Wronskian addition formula and its applications

P. Gaillard + Universityé de Bourgogne, Dijon, France : e-mail: Pierre.Gaillard@u-bourgogne.fr,

+∗ Vladimir Matveev ^{∗} + Université de Bourgogne Laboratoire Gevrey de Mathématique Physique e-mail: Vladimir.Matveev@u-bourgogne.fr

∗ 01.10.01-30.10.02:Max-Planck-Institut für Mathematic, Vivatsgasse 7 , 53111 , Bonn, Germany e-mail: matveev@mpim-bonn.mpg.de

April 22, 2002

Abstract

We associate with every positive integer m the same number of Wronskian determinants of the order m corresponding to particular sequences of sine functions. We show that all these determinants have a form of a product of integer powers of sine and cosine functions multiplied by some integer factor. This result can be interpreted as addition formula for sine functions. We consider various applications of this formula including Wronskian representations for the solutions of the Schrödinger operators with the DPT potentials and the solutions of the Cauchy initial problem for the KdV hierarchy with Darboux-Pöshl-Teller initial conditions.

∗The second author wishes to thank Max-Planck-Institut where this work was completed for hospitality

1 Wronskian addition formula for sine functions

Let n and d > 0 be some nonegative integers, m = n+d, ^{1} q and x_{0} be some complex valued parameters , and φ_{p}(x) are the functions defined as follows:

φ_{p}(x) := sin qa_{p}(x − x_{0}), a_{p} = p, if 1 ≤ p ≤ m − n;  (1)
a_{m−n+j} = m − n + 2j, j = 1, …, n.

We use below the standard notation

W (f_{1}, f_{2}, …, f_{m}) := det ||∂_{x}^{j−1}f_{k}(x)||; j, k = 1, …, m,

for Wronskian determinant of any m functions and the short notation W_{mn} = W_{mn}(q, x, x_{0}) := W (φ_{1}, ..., φ_{m}), where φ_{x} were defined in (1). With these notations we have the following formulas. ^{2}

W_{m0} = (−2q)^{\frac{m(m−1)}{2}} ∏_{k=0}^{m−1} k! · sin^{\frac{m(m+1)}{2}} q(x − x_{0}),  (2)

W_{mn} = c_{mn} · sin^{\frac{m(m+1)}{2}} q(x − x_{0}) cos^{\frac{n(n+1)}{2}} q(x − x_{0}),  (3)

c_{mn} = (−2q)^{\frac{m(m−1)}{2}} · 2^{n^{2}} ∏_{p=0}^{d−1} p! ∏_{k=1}^{n} \frac{(d + 2k − 1)!}{(2k − 1)!}(k − 1)!, m ≥ 2, d ≥ 1

We note Ŵ_{mn} the determinant obtained from W_{mn} replacing sin functions φ_{j}(x) by the sinh functions g_{j}(x) of the same argument:

Ŵ_{mn} := W (g_{1}, …, g_{m}),
g_{p}(x) := sinh qa_{p}(x − x_{0}), a_{p} = p, if 1 ≤ p ≤ m − n;
a_{m−n+j} = m − n + 2j, j = 1, …, n.

Then the following hyperbolic versions of (2) and (3) holds

Ŵ_{m0} = (2q)^{\frac{m(m−1)}{2}} ∏_{k=0}^{m−1} k! · sinh^{\frac{m(m+1)}{2}} q(x − x_{0}),  (4)

1We shall not consider the case d=0, m=n, since it is equivalent to the case n=0, d=m modulo replacing q by 2q in the corresponding formulas

2product 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.

Ŵ_{mn} = |c_{mn}| · sinh^{\frac{m(m+1)}{2}} q(x − x_{0}) cosh^{\frac{n(n+1)}{2}} q(x − x_{0}),  (5)

where c_{mn} is the same as before. Replacing (1) by the system of functions

h_{p}(x) := cos qa_{p}(x − x_{0}), a_{p} = p, if p is odd h_{p}(x) := sin qa_{p}(x − x_{0}), a_{p} = p, if p is even, 1 ≤ p ≤ m − n; a_{m−n+j} = m − n + 2j, j = 1, …, n. h_{m−n+j}(x) := cos qa_{m−n+j}(x − x_{0}), if m − n is odd h_{m−n+j}(x) := sin qa_{m−n+j}(x − x_{0}), if m − n is even,

we get for their Wronskian

w_{mn} := W (h_{1}, …, h_{m}),  (6)

the following analog of (2) and (3):

w_{m0} = s_{m0}(2q)^{\frac{m(m−1)}{2}} ∏_{k=0}^{m−1} k! · cos^{\frac{m(m+1)}{2}} q(x − x_{0}),  (7)

w_{mn} = s_{mn}|c_{mn}| · cos^{\frac{m(m+1)}{2}} q(x − x_{0}) sin^{\frac{n(n+1)}{2}} q(x − x_{0}),  (8)

where c_{mn} has the same meaning as before and the sign factor s_{mn} is given by the formula

s_{mn} = (−1)^{\frac{(m−1)((m−n)^{2}+n)}{2}} .

This sign factor depends only on the values of m and n modulo 4. In particular s_{mn} = 1 if m = 1(mod 4), s_{mn} = −1 if m = 3(mod 4), and

s_{00} = 1, s_{01} = −1, s_{02} = −1, s_{03} = 1, s_{20} = 1, s_{21} = −1, s_{22} = −1, s_{23} = 1,

where the values of the indices m and n are reduced modulo 4.

Finally, replacing in the latter formulas sin and cos functions h_{j}(x) by sinh and cosh functions l_{j}(x) of the same arguments and using the notation ŵ_{mn} for their Wronskian we get the formulas:

ŵ_{mn} := W (l_{1} …, l_{m}), l_{p}(x) := cosh qa_{p}(x − x_{0}), a_{p} = p, if p is odd l_{p}(x) := sinh qa_{p}(x − x_{0}), a_{p} = p, if p is even, 1 ≤ p ≤ m − n; a_{m−n+j} = m − n + 2j, j = 1, …, n. l_{m−n+j}(x) := cosh qa_{m−n+j}(x − x_{0}), if m − n is odd l_{m−n+j}(x) := sinh qa_{m−n+j}(x − x_{0}), if m − n is even

ŵ_{m0} = (2q)^{\frac{m(m−1)}{2}} ∏_{k=0}^{m−1} k! · cosh^{\frac{m(m+1)}{2}} q(x − x_{0}),  (9)

ŵ_{mn} = |c_{mn}| · cosh^{\frac{m(m+1)}{2}} q(x − x_{0}) sinh^{\frac{n(n+1)}{2}} q(x − x_{0}).  (10)

Proof The proof of (2),(3) ,(5) (8) and (39) is obtained by use of the following propositions concerning Wronskians of n smooth functions. Proposition 1.

W (f · f_{1}, …, f · f_{n}) = f^{n}(x)W (f_{1}, f_{2}, …, f_{n}),  (11)

where f is any smooth function of x. This statement was first proved by Frobenius [7].

Another well known result which we use below is a formula of change of independent variable in a Wronskian. Let t = f(x) be a new variable so that f_{j}(x) = f̃_{j}(t) . Than we have the following statement. Proposition 2.

W (f_{1}, …, f_{n}) = (f′(x))^{\frac{n(n−1)}{2}} · W (f̃_{1}, f̃_{2}, …, f̃_{n}).  (12)

In particular, this proposition exhibits the dependence of the RHS of (3) on the parameter q:

W_{mn}(q, x, x_{0}) = W_{mn}(1, q(x − x_{0}), 0)q^{\frac{m(m−1)}{2}}

Therefore it is enough to prove (2) and (3) in the case q = 1, x_{0} = 0. We assume that this restriction is satisfied until the end of this section.

Formulas (5) (8) and (39) represent the trivial consequence of (2) and (3).We wrote them explicitely only by the reason of further applications to the differential equations considered below. It is enough to replace q by iq in order to obtain (5) from (3). Replacing x_{0} by x_{0} + π/2q we switch from (3) to (8). Replacing in q by iq in (8) we obviously get (39).

The proof of (3) can be obtained now in two steps. First, using the expansion

sin jx = sin x[(2 cos x)^{j−1}−(j−2) (2 cos x)^{j−3}+\frac{(j − 3)(j − 4)}{2}(2 cos x)^{j−5}+…] =

= u_{j}(cos x) sin x,  (13)

where u_{j}(t) is the Tchebyshev polynomial, we can drop out of the Wronskian in the RHS of (3) the factor sin^{m} x by virtue of Proposition 1.

Next, we make the change of variable t = cos x and use the formula (12). This leads us to appearance of the extra factor (− sin x)^{\frac{m(m−1)}{2}} and a new Wronskian W_{c}:

W_{c} := W (1, u_{2}(t), …, u_{m−n}(t), u_{m−n+2}(t), u_{m−n+4}(t), … u_{m+n}(t)), (14)

where the functions u_{j}(t) are Tchebyshev polynomials:

u_{j}(t) = \frac{sin jx}{sin x} =

= ∑_{k=0}^{[ \frac{n−1}{2} ]} (−1)^{k}C_{j−k−1}^{k} · (2t)^{j−1−2k}, t = cos x.

Now W_{mn} can be written as

W_{mn} = sin^{m} x · (− sin x)^{\frac{m(m−1)}{2}} W_{c} = −(− sin x)^{\frac{m(m+1)}{2}} W_{c},  (15)

It is easy to see that W_{c} is equal to the Wronskian obtained by replacing the Tchebyshev polynomials by their first terms i.e. by the terms containing highest powers of t. In the case n = 0 this leads to the following formula.

W_{c} = W_{p}, W_{p} := W (1, 2t, (2t)^{2}, …, (2t)^{m−1}).  (16)

Wronskian matrix corresponding to W_{p} has (in this particular case) the upper triangular form and hence W_{c} is equal to the product of its diagonal elements. Therefore, in the case n = 0 we get the following expression for W_{c}:

W_{c} = 2^{\frac{m(m−1)}{2}} · ∏_{j=1}^{m−1} j!

Substitution of this expression into (15) completes the proof of (3) for W_{m0}. In the case when m > n, n = 1, W_{c} reduces to the following expression

W_{c} = W (1, 2t, (2t)^{2}, …, (2t)^{m−2}, (2t)^{m}) = t · 2^{\frac{m(m−1)}{2} +1} m! ∏_{k=1}^{m−2} k!,

which is again in agreement with the general formula (3).

In the case m > n > 1 it is easy to show that the Wronskian (14) is the same as obtained by replacing all u_{j}(t) by their terms containing highest powers of t i.e.

W_{c} = W_{p}, W_{p} :=
= W (1, 2t, …, (2t)^{m−n−1}, (2t)^{m−n+1}, (2t)^{m−n+3}, …, (2t)^{m+n−1}). (17)

Wronskian matrix corresponding to W_{p} is not upper triangular any more. Nevertheless it can be always reduced to the upper triangular form by the elementary transformations of the right columns or of the upper rows. This can be done in such a way that the diagonal elements of the reduced matrix are proportional to the same powers of t as the diagonal elements of the non reduced matrix. For instance, in the case n = 2 it is easy to obtain this way the formula

W_{m2} = 2^{3} · (−2)^{\frac{m(m−1)}{2}} · \frac{(m − 1)!(m + 1)!}{3} ∏_{j=1}^{m−3} j! sin^{\frac{m(m+1)}{2}} x · cos^{3} x,

which is again in agreement with (3). The possibility of similar transformation of W_{c} reducing it to determinant of upper triangular matrix might be proved for any value of n, m > n ≥ 1. The related procedure preserves the structure of the principal diagonal i.e. we have the same powers of t at the same places as before but with different constant coefficients. These coefficients are certainly different from zero since W_{c} is obviously different from zero everywhere except the point t = 0. The above arguments show that W_{mn} really has the same functional structure as the RHS of (3). To complete the proof it is necessary to check that the coefficient c_{nm} is really the same as in (3). As soon as the correctness of the functional structure of (3) is established, c_{nm} can be calculated as follows.

Both expressions in the LHS and RHS of (3) are the polynomials of e^{ix}. So to check that the constant factor is correct we have to compute the coefficients of the highest power of e^{ix} in both sides of (3) and compare them. The related calculation leads to the following formula for c_{mn}:

c_{nm} = 2^{\frac{m(m−1)+n(n+1)}{2}} (−1)^{\frac{m(m−1)}{2}} V (1, 2, …, d, d + 2, …, d + 2n), d = m − n.

(18)

In the last formula V is a following Vandermond determinant:

V = V (a_{1}, …, a_{m}) = ∏_{m≥j>k≥1}(a_{j} − a_{k}),

where a_{j} were defined by (1). The short calculation gives the following formula for V:

V = 2^{\frac{n(n−1)}{2}} ∏_{p=0}^{p=d−1} p! ∏_{k=1}^{k=n} \frac{(d + 2k − 1)!}{(2k − 1)!}(k − 1)!

Substitution of the last expression in (16) completes the calculation of c_{mn} and hence also the proof of (3) □.

The following list of the first sixteen Wronskians W_{mn} with m ≤ 6 illustrates well the behavior of the coefficients c_{mn}, especially that of c_{m,m−1}, when m increases.

W (sin x, sin 2x) = −2 sin^{3} x, W (sin x, sin 3x) = −8 cos x sin^{3} x, W (sin x, sin 2x, sin 3x) = −2^{4} · sin^{6} x, W (sin x, sin 2x, sin 4x) = −2^{5} · 3 sin^{6} x cos x, W (sin x, sin 3x, sin 5x) = −2^{10} · sin^{6} x cos^{3} x, W (sin x, sin 2x, sin 3x, sin 4x) = 2^{8} · 3 sin^{10} x, W (sin x, sin 2x, sin 3x, sin 5x) = 2^{11} · 3 · cos x sin^{10} x, W (sin x, sin 2x, sin 4x, sin 6x) = 2^{13} · 3 · 5 sin^{10} x cos^{3} x, W (sin x, sin 3x, sin 5x, sin 7x) = 2^{20} · 3 sin^{10} x cos^{6} x, W (sin x, sin 2x, sin 3x, sin 4x, sin 5x) = 2^{15} · 3^{2} · sin^{15} x, W (sin x, sin 2x, sin 3x, sin 4x, sin 6x) = 2^{16} · 3^{2} · 5 cos x sin^{15} x, W (sin x, sin 2x, sin 3x, sin 5x, sin 7x) = 2^{21} · 3^{2} · 5 sin^{15} x cos^{3} x, = 94.371.840 sin^{15} x cos^{3} x, W (sin x, sin 2x, sin 4x, sin 6x, sin 8x) = 2^{24} · 3^{2} · 5 · 7 cos^{6} x sin^{15} x = 5.284.823.040 · cos^{6} x sin^{15} x, W (sin x, sin 3x, sin 5x, sin 7x, sin 9x) = 2^{35} · 9 cos^{10} x sin^{15} x = 309.237.645.312 · cos^{10} x sin^{15} x, W (sin x, sin 2x, sin 3x, sin 4x, sin 5x, sin 6x) = −2^{23} · 3^{3} · 5 · sin^{21} x = −1.132.462.080 sin^{21} x, W (sin x, sin 2x, sin 3x, sin 4x, sin 5x, sin 7x) = −3^{4} · 5 · 2^{25} · cos x sin^{21} x = = −13.589.544.960 cos x sin^{21} x.

2 Wronskian representations of the DPT potentials and beyond

Below we need the following two propositions. Let v(x) is any potential and f_{1}(x, E_{1}), f_{2}(x, E_{2}), …, f_{m}(x, E_{m}) are any solutions of the 1D Schrödinger (Sturm-Liouville) equation

−f′′ + v(x)f = Ef,  (19)

corresponding to E = E_{1}, E_{2}, …, E_{m}. Proposition 3 The solution ψ of the Schrödinger equation

−ψ′′ + uψ = Eψ, u = v − 2∂_{x}^{2} log W (f_{1}, f_{2}, …, f_{m}))  (20)

is given by the formula

ψ(x, E) = \frac{W (f_{1}, f_{2}, …, f_{m}, f(x))}{W (f_{1}, f_{2}, …, f_{m})}.  (21)

In the case m = 1, this proposition was found and proved by Darboux [5], and in general case by Crum [4]. Remarkably, their result can be extended to the partial differential equations and their non Abelian and lattice versions, [15] which founds many applications in modern theory of solitons [16, 17]. Below we need this extension only for the particular case of the scalar evolution PDE of the following form:

∂_{t}f = ∑_{j=0}^{N} v_{m}(x, t)∂_{x}^{j}f(x, t).  (22)

Let f_{1}, f_{2}, …, f_{m} be any m linearly independent solutions of (22) different from f. Then the following statement holds. Proposition 4.^{3} The RHS of (21) satisfy to the following PDE

∂_{t}ψ = ∑_{j=0}^{N} u_{m}(x, t)∂_{x}^{j}ψ(x, t),  (23)

where the coefficients u_{m}(x, t) can be explicitely 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−1} = 0, u_{N−2} = v_{N−2} + Nv_{N}∂_{x}^{2} log W (f_{1}, f_{2}, …, f_{m})).

(24)

The RHS of (21) represents the action of the differential operator of order N on the function f :

ψ = D(m)f := (∂_{x}^{m} + ∑_{k=0}^{m−1} g_{k}(x, t)∂_{x}^{k})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 = (j_{1}, j_{2}, …, j_{m}) be any permutation of the numbers (1, 2, …, m). To each permutation corresponds the following factorization of D(m):

D(m) = D_{j_{m}} · D_{j_{m−1}} ⋯ D_{j_{1}}, D_{j_{k}} = ∂_{x} − σ_{j_{k}},

3See [16, 17] for the proof and further details

σ_{j_{k}} := ∂_{x} log \frac{W (f_{j_{1}}, f_{j_{2}}, …, f_{j_{k}})}{W (f_{j_{1}}, f_{j_{2}}, …, f_{j_{k−1}})}.  (25)

Assume now that J = (1, 2, ..., m), v = 0, q and x_{0} are real, and f_{j} = φ_{j}, where φ_{j} are defined in (1). Combining formula (3) with Proposition 3 we get the following result. The function ψ corresponding to the case f_{j} = φ_{j}, v = 0, solves the Schrödinger equation with real singular periodic potential u_{mn}:

−ψ′′ + U_{mn}(x)ψ = Eψ,  (26)

where

U_{m,n} = −∂_{x}^{2} log W_{mn} = \frac{q^{2}m(m + 1)}{sin^{2} q(x − x_{0})} + \frac{q^{2}n(n + 1)}{cos^{2} q(x − x_{0})}, m > n. (27)

Replacing W_{mn} by Ŵ_{mn} we obtain the real potential Ũ_{m,n} exponentially decreasing at infinity:

Ũ_{m,n} = −∂_{x}^{2} log Ŵ_{mn} = \frac{q^{2}m(m + 1)}{sinh^{2} q(x − x_{0})} − \frac{q^{2}n(n + 1)}{cosh^{2} q(x − x_{0})}, m > n. (28)

General solution of the related Schrödinger equation (26) is obtained from (21) by choosing f_{j}(x) = g_{j}(x). Quite similarly we have

u_{mn} := −2∂_{x}^{2} log w_{mn} = \frac{q^{2}m(m + 1)}{cos^{2} q(x − x_{0})} + \frac{q^{2}n(n + 1)}{sin^{2} q(x − x_{0})}, m > n. (29)

where w_{mn} was defined in (6) and the solution ψ(x, E) of the related Schrödinger equation can be obtained by choosing f_{j} = h_{j} in (21).

It is clear as well that

û_{mn} := 2∂_{x}^{2} log ŵ_{mn} = −\frac{q^{2}m(m + 1)}{cosh^{2} q(x − x_{0})} + \frac{q^{2}n(n + 1)}{sinh^{2} q(x − x_{0})}, m > n, (30)

and the solutions of the related Schrödinger equation are obtained from (21) by choosing f_{j}(x) = l_{j}(x). In the case n = 0, û_{mn} is a nonsingular real reflectionless potential and the discrete negative eigenvalues of the related Schrödinger operator on the real line are E_{j} = −q^{2}j^{2}.

The integrability of (26) with potential (27) with q = 1, x_{0} = 0 was discovered first time by Darboux in 1882 and rediscovered for its hyperbolic version (28) by Pöshl and Teller in 1933. Therefore, it is reasonable to call (27) , (28), (29) and (30) Darboux-Pöshl-Teller or shortly DPT potentials.

It seems, that the Wronskian determinant representations for DPT potentials and the related solutions of the Schrödinger equations written in Wronskian form were missing in the literature.

The solution of (26) found by Darboux [5] corresponds to particular factorization (25) with J = (1, 2, …, m) i.e. the related function ψ can be written as

ψ = D_{m} · D_{m−1} ⋯ D_{1}f, D_{k} = ∂_{x} − σ_{k}.

The related functions σ_{k} now can be computed combining (3) and (25), where, of course, the constant factor in the RHS of (3) can be ignored.

σ_{j} = qj cot q(x − x_{0}), j ≤ m − n (31)
σ_{m−n+k} = (m − n + k)q cot q(x − x_{0}) + kq tan q(x − x_{0}), k = 1, ..., n. (32)

Original derivation of (31) by Darboux [5] was based on the observation that the functions sin^{m}(x − x_{0}) cos^{n} q(x − x_{0}) represent the particular solutions of the Schrödinger equation (26) with potential U_{m−1,n−1} corresponding to E = q^{2}(m + n)^{2} and was not using the general Crum’s formulas as we did.

The advantage of 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 4. The j-th equation of the KdV hierarchy can be obtained as a compatibility condition of (19) and the following evolution equation

\frac{∂f}{∂t_{j}} = c_{j} \frac{∂^{2j+1}f}{∂x^{2j+1}} − \frac{(2j + 1)c_{j}}{2} v(x, t_{j})∂_{x}^{2j−1}f + ∑_{p=0}^{2j−2} v_{p}(x, t)∂_{x}^{p}f.  (33)

In particular for j = 1 and c_{j} = −4 compatibility of (19) and (33) implies that v satisfies the KdV equation

v_{t} = 6 v v_{x} − v_{xxx}  (34)

In general the choice of c_{j} is arbitrary : it fixes normalization of the KdV hierarchy. Now according to the Proposition 4 for any given solution v(x, t_{j}) of j-th KdV equation the sequence f_{1}, …, f_{m} of the common solutions of (19) and (33), generates the new solution u(x, t_{j}) of the same equation:

u(x, t_{j}) = v(x, t_{j}) − 2∂_{x}^{2} log W (f_{1}, f_{2}, …, f_{m}).  (35)

In particular, in the case

v = 0, f_{p} = sin(qp(x−x_{0})+(−1)^{j}c_{j}q^{2j+1}p^{2j+1}t_{j}), p = 1, …, d, d = m−n,

f_{d+l} = sin[q(d+2l)(x−x_{0})+(−1)^{j}c_{j}q^{2j+1}(d+2l)^{2j+1}t_{j}], l = 1, …, n, m ≥ n,

the RHS of (35) is the solution of the j-th KdV equation with the initial data equal to the DPT potential (27). Therefore, we obtained a very simple description of the isospectral deformations of the DPT potentials generated by the KdV hierarchy. In particular the solution of (34) with the initial condition (27) is given by the formula ( 35) with v = 0 and

f_{p} = sin(qp(x − x_{0}) + 4q^{3}p^{3}t), p = 1, …, d, d = m − n
f_{d+l} = sin[q(d + 2l)(x − x_{0}) + 4q^{3}(d + 2l)^{3}t], l = 1, …, n, m ≥ n.

Similar formulas for the action of KdV and higher KdV flows on the potentials (29),(28),(30) are also obvious from the previous discussion. For instance, the solutions of the KdV equation (34) with the initial condition v(x, 0) = û_{mn}(x) defined by (30) is given by the formula

v(x, t) = −2∂_{x}^{2} log W (L_{1}, …, L_{m}),  (36)

where the functions L_{j}(x, t) are defined as follows:

L_{p}(x, t) := cosh qa_{p}(x − x_{0} − 4q^{2}a_{p}^{2}t), a_{p} = p, if p is odd L_{p}(x, t) := sinh qa_{p}(x − x_{0} − 4q^{2}a_{p}^{2}t), a_{p} = p, if p is even, 1 ≤ p ≤ m − n; a_{m−n+j} = m − n + 2j, j = 1, …, n. L_{m−n+j}(x, t) := cosh qa_{m−n+j}(x − x_{0} − 4q^{2}a_{m−n+j}^{2}t), if m − n is odd L_{m−n+j}(x) := sinh qa_{m−n+j}(x − x_{0} − 4q^{2}a_{m−n+j}^{2}t), if m − n is even.

3 Differential identities

Formulas (2) and (3) lead to some differential identities. Let D̂ be a differential operator defined for any integer m > 2 by the formula

(D̂_{m}f)(x) := ( ∏_{k=2}^{m} ∂_{x_{k}}^{k−1}) f(x_{1}, x_{2}, …, x_{m})|_{x_{1}=x_{2}=...=x_{m}=x}  (37)

Let Q_{m} be the following determinant: Q_{m} = det a_{j,k}, a_{j,k} = sin jx_{k}, 1 ≤ j, k ≤ m. Clearly, we have

D̂_{m}Q_{m} = det \frac{∂^{j−1} sin kx}{∂x^{j−1}} = W (sin x, sin 2x, …, sin mx) = W_{m0}. (38)

The determinant Q_{m} can be easily computed by use of (13) and further obvious reduction to a Vandermond determinant:^{4}

Q_{m} = 2^{\frac{m(m−1)}{2}} ∏_{k=1}^{m} sin x_{k} ∏_{k>j}(cos x_{k} − cos x_{j})  (39)

Now, substituting this expression for Q_{m} in (44) and taking into account the formula (3) for W_{m0}, we get the following identity:

D̂_{m} [( ∏_{k=1}^{m} sin x_{k}) · (∏_{j>k}(cos x_{k} − cos x_{j}))]

= ∏_{k=1}^{m−1} k! sin^{\frac{m(m+1)}{2}} x  m ≥ 2  (40)

Similar identities might be obtained along the same lines , comparing (3) with D̂_{m} det B , where B_{jk} = φ_{p}(x_{j}), j, p = 1, …, m, and the functions φ_{p}(x) were defined in (1). For instance for n = 1, q = 1, x_{0} = 0 this leads to the following identity:

D̂_{m} [( ∏_{k=1}^{m} sin x_{k}) · (∑_{j=1}^{m} cos x_{j}) · (∏_{k>j}(cos x_{j} − cos x_{k}))]

= m! ( ∏_{k=1}^{m−2} k!) cos x · sin^{\frac{m(m+1)}{2}} x,  (41)

where the operator D̂_{m} is defined by (37).

It is possible to generalize and unify the identities above in a single formula (47) using a slightly different approach.

It is obvious that the Wronskian

W̃_{mn} := W (1, cos x, cos 2x, cos dx, cos(d + 2)x, cos(d + 4)x, …, cos(d + 2n)x)

is proportional to W_{mn}:

W̃_{mn} = (−1)^{m}d! ∏_{k=1}^{n}(d + 2k) · W_{mn},

W̃_{m0} = (−1)^{m}m! · W_{m0},  (42)

4The same expression, (up to a different constant normalization factor and the sign factor ), as the RHS of (39) recently appeared [24] (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_{q} when q → ∞.

This follows immediately from the expansion of W̃_{mn} by the elements of its first column. Assuming that n = 0, and introducing the determinant

G_{m} := det B, B_{jk} = cos j x_{k}, j = 0, 1, …, m; k = 1, 2, … m + 1, (43)

we obviously have the equality

W̃_{m0} = D̂_{m+1}G_{m},  (44)

where D̂_{m} is defined by (37). Now G_{m} can be easily computed using the expansion

cos jx_{k} = 2^{j−1} cos^{j} x_{k} + sum of the lower powers of cos x_{k}.

Replacing cos jx_{k} by the first term of this expansion in G_{m} we get the Vandermond determinant equal to G_{m}. Therefore, for G_{m} the following expression holds

G_{m} = 2^{\frac{m(m−1)}{2}} ∏_{m+1≥j>k≥1}(cos x_{j} − cos x_{k}).  (45)

Comparing (45) and (44) with explicit expression for W̃_{m0}, following immediately from (42), and replacing m by m−1 we obtain the following identity :

D̂_{m}V_{m} = ( ∏_{k=1}^{m−1} k!) · sin^{\frac{m(m−1)}{2}} x, V_{m} := ∏_{m≥k>j≥1}(cos x_{j} − cos x_{k}). (46)

Replacing D̂_{m} by any operator D̃ obtained by removing from D̂_{m} some partial differentiations we have the obvious relation D̃ · G_{m−1} = 0. Combining this relation with the fact that V_{m} is proportional to G_{m−1}, we get the following identity:

D̂_{m}(V_{m} · f(x_{1}, …, x_{m})) = f(x_{1}, …, x_{m})|_{x_{1}=x_{2}=...=x_{m}=x} D̂_{m}(V_{m}), (47)

where explicit expression for D̂_{m}(V_{m}) is given by (46). It is also obvious that

D̃(V_{m} · f(x_{1}, …, x_{m})) = 0,  (48)

where again D̃ means the differential operator obtained from D̂_{m} by removing some partial differentiations.

The identities (40,41) now can be considered as immediate consequence of (47) and (46) corresponding to particular choices of the function f(x_{1}, x_{2}, ...x_{m}) in (47).

4 Additional comments

• Initial motivation of this article was the following question. How to select the solutions f_{j} of the free Schrödinger equation −f′′_{j} = E_{j}f_{j} in order to get the DPT potential from Crum formula, (20) with v = 0. The choice of E_{j} was clear from [5] but at every point E_{j} we have the generic solution A_{j} sin(\sqrt{E_{j}}x+x_{j}), where A_{j} play no role in computing ∂_{x}^{2} log W (f_{1}, …, f_{m}). In general, there is no compact expression for the Wronskian provided that the phases are chosen arbitrary. It was also clear that some selection of phases x_{j} leading to DPT potential via (20) should exist but the fact that trivial phases provide the right answer apriory was not obvious and it was necessary to prove (3), (at least modulo constant normalization factor c_{mn}), in order to confirm it.

• There is some similarity of (3), with n = 0 with addition formulas for elliptic functions obtained by Kiepert, Brioschi, Hermite, Frobenius and Stickelberger [6, 23] 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.

• Taking f = e^{ikx}, k^{2} = E in the formula (21) for the solutions of (26 ) we see that the related function 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 potentials [3]. In particular, all the DPT potentials (27),(29), (28),(30) might be obtained from the general m-gap hyperelliptic potentials by taking appropriate limit in Its-Matveev formula ([3]), which is somehow technically less convinient with respect to direct algebraic approach used here.

• After Darboux, integrability of the hyperbolic version of the same potentials was rediscovered [18] by Pöschl and Teller (father of the US hydrogenous bombe). These authors certainly did not observed that the same (and more general) equation was already solved by Darboux but they found important applications of the DPT potentials in quantum theory of two atomic molecules. Later the Schrödinger equation with the same potentials, (mainly called Pöschl-Teller potentials in quantum-mechanical literature,) were discussed by many authors for different boundary conditions in connection with different physical applications, see for instance [1, 2, 14, 10, 12, 11].

• Using the change of variable reducing the Schrödinger equation with DFT potential to the Fuchs equation with 3 singular points Darboux found another form of the solution of (26):

y = sin^{m} x cos^{n} xF (α, β, γ, sin^{2} x),  (49)

γ = \frac{m + 1}{2}, α = \frac{m + n}{2} + \frac{1}{2}\sqrt{λ},

β = \frac{m + n}{2} − \frac{1}{2}\sqrt{λ},

where F is any solution of the Gauss hypergeometrical equation. This form of the solution was also rediscovered several times by some of the aforementioned authors.

• Darboux [8] also found the integrable elliptic generalization of the DPT potentials also representing the 4-parametric generalization of the Lame equation. This elliptic Darboux potential has the following form:

u(x) = \frac{m(m − 1)k^{2}cn^{2}x}{dn^{2}x} + \frac{q(q − 1)dn^{2}x}{cn^{2}x} + \frac{p(p − 1)}{sn^{2}x} + n(n − 1)k^{2}sn^{2}x

(50)

In (50) sn, cn and dn are usual Jacobi elliptic functions of modulus k and m, q, p, n are arbitrary non negative integers. In the case m=q=p=0, this equation reduces to Lamé equation first integrated by Hermite. The potentials of the form (50) were rediscovered relatively recently by Trebich and Verdier [22] as a special reduction of the general formula [9, 13, 3] describing all finite-gap periodic or quasiperiodic potentials . More general connections between the finite gap potentials and Fuchsian equations were recently discussed by Smirnov [20].

In the limit when one of the periods of the elliptic functions tends to infinity the Darboux-Vedrier-Trebich potential reduces to the Darboux-Pöschl-Teller potential

• The DPT potentials and the Darboux-Verdier-Trebich potentials recently found some applications to the algebraic Bethe Anzatz construction for trigonometric case of the quantum Calogero-Moser multi particle system. [21]

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