Symmetry, Integrability and Geometry: Methods and Applications On Darboux’s Approach to R-Separability of Variables ⋆

We discuss the problem of $R$-separability (separability of variables with a factor $R$) in the stationary Schr\"odinger equation on $n$-dimensional Riemann space. We follow the approach of Gaston Darboux who was the first to give the first general treatment of $R$-separability in PDE (Laplace equation on ${\mathbb E}^3$). According to Darboux $R$-separability amounts to two conditions: metric is isothermic (all its parametric surfaces are isothermic in the sense of both classical differential geometry and modern theory of solitons) and moreover when an isothermic metric is given their Lam\'e coefficients satisfy a single constraint which is either functional (when $R$ is harmonic) or differential (in the opposite case). These two conditions are generalized to $n$-dimensional case. In particular we define $n$-dimensional isothermic metrics and distinguish an important subclass of isothermic metrics which we call binary metrics. The approach is illustrated by two standard examples and two less standard examples. In all cases the approach offers alternative and much simplified proofs or derivations. We formulate a systematic procedure to isolate $R$-separable metrics. This procedure is implemented in the case of 3-dimensional Laplace equation. Finally we discuss the class of Dupin-cyclidic metrics which are non-regularly $R$-separable in the Laplace equation on ${\mathbb E}^3$.


Introduction
One of the highlights of Darboux's research on the whole is a memoir [11] devoted mainly to orthogonal coordinates in Euclidean spaces. The fundamental monograph [13] includes much of the material of [11]. The last fifty pages of the third and last part of the memoir [12] are nothing else but the first general treatment of the R-separability of variables (separability of variables with a factor R) in a PDE. Remark 2. If R = 1 or more generally if R = i r i (x i ), we replace the term "R-separability" by the term "separability".

R-separability in the Schrödinger equation
We assume that a Riemann space R n admits local orthogonal coordinates u = (u 1 , . . . , u n ) in which the metric has the following form (1.4) H.P. Robertson was the first to consider the stationary Schrödinger equation on R n equipped with orthogonal coordinates ∆ψ + k 2 − V ψ = 0, (1.5) where is the Laplace-Beltrami operator on R n , k is a scalar and V = V (u) is a potential function [27]. We adapt the Definition 1 to the case of equation (1.5) as follows. (1. 8) In the context of the R-separability in the Schrödinger equation the following problem seems to be fundamental.
R-separability problem. Let R n be a Riemann space with a metric ds 2 = g ij dx i dx j , where (x i ) are local coordinates. We assume that R n admits orthogonal coordinates and we are given a class of R-separable metrics (1.4). By R-separability problem, we mean the problem of isolating those metrics of the class which are equivalent to the metric ds 2 .
Remark 3. As is well known a generic R n for n > 3 does not admit orthogonal coordinates [1, p. 470]. Any analytic R 3 always admits orthogonal coordinates [5] and even more any R 3 of C ∞ -class also admits orthogonal coordinates [15]. Also the problem when a given metric is diagonalizable seems to be very difficult [30].
Robertson proved in [27] that any n-dimensional Stäckel metrics satisfying the so called Robertson condition is separable in the Schrödinger equation (1.5) (see also (3.1) of this paper). The corresponding R-separability problem for n-dimensional Euclidean space has been solved by L.P. Eisenhart [16].

Darboux's R-separability problem
Gaston Darboux was interested in R-separability of variables in the Laplace equation on E 3 . His pioneering research in the field of R-separability [9,10,12,13] has been almost completely forgotten. It can be interpreted as an advanced attempt to solve the following specific Rseparability problem.
Here we do not use the original Darboux's notation dating back to Lamé writings. Instead, we apply the notation used in this paper. Theorem 1. The 3-dimensional diagonal metric if and only if the following two conditions are satisfied where G (i) does not depend on u i and f i depends only on u i , for appropriately chosen functions q i (u i ). Moreover, the resulting separation equations are Remark 4. Theorem 1 has never been explicitly stated by Darboux. Actually he applied this theorem in many places of his research. E.g. (3) in Chapter IV of [13] is a special case of (1.11) while (69) in Chapter V of [13] is a special case of (1.12).
In view of the Theorem 1 the question of R-separability of variables in the Laplace equation on E 3 amounts to the following R-separability problem: to isolate all the metrics with Lamé coefficients (1.11) which are flat and which satisfy (1.12). In other words, firstly, one has to find (classify) all solutions to the Lamé equations (i, j, k = (1, 2, 3), (2, 3, 1), (3,1,2)) under the ansatz (1.11) and, secondly, to select among them those satisfying the constraint (1.12). Darboux was successful in solving the Lamé equations under the ansatz (1.11). However as a rule he paid no closer attention to the question of separation equations and thus with one exception the constraint (1.12) was not the subject of his detailed analysis. This exceptional case not covered by the modern treatments of R-separability in the Laplace equation on E 3 is one of the Dupin-cyclidic metrics [13, pp. 283-286]. Indeed, Dupin-cyclidic metrics are non-regularly R-separable in the Laplace equation on E 3 and cannot be treated by the standard techniques discussed e.g. in [3]. For a discussion of regular and non-regular R-separability see [23].
Definition 3. A surface in E 3 is isothermic if, away from umbilics, its curvature net can be conformally parametrized.
Another important Darboux's result is as follows.
Theorem 2. If the metric (1.9) is R-separable in the Laplace equation on E 3 , then all the corresponding parametric surfaces are isothermic.

The class of isothermic surfaces is conformally invariant and in particular includes
• planes and spheres, • surfaces of revolution, • quadrics, • tori, cones, cylinders and their conformal images, i.e. Dupin cyclides, • cyclides or better Darboux-Moutard cyclides, • constant mean curvature surfaces and in particular minimal surfaces.
Apart from the fact that the Theorem 2 is a necessary condition for R-separability in the Laplace equation on E 3 , it is an interesting connection between the linear mathematical physics (separation of variables) and the non-linear mathematical physics (solitons). Indeed, the current interest in isothermic surfaces is mainly due to the fact that their geometry is an important example of the so called integrable or soliton geometry [8,28,4,19].

Aims and results of the paper
In this paper we extend the original Darboux's approach to R-separability of variables in the Laplace equation on E 3 to the case of the stationary Schrödinger equation on n-dimensional Riemann space R n admitting orthogonal coordinates. The Darboux's Theorem 1 is generalized as Theorem 3. Correspondingly 3-dimensional isothermic metrics (1.11) are generalized to n-dimensional isothermic metrics (2.3) while 3dimensional constraint (1.12) is generalized to n-dimensional constraint (2.11) which we call R-equation.
We distinguish a subclass (2.9) of isothermic metrics which we call the binary metrics. A representative example of the binary metric is n-elliptic metric (2.10). In the case of a binary metric the R-equation assumes the simpler form (2.12).
The approach is illustrated by examples of the Section 3. Here we discuss two standard results and two less standard results. These are 1) Robertson paper revisited (Subsection 3.1), 2) the nelliptic metric (Subsection 3. The main result of the paper encoded in Theorem 3 suggests the following procedure to identify a given n-dimensional diagonal metric (1.4) as R-separable in n-dimensional Schrödinger equation. The procedure in question consists of three steps.
Firstly, we have to prove or disprove that the metric is isothermic. If the metric is not isothermic, then it is not R-separable. Suppose it is isothermic. As a result this step predicts R-factor and p i coefficients in the separation equations. Secondly, we set up the corresponding R-equation which we treat as an equation for q i coefficients in the separation equations. Thirdly, we attempt to solve the R-equation. Any solution to the R-equation concludes the procedure: R-separability of the starting metric is proved and in particular the corresponding separation equations are explicitly constructed. Notice that unknowns q i enter into R-equation linearly and this is the right place to introduce (linearly) extra parameters (separation constants) into the separation equations. In Subsection 2.3 we introduce remarkable algebraic identities (Bôcher-Ushveridze identities) which can be successfully applied in solving R-equation. This is a remarkably simple procedure and its implementation in the case of 3-dimensional Laplace equation is discussed in Subsection 4.1 together with the relevant examples.
Gaston Darboux found a class of Dupin-cyclidic metrics which are R-separable in the Laplace equation on E 3 . These are non-regularly R-separable and can not be covered by the modern standard approaches. In Subsection 4.3 we re-derive this remarkable result. The original Darboux's calculations are long and rather difficult to control. Here we simplify the derivation using the standard Riemannian tools (Ricci tensor and Cotton-York criterion of conformal flatness).

The main result
Here we extend Darboux's Theorem 1 valid for 3-dimensional Laplace equation (1.10) to the case of n-dimensional stationary Schrödinger equation (1.5). Correspondingly, we extend the Definition 4 of isothermic metrics to n-dimensional case.

1)
• second condition of R-separability (called R-equation) B. The metric (1.4) satisfies the first condition of R-separability if and only if it can be cast into the form 2) are satisfied then the corresponding separation equations read Proof . A. R-separability implies (2.1) and (2.2). Indeed, we insert ψ = R i ϕ i into (1.5) and make use of (1.6). This results in for an arbitrary choice of solutions ϕ i . Let (ϕ i1 , ϕ i2 ) be a basis in the solution space of the corresponding equation. We put Thus for each ϕ i (λ i = 0) we have Certainly, without loss of generality we can replace F (i) by 3) and vice-versa.
Notice that (2.3) for n = 3 gives the isothermic metric of Definition 4.

Definition 5. The metric (2.3) is called isothermic.
We introduce now an important sub-class of isothermic metrics. Given n
Example. The n-elliptic coordinates on E n [20,22,17,33]. We choose n real numbers b i such that b 1 > b 2 > · · · > b n . The n-elliptic coordinates λ = λ 1 , λ 2 , . . . , λ n satisfy inequalities The following formulae give rise to a diffeomorphism onto any of 2 n open n-hyper-octants of E n equipped with the standard Cartesian coordinates x = (x 1 , x 2 , . . . , x n ) The corresponding n-elliptic metric is The n-elliptic metric is binary thus isothermic.

R-equation
Having found the general formulae (2.3) and (2.9) for isothermic metrics which -ex definitionesatisfy the 1st condition of R-separability, we are in a position to claim that the various questions of R-separability amount to the 2nd condition of R-separability (2.2) which we call R-equation.
B. The binary metric (2.9) is R-separable in the Schrödinger equation if and only if Proof . Indeed, both (2.11) and (2.12) are R-equations rewritten in terms of the corresponding metric.
Remark 7. Notice that the linear operators acting on R −1 in (2.11) and (2.12) also define the separation equations (2.4).

Bôcher-Ushveridze identities
Gaston Darboux was the first to discuss the so called triply conjugate coordinates in E 3 [12]. These constitute a projective generalization of orthogonal coordinates in E 3 . In this context he introduced the following system of three equations for a single unknown M ( and gave a general solution to it in the form where m i (x i ) are arbitrary functions (see formulae (40), (41) and (42) in [12]). A generalization of (2.13) and (2.14) is straightforward. Consider in R n the following system of n 2 PDEs for a single unknown M (x 1 , x 2 , . . . , x n ) This is the overdetermined system of PDEs which is an example of the so called linear Darboux-Manakov-Zakharov system [34]. Fortunately (2.15) is involutive (see Proposition 1 in [34]). Its general solution reads .
for m = n − 1, homogeneous polynomial of degree and homogeneity = d for m ≥ n.
where σ i are elementary symmetric polynomials: . . and f l 1 l 2 ...l d are constants defined uniquely by r.h.s. of (2.17). In particular The identities (2.18) and in particular the identities (2.16) we call the Bôcher-Ushveridze identities. Certainly, both sides of any Bôcher-Ushveridze identity is a particular solution to the Euler-Poisson-Darboux system. Notice also that functions m i (x i ) are not defined by M uniquely. As we shall see both the Euler-Poisson-Darboux system and the Bôcher-Ushveridze identities can be applied in discussing R-equation.

Examples
In this section we discuss two standard results and two less standard results within the developed approach. In all cases the approach offers alternative and much simplified proofs or derivations.

Robertson paper revisited
Here we present the essence of Howard Percy Robertson fundamental paper [27]. Our aim is to re-derive the basic formulae (A), (B), (C) and (9) of the paper using earlier stated results.
In (1) of [27] we put k = 1 and replace E by k 2 . Notice that e.g. Robertson's h i is our H −2 i . The paper deals with the case R = 1.

R-equation (2.2) is now the functional constraint which is bilinear in H
(3.1) We decompose q i as follows where v i are arbitrary. Inserting (3.2) into (3.1) gives and decompose (Q i ) in this basis as follows where the coefficients of the decomposition are arbitrary constants.
Remark 9. (3.3) and (3.6) introduce (non-uniquely!) an n × n matrix q = [q ij (u i )]. We assume that q is non-singular everywhere. It is called the Stäckel matrix. Notice that the co-factor Q ij of q ij does not depend on u i .

We collect (3.3) and (3.6) as
Inverting of (3.8) yields It is clear that the metric satisfies (3.1) or the 2-nd condition of R-separability (2.2). Finally we demand the metric (3.10) has to satisfy the first condition of R-separability (2.1) and thus (2.8) is which means that (3.12) exactly conforms to (2.8). As a result of (3.12), (3.2) and (3.7) the separation equations are

The n-elliptic metric
It is well known that n-elliptic metric (2.10) is separable (R = 1) in the Schrödinger equation with an appropriately chosen potential function. An indirect proof consists in showing that (2.10) is the Stäckel metric (in this case the Robertson condition is satisfied) and Eisenhart stated it without proof in [16, p. 302].
Theorem 5. The n-elliptic metric (2.10) is separable in the Schrödinger equation with a potential function where v i (λ i ) are arbitrary functions, i.e. V (λ) is an arbitrary solution to the Euler-Poisson-Darboux system (2.15). The corresponding separation equations are Proof . Indeed, from example (Subsection 2.1) we know that the metric (2.10) is isothermic. Again R-equation is reducible to the functional constraint We put where k m = const and v i (λ i ) are arbitrary functions. Now the Bôcher-Ushveridze identity (2.16) implies the statement.

Remarkable example of Kalnins-Miller revisited
Our setting can be easily extended to the pseudo-Riemannian case. Consider the following metric where λ 1 > λ 2 > λ 3 > 0. It is 3-dimensional Minkowski metric. Indeed, on replacing λ i by t, x and y t = 1 9 we arrive at Certainly, any metric conformally equivalent to (3.14) is an isothermic metric and thus satisfies the 1st condition of R-separability (2.1). Kalnins and Miller proved that the metric is R-separable in the Helmholtz equation (1.7) [21, p. 472]. We re-derive this remarkable result within our approach. First of all it is easy to predict R-factor (see (2.9)) and the form of the separation equations The point is that (3.16) is harmonic with respect of (3.15). Again R-equation is reducible to the functional constraint .

Fixed energy R-separation revisited
In order to treat R-separability in the Schrödinger equation (1.8) a pretty complicated formalism was proposed in [7]. Presumably some part of the formalism of [7] can be simplified according to the following result.
Proposition 1. Any isothermic metric which is R-separable in n-dimensional Laplace equation is R-separable in n-dimensional Schrödinger equation with k = 0 for an appropriately chosen potential function.
Proof . Consider the isothermic metric given by (2.3). R-separability of (2.3) in n-dimensional Laplace equation implies that R-equation simplifies to We put where v i (u i ) are arbitrary functions and defineq i = q i − v i . Then (3.17) can be rewritten as 4 R-separability in 3-dimensional case 4.1 Procedure to detect R-separable metrics Here we describe a simple procedure to identify a given 3-dimensional diagonal metric as Rseparable in 3-dimensional Laplace equation.

Proposition 2.
In the 3-dimensional case any isothermic metric is binary.
Proof . Indeed, we put n = 3 in (2.3) and hence we deduce the following expressions for Lamé coefficients H i or more explicitly Now see (2.9).
To simplify notation we rewrite (4.1) as

Examples
Here we present five examples proving efficiency of our procedure.

Spherical metric
The spherical metric ds 2 = dr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2 is isothermic. It is easily seen that in this case. Equation (4.7) reads and can be easily solved The resulting separation equations read

Toroidal metric II
Interestingly, the metric (4.8) can be identified as isothermic in two ways. It was shown implicitly in [7]. Indeed, we rewrite (4.8) as follows Metric (4.11) suggests the following identifications Again (4.9) holds. Hence equation (4.5) is satisfied if and only if A general solution to (4.12) is The resulting separation equations read

Cyclidic metric
Consider the following metric We select d = 0 and p = 1/ √ abc. Hence the metric It is isothermic and We readily check the equality Hence equation (4.5) is satisfied if and only if .

Dupin-cyclidic metric
The metric is Dupin-cyclidic [26]. It is R-separable in the Helmholtz equation (1.7) on E 3 (see Theorem 1 in [26]). Here we give an alternative and remarkably simple proof of this result. Metric (4.16) is isothermic and It is easily to verify the equality which is equation (4.4) in this case. Taking into account H 3 = 1 we rewrite (4.17) Certainly, (4.18) is R-equation (2.2) for 3-dimensional Helmholtz equation (1.7). The corresponding separation equations are There are many definitions (not necessarily equivalent) of Dupin cyclides (see [6, p. 148]). We select the following one.

Definition 9.
A Dupin cyclide is a regular parametric surface in E 3 whose both principal curvatures are constant along their curvature lines.
Let us recall the celebrated theorem of Dupin (see [18, p. 609]). Theorem 6. Let u = (u 1 , u 2 , u 3 ) be orthogonal coordinates in E 3 . Two arbitrary parametric surfaces u i = const and u j = const (i = j) intersect in a curvature line of each. A natural question arises as to when the isothermic metric (4.2) satisfies (4.19)? With no difficulty we prove the following result. Certainly, one solution to (4.20) is provided by the metric (4.16). On performing re-scaling in (4.16) u 1 = c cos u, u 2 = a cosh v, u 3 = w we arrive at Obviously, (4.21) is isothermic and from (4.22)-(4.24) we have  where the constants α i , β i and γ i satisfy identities Proof . 1) Suppose we are given the metric (4.21) whose Lamé coefficients are (4.27) and M is an arbitrary function. Then off-diagonal components of its Ricci tensor are 2 We identify (4.32) as R-equation