤𝑛,_{𝑚−𝑛+𝑗}(𝑥)=cosh𝛼𝑎_{𝑚−𝑛+𝑗}𝑥−𝑥₀𝑙,if𝑚−𝑛isodd,_{𝑚−𝑛+𝑗}(𝑥)=sinh𝛼𝑎_{𝑚−𝑛+𝑗}𝑥−𝑥₀,if𝑚−𝑛iseven. (31)

Theorem 5. Consider the following: 𝑤_𝑚0=(2𝛼)^{𝑚(𝑚−1)/2𝑚−1}_𝑘=0𝑘!⋅cosh^{𝑚(𝑚+1)/2}𝛼𝑥−𝑥₀,𝑤_𝑚𝑛=𝑐_𝑚𝑛⋅cosh^{𝑚(𝑚+1)/2}𝛼𝑥−𝑥₀×sinh^{𝑛(𝑛+1)/2}𝛼𝑥−𝑥₀,𝑚≥2. (32)(33)

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 notation 𝑞=𝑒^𝑖𝛼. We define the functions 𝑓_𝑗 by the formula 𝑓_𝑗=sin𝛼𝑗𝑥,with𝑗∈ℤ. (34) Then it is easy to see that 𝑊_𝑚𝑛=𝑊𝑓₁,𝑓₂,…,𝑓_𝑚−𝑛,𝑓_{𝑚−𝑛+2},𝑓_{𝑚−𝑛+4},…,𝑓_𝑚+𝑛. (35) We define the sequence of wronskians 𝑊_{𝑚−𝑘,𝑛} of order 𝑚−𝑘 by the formulas 𝑊_{𝑚−1,𝑛}=𝑊𝑓₁,𝑓₂,…,𝑓_{𝑚−𝑛−1},𝑓_{𝑚−𝑛+1},𝑓_{𝑚−𝑛+3},…,𝑓_{𝑚+𝑛−1},𝑊_{𝑚−𝑘,𝑛}=𝑊𝑓₁,𝑓₂,…,𝑓_{𝑚−𝑛−𝑘},𝑓_{𝑚−𝑛−𝑘+2},𝑓_{𝑚−𝑛−𝑘+4},…,𝑓_{𝑚−𝑘+𝑛},𝑊_𝑛,𝑛=𝑊𝑓₂,𝑓₄,…,𝑓_2𝑛. (36) It is obvious that 𝑊_𝑛𝑛(𝛼)=𝑊_𝑛0(2𝛼). We consider the functions equal to 𝜑_𝑗 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: 𝑊_{𝑚−𝑘,𝑛}=𝑓₁^𝑚−𝑘(−2𝛼)^{𝑚−𝑘−1}()!×𝑚−𝑘−1−𝑛_𝑛𝑗=1𝑚−𝑘−1−𝑛+2𝑗⋅𝑊_{𝑚−𝑘−1,𝑛}. (37) For brevity, we replace 𝑥−𝑥₀ by 𝑥 in the following.
We calculate the wronskian 𝑊_𝑛𝑛=𝑊(𝑓₂,…,𝑓_2𝑛).
For this, we use the expansion sin𝑝𝑥=sin𝑥^{[(𝑝−1)/2]}_𝑘=0(−1)^𝑘𝐶^𝑘_{𝑝−𝑘−1}⋅(2cos𝑥)^{𝑝−1−2𝑘}=sin𝑥(2cos𝑥)^𝑝−1−𝑝−2(2cos𝑥)^𝑝−3+⋯. (38) Therefore 𝑊_𝑛𝑛 can be reduced to 𝑊_𝑛𝑛=𝑊sin2𝛼𝑥,2sin2𝛼𝑥cos2𝛼𝑥,…,sin2𝛼𝑥(cos2𝛼𝑥)^𝑛−1, (39) 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 column 𝑝 the term 2^𝑝−1 for 2≤𝑝≤𝑛, so we can drop out the factor 2^{𝑛(𝑛−1)/2}; 𝑊_𝑛𝑛 can be written as 𝑊_𝑛𝑛=2^{𝑛(𝑛−1)/2}𝑊×sin2𝛼𝑥,sin2𝛼𝑥cos2𝛼𝑥,…,sin2𝛼𝑥(cos2𝛼𝑥)^𝑛−1. (40) Then we use the two following statements.
Proposition 6. For any smooth functions 𝑓,𝜑₁,…,𝜑_𝑛, the wronskian 𝑊(𝑓⋅𝜑₁,…,𝑓⋅𝜑_𝑛) verify 𝑊𝑓⋅𝜑₁,…,𝑓⋅𝜑_𝑛=𝑓(𝑥)^𝑛𝑊𝜑₁,…,𝜑_𝑛, (41)first proved by Frobenius [19]. The second one is a well-known formula of change of independent variable in a wronskian.
Proposition 7. For 𝑡=𝜑(𝑥) and 𝑓^∼_𝑗 defined by 𝑓_𝑗(𝑥)=𝑓^∼_𝑗(𝑡): 𝑊𝑓₁,…,𝑓_𝑛=𝜑^′(𝑥)^{𝑛(𝑛−1)/2}⋅𝑊𝑓^∼₁,𝑓^∼₂,…,𝑓^∼_𝑛. (42) Then we can factorize (sin(2𝛼𝑥))^𝑛 in the preceding determinant 𝑊_𝑛 to get 𝑊_𝑛𝑛=2^{𝑛(𝑛−1)/2}(sin(2𝛼𝑥))^𝑛𝑊1,cos2𝛼𝑥,…,(cos2𝛼𝑥)^𝑛−1. (43) 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 𝑊_𝑛 can be written as 𝑊_𝑛𝑛=2^{𝑛(𝑛−1)/2}(−2𝛼sin2𝛼𝑥)^{𝑛(𝑛−1)/2}×(sin(2𝛼𝑥))^𝑛𝑊1,𝑡,…,𝑡^𝑛−1. (44) The wronskian 𝑊(1,𝑡,…,𝑡^𝑛−1) has the upper triangular form and thus is equal to the product of its diagonal elements; in other words, 𝑊1,𝑡,…,𝑡^𝑛−1=^𝑛−1_𝑗=1𝑗!. (45) So 𝑊_𝑛 can be written as 𝑊_𝑛𝑛=2^{𝑛(𝑛−1)}(−𝛼)^{𝑛(𝑛−1)/2𝑛−1}_𝑗=1𝑗!⋅(sin2𝛼𝑥)^{𝑛(𝑛+1)/2}. (46) Expanding sin2𝛼𝑥, we get finally 𝑊_𝑛𝑛=2^{𝑛(𝑛−1)+(𝑛(𝑛+1)/2)}(−𝛼)^{𝑛(𝑛−1)/2}×^𝑛−1_𝑗=1𝑗!⋅(sin𝛼𝑥)^{𝑛(𝑛+1)/2}(cos𝛼𝑥)^{𝑛(𝑛+1)/2}. (47) In a second step, we establish a recursion relation between the wronskians 𝑊_𝑚𝑛 and 𝑊_{𝑚−1,𝑛} (here 𝑚>𝑛).
We can factorize 𝜑^𝑚₁ in 𝑊_𝑚𝑛 to get 𝑊_𝑚𝑛=𝜑^𝑚₁⋅1𝜑₂𝜑₁⋯𝜑_𝑚𝜑₁𝜑₁⁽¹⁾𝜑₁𝜑₂⁽¹⁾𝜑₁⋯𝜑_𝑚⁽¹⁾𝜑₁𝜑⋮⋮⋮⋮_1(𝑚−1)𝜑₁𝜑₂^(𝑚−1)𝜑₁⋯𝜑_𝑚^(𝑚−1)𝜑₁. (48) It can be written as 𝑊_𝑚𝑛=𝜑^𝑚₁⋅𝜑₂𝜑₁⁽¹⁾𝜑₃𝜑₁⁽¹⁾⋯𝜑_𝑚𝜑₁⁽¹⁾𝜑⋮⋮⋮⋮₂𝜑₁^(𝑚−1)𝜑₃𝜑₁^(𝑚−1)⋯𝜑_𝑚𝜑₁^(𝑚−1). (49) In a shortened way we get the recursion relation 𝑊_𝑚𝑛=𝜑^𝑚₁𝜑⋅𝑊²𝜑₁⁽¹⁾𝜑,…,^𝑚𝜑₁⁽¹⁾. (50) We can evaluate the quotient 𝜑_𝑝/𝜑₁ in function of 𝑞; namely, 𝜑_𝑝𝜑₁=𝑞^{𝑎𝑝(𝑥)}−𝑞^{−𝑎𝑝(𝑥)}𝑞^(𝑥)−𝑞^−(𝑥)=𝑞^{(𝑎𝑝−1)(𝑥)}+𝑞^{(𝑎𝑝−3)(𝑥)}+⋯+𝑞^{−(𝑎𝑝−1)(𝑥)}. (51) So the 𝑗th row of the column 𝑝 of the determinant 𝑊((𝜑₂/𝜑₁)⁽¹⁾,…,(𝜑_𝑚/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)=𝑞^2𝑥+1+𝑞^−2𝑥, and the derivations eliminate the constant term).
Then each term (𝜑_𝑝/𝜑₁)⁽¹⁾ in the determinant 𝑊((𝜑₂/𝜑₁)⁽¹⁾,…,(𝜑_𝑚/𝜑₁)⁽¹⁾) can be replaced by (𝑞^{(𝑎𝑝−1)(𝑥)}+𝑞^{−(𝑎𝑝−1)(𝑥)})⁽¹⁾, which can be rewritten as 𝛼𝑖(𝑎_𝑝−1)(𝑞^{(𝑎𝑝−1)(𝑥)}−𝑞^{−(𝑎𝑝−1)(𝑥)})=−2𝛼(𝑎_𝑝−1)sin((𝑎_𝑝−1)𝑥) for 2≤𝑝≤𝑚.
Let Φ_𝑝 be 𝜑_𝑝/𝜑₁. It is obvious that 𝑎_𝑝(𝑚,𝑛)−1=𝑎_𝑝−1(𝑚−1,𝑛), for 2≤𝑝≤𝑚. So the sequence (Φ₂,…,Φ_𝑚−𝑛,Φ_{𝑚−𝑛+1},…,Φ_𝑚) can be replaced by (𝑓₁,…𝑓_{𝑚−𝑛−1},𝑓_{𝑚−𝑛+1},…,𝑓_{𝑚+𝑛−1}). We get the equality 𝑊_𝑚𝑛=𝑓^𝑚₁𝑊×−2𝛼𝑓₁,…,−2𝛼(𝑚−𝑛−1)𝑓_{𝑚−𝑛−1},−2𝛼(𝑚−𝑛+1)𝑓_{𝑚−𝑛+1},…,−2𝛼(𝑚+𝑛−1)𝑓_{𝑚+𝑛−1}. (52) We can drop out the constants appearing in each column to get 𝑊_𝑚𝑛=𝑓^𝑚₁(−2𝛼)^𝑚−1()!×𝑚−1−𝑛_𝑛𝑗=1𝑚−1−𝑛+2𝑗⋅𝑊_{𝑚−1,𝑛}. (53) We can make these transformations for each integer 𝑘 such that 0≤𝑘≤𝑚−𝑛−1, so we get the recursion relation 𝑊_{𝑚−𝑘,𝑛}=𝑓₁^𝑚−𝑘(−2𝛼)^{𝑚−𝑘−1}()!×𝑚−𝑘−1−𝑛_𝑛𝑗=1𝑚−𝑘−1−𝑛+2𝑗⋅𝑊_{𝑚−𝑘−1,𝑛}. (54) We apply the recursion relation (54) (𝑚−𝑛) times and we get the expression 𝑊_𝑚𝑛=sin𝛼(𝑥)^{(𝑚−𝑛)(𝑚+𝑛+1)/2}(−2𝛼)^{(𝑚−𝑛)(𝑚+𝑛−1)/2}⋅^{𝑚−𝑛−1}_𝑗=0𝑗!^{𝑚−𝑛𝑛}_𝑘=1𝑗=1𝑘+2𝑗−1⋅𝑊_𝑛,𝑛. (55) To compute 𝑊_𝑚𝑛 it is now sufficient to replace 𝑊_𝑛𝑛 given by (47) in 𝑊_𝑚,𝑛 defined by (55). So we get 𝑊_𝑚𝑛=(−1)^{𝑚(𝑚−1)/2}2^{((𝑚(𝑚−1)/2)+(𝑛(𝑛+1)/2)+(𝑛(𝑛−1)/2))}𝛼^{𝑚(𝑚−1)/2}⋅^𝑛−1_𝑗=1𝑗!^{𝑚−𝑛−1}_𝑗=1𝑗!^{𝑚−𝑛𝑛}_𝑘=1𝑗=1!⋅()𝑘+2𝑗−1sin𝛼𝑥^{𝑚+⋯+𝑛+1+(𝑛(𝑛+1)/2)}(cos𝛼𝑥)^{𝑛(𝑛+1)/2}. (56) We can rearrange these terms and get 𝑊_𝑚𝑛=(−2𝛼)^{𝑚(𝑚−1)/2}2^𝑛2⋅^𝑛−1_𝑗=1𝑗!^{𝑚−𝑛−1}_𝑗=1𝑗!^{𝑚−𝑛𝑛}_𝑘=1𝑗=1!⋅()𝑘+2𝑗−1sin𝛼𝑥^{𝑚(𝑚+1)/2}(cos𝛼𝑥)^{𝑛(𝑛+1)/2}. (57) It is easy to verify the identity ^{𝑚−𝑛𝑛}_𝑘=1𝑗=1𝑘+2𝑗−1!=^𝑛_𝑗=1𝑚−𝑛+2𝑗−1!!.2𝑗−1 (58) Finally, we get 𝑊_𝑚𝑛=(−2𝛼)^{𝑚(𝑚−1)/2}2^𝑛2⋅^{𝑚−𝑛−1}_𝑗=1𝑗!^𝑛_𝑗=1𝑚−𝑛+2𝑗−1!!×()2𝑗−1𝑗−1!⋅sin𝛼𝑥^{𝑚(𝑚+1)/2}(cos𝛼𝑥)^{𝑛(𝑛+1)/2}. (59) Taking into account the notation 𝑑=𝑚−𝑛, we recover exactly the formula (24).
In the case where 𝑛=0, 𝑊_𝑚0 can be computed with the obvious relation 𝑊_𝑛0(2𝛼)=𝑊_𝑛𝑛(𝛼). So we have 𝑊_𝑚0=(−2𝛼)^{𝑚(𝑚−1)/2𝑚−1}_𝑗=1𝑗!(sin𝛼𝑥)^{𝑚(𝑚+1)/2}. (60) Thus, one gets the results of Theorem 2, and consequently those of Theorems 3, 4 and 5.

Example 8. The following example of wronskian 𝑊_𝑚𝑛 illustrates well the behavior of the coefficients 𝑐_𝑚𝑛: 𝑊(sin𝑥,sin3𝑥,sin5𝑥,sin7𝑥,sin9𝑥)=2³⁵⋅9cos¹⁰𝑥sin¹⁵𝑥=309237645312cos¹⁰𝑥sin¹⁵𝑥. (61) 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.

Let 𝑣(𝑥) be any potential and 𝑓₁(𝑥,𝐸₁),𝑓₂(𝑥,𝐸₂),…,𝑓_𝑚(𝑥,𝐸_𝑚) be any solutions of the 1D Schrödinger (Sturm-Liouville) equation −𝑓^′′+𝑣(𝑥)𝑓=𝐸𝑓, (62) corresponding to 𝐸=𝐸₁,𝐸₂,…,𝐸_𝑚.

Proposition 9. The solution 𝜓 of the Schrödinger equation −𝜓^′′+𝑢𝜓=𝐸𝜓,𝑢=𝑣−2𝜕²_𝑥log𝑊𝑓₁,𝑓₂,…,𝑓_𝑚 (63) is given by the formula 𝜓()=𝑊𝑓𝑥,𝐸¹,𝑓₂,…,𝑓_𝑚,𝑓(𝑥)𝑊𝑓₁,𝑓₂,…,𝑓_𝑚. (64)

In the case 𝑚=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: 𝜕_𝑡𝑓=^𝑁_𝑗=0𝑣_𝑚(𝑥,𝑡)𝜕^𝑗_𝑥𝑓(𝑥,𝑡). (65) Let 𝑓₁,𝑓₂,…,𝑓_𝑚 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: 𝜕_𝑡𝜓=^𝑁_𝑗=0𝑢_𝑚(𝑥,𝑡)𝜕^𝑗_𝑥𝜓(𝑥,𝑡), (66) where the coefficients 𝑢_𝑚(𝑥,𝑡) can be explicitly written in terms of 𝑣_𝑗 and 𝑓_𝑘(𝑥,𝑡). In particular, assuming that 𝑣_𝑁=const and 𝑣_𝑁−1=0 we obtain the following formulas for 𝑢_𝑁,𝑢_𝑁−1,𝑢_𝑁−2: 𝑢_𝑁=𝑣_𝑁,𝑢_𝑁−1=𝑣_𝑁−1𝑢=0,_𝑁−2=𝑣_𝑁−2+𝑁𝑣_𝑁𝜕²_𝑥log𝑊𝑓₁,𝑓₂,…,𝑓_𝑚. (67) The RHS of (64) represents the action of the differential operator of order 𝑚 on the function 𝑓: 𝜓=𝐷(𝑚)𝑓∶=𝜕^𝑚_𝑥+^𝑚−1_𝑘=0𝑔_𝑘(𝑥,𝑡)𝜕^𝑘_𝑥𝑓. (68)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,…,𝑗_𝑚) be any permutation of the numbers (1,2,…,𝑚). To each permutation corresponds the following factorization of 𝐷(𝑚): 𝐷(𝑚)=𝐷_𝑗𝑚⋅𝐷_𝑗𝑚−1⋯𝐷_𝑗1,𝐷_𝑗𝑘=𝜕_𝑥−𝜎_𝑗𝑘,𝜎_𝑗𝑘∶=𝜕_𝑥𝑊𝑓log^𝑗1,𝑓_𝑗2,…,𝑓_𝑗𝑘𝑊𝑓_𝑗1,𝑓_𝑗2,…,𝑓_𝑗𝑘−1. (69)(70) Assume now that 𝐽=(1,2,...,𝑚), 𝑣=0, and 𝑓_𝑗=𝜑_𝑗, where 𝜑_𝑗 are defined in (19). Combining formula (24) with Proposition 9 we get the following result. The function 𝜓 corresponding to the case 𝑓_𝑗=𝜑_𝑗,𝑣=0 solves the Schrödinger equation with singular periodic potential 𝑈_𝑚,𝑛: −𝜓^′′+𝑈_𝑚,𝑛(𝑥)𝜓=𝐸𝜓, (71) where 𝑈_𝑚,𝑛=−2𝜕²_𝑥log𝑊_𝑚𝑛=𝛼²𝑚(𝑚+1)sin²𝛼(𝑥)+𝛼²𝑛(𝑛+1)cos²𝛼(𝑥),𝑚>𝑛. (72) Replacing 𝑊_𝑚𝑛 by 𝑊_𝑚𝑛 we obtain the potential 𝑈^∼_𝑚,𝑛 exponentially decreasing at infinity: 𝑈^∼_𝑚,𝑛=−2𝜕²_𝑥log𝑊_𝑚𝑛=𝛼²𝑚(𝑚+1)sinh²𝛼(𝑥)−𝛼²𝑛(𝑛+1)cosh²𝛼(𝑥),𝑚>𝑛. (73) General solution of the related Shrödinger equation (71) is obtained from (64) by choosing 𝑓_𝑗=𝑔_𝑗. Quite similarly we have 𝑢_𝑚𝑛∶=−2𝜕²_𝑥log𝑤_𝑚𝑛=𝛼²𝑚(𝑚+1)cos²𝛼𝑥−𝑥₀+𝛼²𝑛(𝑛+1)sin²𝛼𝑥−𝑥₀,𝑚>𝑛, (74) where 𝑤_𝑚𝑛 was defined in (2) and the solution 𝜓(𝑥,𝐸) of the related Schrödinger equation can be obtained by choosing 𝑓_𝑗=ℎ_𝑗 in (64).

It is clear as well that 𝑢_𝑚𝑛∶=−2𝜕²_𝑥log𝑤_𝑚𝑛𝛼=−²𝑚(𝑚+1)cosh²𝛼𝑥−𝑥₀+𝛼²𝑛(𝑛+1)sinh²𝛼𝑥−𝑥₀,𝑚>𝑛, (75) and the solutions of the related Schrödinger equation are obtained from (64) by choosing 𝑓_𝑗=𝑙_𝑗. 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 𝐸_𝑗=−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 𝜓=𝐷_𝑚⋅𝐷_𝑚−1⋯𝐷₁𝑓,𝐷_𝑘=𝜕_𝑥−𝜎_𝑘. (76) The related functions 𝜎_𝑘 now can be computed combining (24) and (70), where, of course, the constant factor in the RHS of (24) can be ignored: 𝜎_𝑗=𝛼𝑗cot𝛼𝑥−𝑥₀𝜎,𝑗≤𝑚−𝑛,_{𝑚−𝑛+𝑘}=(𝑚−𝑛+𝑘)𝛼cot𝛼𝑥−𝑥₀−𝑘𝛼tan𝛼𝑥−𝑥₀,𝑘=1,…,𝑛. (77) Original derivation [10] by Darboux of (77) was based on the observation that the functions sin^𝑚(𝑥−𝑥₀)cos^𝑛𝛼(𝑥−𝑥₀) represent the particular solutions of the Schrödinger equation (71) with potential 𝑈_{𝑚−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 Proposition 10. The 𝑗th equation of the KdV hierarchy can be obtained as a compatibility condition of (62) and the following evolution equation: 𝜕_𝑡𝑗𝑓=𝑐_𝑗𝜕_𝑥^2𝑗+1𝑐𝑓−2𝑗+1_𝑗2𝑣𝑥,𝑡_𝑗𝜕_𝑥^2𝑗−1𝑓+^2𝑗−2_𝑝=0𝑣_𝑝𝑥,𝑡_𝑗𝜕^𝑝_𝑥𝑓. (78) In particular for 𝑗=1, 𝑐_𝑗=−4, and 𝑣₀=3𝑣_𝑥 compatibility of (62) and (78) implies that 𝑣 satisfies the KdV equation 𝑣_𝑡=6𝑣𝑣_𝑥−𝑣_𝑥𝑥𝑥. (79) In general the choice of 𝑐_𝑗 is arbitrary: it fixes normalization of the KdV hierarchy. Now according to Proposition 10 for any given solution 𝑣(𝑥,𝑡_𝑗) of 𝑗th KdV equation the sequence 𝑓₁,…,𝑓_𝑚 of the common solutions of (62) and (78) generates the new solution 𝑢(𝑥,𝑡_𝑗) of the same equation: 𝑢𝑥,𝑡_𝑗=𝑣𝑥,𝑡_𝑗−2𝜕²_𝑥log𝑊𝑓₁,𝑓₂,…,𝑓_𝑚. (80) In particular, in the case 𝑓𝑣=0,_𝑝(𝑥,𝑡)=sin𝛼𝑝𝑥−𝑥₀+(−1)^𝑗𝑐_𝑗𝛼^2𝑗+1𝑝^2𝑗+1𝑡_𝑗,𝑓𝑝=1,…,𝑑,𝑑=𝑚−𝑛,_𝑑+𝑙()𝛼()𝑥,𝑡=sin𝑑+2𝑙𝑥−𝑥₀+(−1)^𝑗𝑐_𝑗𝛼^2𝑗+1(𝑑+2𝑙)^2𝑗+1𝑡_𝑗,𝑙=1,…,𝑛,𝑚≥𝑛; (81) 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 𝑓_𝑝(𝑥,𝑡)=sin𝛼𝑝𝑥−𝑥₀+4𝛼³𝑝³𝑡,𝑓𝑝=1,…,𝑑,𝑑=𝑚−𝑛,_𝑑+𝑙(𝑥,𝑡)=sin𝛼(𝑑+2𝑙)𝑥−𝑥₀+4𝛼³(𝑑+2𝑙)³𝑡,𝑙=1,…,𝑛,𝑚≥𝑛. (82) 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 𝑣(𝑥,𝑡)=−2𝜕²_𝑥log𝑊𝐿₁,…,𝐿_𝑚, (83) where the functions 𝐿_𝑗(𝑥,𝑡) are defined as follows: 𝐿_𝑝(𝑥,𝑡)∶=cosh𝛼𝑎_𝑝𝑥−𝑥₀−4𝛼²𝑎²_𝑝𝑡,𝑎_𝑝𝐿=𝑝,if𝑝isodd,_𝑝(𝑥,𝑡)∶=sinh𝛼𝑎_𝑝𝑥−𝑥₀−4𝛼²𝑎²_𝑝𝑡,𝑎_𝑝𝑎=𝑝,if𝑝iseven,if𝑑≠01≤𝑝≤𝑚−𝑛,_{𝑚−𝑛+𝑗}𝐿=𝑚−𝑛+2𝑗,if𝑛≠0,𝑗=1,…,𝑛,_{𝑚−𝑛+𝑗}(𝑥,𝑡)∶=cosh𝛼𝑎_{𝑚−𝑛+𝑗}𝑥−𝑥₀−4𝛼²𝑎²_{𝑚−𝑛+𝑗}𝑡,𝐿if𝑚−𝑛isodd,_{𝑚−𝑛+𝑗}(𝑥)∶=sinh𝛼𝑎_{𝑚−𝑛+𝑗}𝑥−𝑥₀−4𝛼²𝑎²_{𝑚−𝑛+𝑗}𝑡,if𝑚−𝑛iseven. (84)

3.2. Differential Identities

Formulas (23) and (24) lead to some nontrivial identities. Let 𝐷 be a differential operator defined by the formula 𝐷𝑓(𝑥)=^𝑚_𝑙=2𝜕_𝑥^𝑙−1_𝑙𝑓𝑥₁,𝑥₂,…,𝑥_{𝑚𝑥1=𝑥2=⋯=𝑥𝑚=𝑥}. (85) Let 𝐷_𝑚 be the following determinant: 𝐷_𝑚=det𝑎_𝑗,𝑘, 𝑎_𝑗,𝑘=sin𝑗𝑥_𝑘, with 1≤𝑗,𝑘≤𝑚.

Clearly, we have 𝐷𝐷_𝑚=det𝜕_𝑥^𝑗−1()sin𝑘𝑥=𝑊sin𝑥,sin2𝑥,…,sin𝑚𝑥=𝑊_𝑚0. (86) The determinant 𝐷_𝑚 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 𝐹_𝛼 when 𝛼→∞): 𝐷_𝑚=2^{𝑚𝑚(𝑚−1)/2}_𝑘=1sin𝑥_{𝑘𝑚≥𝑘>𝑗≥1}cos𝑥_𝑘−cos𝑥_𝑗. (87) Now, substituting this expression with 𝐷_𝑚 in (86) and taking into account the formula (24) for 𝑊_𝑚0, we get the following identity: ^𝑚_𝑙=2𝜕_𝑥^𝑙−1_𝑙^𝑚_𝑘=1sin𝑥_𝑘⋅_{𝑚≥𝑘>𝑗≥1}cos𝑥_𝑗−cos𝑥_{𝑘𝑥1=𝑥2=⋯=𝑥𝑚=𝑥}=^𝑚−1_𝑘=1𝑘!⋅sin^{𝑚(𝑚+1)/2}𝑥,𝑚≥2. (88) It is obvious that the wronskian 𝑊^∼_𝑚𝑛(()()()𝑥)∶=𝑊1,cos𝑥,cos2𝑥,…,cos𝑑𝑥,cos𝑑+2𝑥,cos𝑑+4𝑥,…,cos𝑑+2𝑛 (89) is proportional to 𝑊_𝑚𝑛: 𝑊^∼_𝑚𝑛=(−1)^𝑚(𝑑−1)!^𝑛_𝑘=0(𝑑+2𝑘)⋅𝑊_𝑚𝑛,_∼𝑊_𝑚0=(−1)^𝑚𝑚!⋅𝑊_𝑚0. (90) This follows immediately from the expansion of 𝑊^∼ by the elements of its first column. Assuming that 𝑛=0, and introducing the determinant, 𝐺_𝑚=det𝐵,𝐵_𝑗𝑘=cos𝑗𝑥_𝑘,𝑗,𝑘=0,1,…𝑚,with𝑥₀=𝑥; (91) we obviously have the following formula: 𝑊^∼_𝑚0=^𝑚_𝑗=1𝜕^𝑗_𝑥𝑗𝐺_{𝑚𝑥0=𝑥1=𝑥2=⋯=𝑥𝑚=𝑥}. (92) From the other side, 𝐺_𝑚 can be calculated explicitly and the reduction to a Vandermonde determinant is obvious: 𝐺_𝑚=2^{𝑚(𝑚−1)/2}_{𝑚≥𝑗>𝑘≥0}cos𝑥_𝑗−cos𝑥_𝑘. (93) Comparing these two formulas with explicit expression for 𝑊^∼_𝑚0, following immediately from (90), we obtain the following identity: ^𝑚_𝑙=1𝜕^𝑙_𝑥𝑙⋅_{𝑚≥𝑗>𝑘≥0}cos𝑥_𝑘−cos𝑥_{𝑗𝑥0=𝑥1=𝑥2=⋯=𝑥𝑚=𝑥}=^𝑚_𝑘=1𝑘!⋅sin^{𝑚(𝑚+1)/2}𝑥,𝑚≥2. (94) Similar but more complicated identities might be obtained along the same lines, comparing (24) with explicit expression for the determinant det𝐵,𝐵_𝑝𝑗=𝜑_𝑝(𝑥_𝑗),𝑗,𝑝=1,…,𝑚, where the functions 𝜑_𝑝(𝑥) were defined in (19). For instance for 𝑛=1,𝛼=1,  and  𝑥₀=0 this leads to the following identity: ^𝑚_𝑙=2𝜕_𝑥^𝑙−1_𝑙^𝑚_𝑘=1sin𝑥_𝑘⋅^𝑚_𝑖=1cos𝑥_𝑖⋅_{𝑚≥𝑘>𝑗≥1}cos𝑥_𝑗−cos𝑥_{𝑘𝑥1=𝑥2=⋯=𝑥𝑚=𝑥}=𝑚!^𝑚−2_𝑘=0𝑘!cos𝑥⋅sin^{𝑚(𝑚+1)/2}𝑥. (95) In a more general formulation, using the same arguments as before, we get the identity ∏^𝑚_𝑙=2𝜕_𝑥^𝑙−1_𝑙2^{𝑚𝑚(𝑚−1)/2}∏_𝑘=1sin𝑥_𝑘⋅1cos𝑥₁⋯cos^{𝑚−𝑛−1}𝑥₁cos^{𝑚−𝑛+1}𝑥₁⋯cos^{𝑚+𝑛−1}𝑥₁⋮⋮⋮⋮⋮⋮⋮1cos𝑥_𝑚⋯cos^{𝑚−𝑛−1}𝑥_𝑚cos^{𝑚−𝑛+1}𝑥_𝑚⋯cos^{𝑚+𝑛−1}𝑥_{𝑚𝑥1=𝑥2=⋯=𝑥𝑚=𝑥}=𝑐_𝑚𝑛⋅sin^{𝑚(𝑚+1)/2}𝑥⋅cos^{𝑛(𝑛+1)/2}𝑥, (96) with 𝑐_𝑚𝑛 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)Taking 𝑓=𝑒^𝑖𝑘𝑥,𝑘²=𝐸 in the formula (64) for the solutions of (71) we see that the related function is 𝜓(𝑥,𝐸)=𝑃(𝑥,𝑘)⋅𝑒^𝑖𝑘𝑥 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: 𝑢(𝑥)=𝑚(𝑚−1)𝑘²𝑐𝑛²𝑥𝑑𝑛²𝑥+𝛼(𝛼−1)𝑑𝑛²𝑥𝑐𝑛²𝑥+𝑝𝑝−1𝑠𝑛²𝑥+𝑛(𝑛−1)𝑘²𝑠𝑛²𝑥. (97) 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 Section 1 with [29–31] shows that the DPT trigonometric potentials produce the subfamily of two dimensional Huygens potentials 𝑉(𝑟,𝜑) via a formula 𝑉=1𝑟,𝜑𝑟²𝑚(𝑚+1)sin²𝜑+𝑛(𝑛+1)cos²𝜑, (98)  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): 𝑉2𝑟,𝜑=−𝑟²𝜕²𝜕𝜑²ln𝑊𝜒₁,…,𝜒_𝑚,𝜒_𝑗=sin𝑘_𝑗𝜑+𝛿_𝑗,𝑗=1,…,𝑚;𝑘_𝑗∈ℤ,𝛿_𝑗∈ℝ. (99)

Therefore, taking 𝑘_𝑗=𝑎_𝑗, where 𝑎_𝑗 are defined in (19) and taking 𝛿_𝑗=0, 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].