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 Schrödinger 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 E³)...

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Date:	2011
Main Authors:	Sym, A., Szereszewski, A.
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Language:	English
Published:	Інститут математики НАН України 2011
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Cite this:	On Darboux's Approach to R-Separability of Variables / A. Sym, A. Szereszewski // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 34 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1474132025-02-23T19:39:14Z On Darboux's Approach to R-Separability of Variables Sym, A. Szereszewski, A. We discuss the problem of R-separability (separability of variables with a factor R) in the stationary Schrödinger 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 E³). 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 Lamé 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 E³. This paper is a contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S⁴)”. The full collection is available at http://www.emis.de/journals/SIGMA/S4.html. Our thanks are due to reviewers for critical remarks and notably to the editors for valuable comments which inspired us to deeply revise our preprint. 2011 Article On Darboux's Approach to R-Separability of Variables / A. Sym, A. Szereszewski // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 34 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35J05; 35J10; 35J15; 35Q05; 35R01; 53A05 DOI: http://dx.doi.org/10.3842/SIGMA.2011.095 https://nasplib.isofts.kiev.ua/handle/123456789/147413 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We discuss the problem of R-separability (separability of variables with a factor R) in the stationary Schrödinger 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 E³). 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 Lamé 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 E³.
format	Article
author	Sym, A. Szereszewski, A.
spellingShingle	Sym, A. Szereszewski, A. On Darboux's Approach to R-Separability of Variables Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Sym, A. Szereszewski, A.
author_sort	Sym, A.
title	On Darboux's Approach to R-Separability of Variables
title_short	On Darboux's Approach to R-Separability of Variables
title_full	On Darboux's Approach to R-Separability of Variables
title_fullStr	On Darboux's Approach to R-Separability of Variables
title_full_unstemmed	On Darboux's Approach to R-Separability of Variables
title_sort	on darboux's approach to r-separability of variables
publisher	Інститут математики НАН України
publishDate	2011
url	https://nasplib.isofts.kiev.ua/handle/123456789/147413
citation_txt	On Darboux's Approach to R-Separability of Variables / A. Sym, A. Szereszewski // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 34 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT syma ondarbouxsapproachtorseparabilityofvariables AT szereszewskia ondarbouxsapproachtorseparabilityofvariables
first_indexed	2025-11-24T16:48:50Z
last_indexed	2025-11-24T16:48:50Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 095, 21 pages On Darboux’s Approach to R-Separability of Variables? Antoni SYM † and Adam SZERESZEWSKI ‡ † Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Poland E-mail: Antoni.Sym@fuw.edu.pl ‡ Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Poland E-mail: aszer@fuw.edu.pl Received February 18, 2011, in final form October 02, 2011; Published online October 12, 2011 http://dx.doi.org/10.3842/SIGMA.2011.095 Abstract. We discuss the problem of R-separability (separability of variables with a fac- tor R) in the stationary Schrödinger 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 E3). 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 more- over when an isothermic metric is given their Lamé 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 deriva- tions. 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 E3. Key words: separation of variables; elliptic equations; diagonal n-dimensional metrics; isothermic surfaces; Dupin cyclides; Lamé equations 2010 Mathematics Subject Classification: 35J05; 35J10; 35J15; 35Q05; 35R01; 53A05 1 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. 1.1 R-separability setting Let Λψ = 0 (1.1) be a linear PDE in variables x = ( x1, x2, . . . , xn ) for an unknown (function) ψ(x) and of order N . ?This paper is a contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”. The full collection is available at http://www.emis.de/journals/SIGMA/S4.html mailto:Antoni.Sym@fuw.edu.pl mailto:aszer@fuw.edu.pl http://dx.doi.org/10.3842/SIGMA.2011.095 http://www.emis.de/journals/SIGMA/S4.html 2 A. Sym and A. Szereszewski Definition 1. PDE (1.1) is R-separable (into ODEs) or x-variables are R-separable in equa- tion (1.1) if there exist a non-zero function R(x) and n linear ODEs Liϕi = 0, i = 1, 2, . . . , n, (1.2) each of order νi ≤ N and for a function ϕi(x i) such that the following implication holds if Liϕi = 0, i = 1, 2, . . . , n, then ψ(x) = R(x) ∏ i ϕi ( xi ) solves (1.1). (1.3) Also we say that R-separation occurs in equation (1.1). Equations (1.2) are called separation equations. Remark 1. Following Darboux we assume that coefficients of each equation (1.2) are just functions of variable xi not necessarily dependent on extra variables (parameters). This freedom from the Stäckel imperative is essential. Hence if (1.3) holds then we have a family of solutions to (1.1) depending at least on ∑ i νi parameters. Remark 2. If R = 1 or more generally if R = ∏ i ri(x i), we replace the term “R-separability” by the term “separability”. 1.2 R-separability in the Schrödinger equation We assume that a Riemann space Rn admits local orthogonal coordinates u = (u1, . . . , un) in which the metric has the following form ds2 = n∑ i=1 H2 i ( dui )2 . (1.4) H.P. Robertson was the first to consider the stationary Schrödinger equation on Rn equipped with orthogonal coordinates ∆ψ + ( k2 − V ) ψ = 0, (1.5) where ∆ = h−1 n∑ i=1 ∂ ∂ui h H2 i ∂ ∂ui , h = H1H2 · · ·Hn is the Laplace–Beltrami operator on Rn, 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. Definition 2. The Schrödinger equation is R-separable or metric (1.4) and potential V are R-separable in the Schrödinger equation if there exist 2n + 1 functions R(u) and pi(u i), qi(u i) (i = 1, 2, . . . , n) such that the following implication holds ϕ′′i + piϕ ′ i + qiϕi = 0, i = 1, 2, . . . , n ⇒ ψ(u) = R(u) ∏ i ϕi ( ui ) solves (1.5). (1.6) Particular cases of equation (1.5) are a) n-dimensional Laplace equation (k = 0 and V = 0) ∆ψ = 0, On Darboux’s Approach to R-Separability of Variables 3 b) n-dimensional Helmholtz equation (V = 0) ∆ψ + k2ψ = 0, (1.7) c) n-dimensional Schrödinger equation with k = 0 ∆ψ = V ψ. (1.8) In the context of the R-separability in the Schrödinger equation the following problem seems to be fundamental. R-separability problem. Let Rn be a Riemann space with a metric ds2 = gijdx idxj , where (xi) are local coordinates. We assume that Rn 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 ds2. Remark 3. As is well known a generic Rn for n > 3 does not admit orthogonal coordinates [1, p. 470]. Any analytic R3 always admits orthogonal coordinates [5] and even more any R3 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]. 1.3 Darboux’s R-separability problem Gaston Darboux was interested in R-separability of variables in the Laplace equation on E3. 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 R- separability 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 ds2 = H2 1 (u) ( du1 )2 +H2 2 (u) ( du2 )2 +H2 3 (u) ( du3 )2 (1.9) is R-separable in 3-dimensional Laplace equation( 3∑ i=1 ∂i H1H2H3 H2 i ∂i ) ψ = 0 (1.10) if and only if the following two conditions are satisfied i) H1 = G(2)G(3) R2f1 , H2 = G(1)G(3) R2f2 , H3 = G(1)G(2) R2f3 , (1.11) where G(i) does not depend on ui and fi depends only on ui, ii) 3∑ i=1 G2 (i)f 2 i [ ∂2iR −1 + f ′i fi ∂iR −1 + qiR −1 ] = 0 (1.12) for appropriately chosen functions qi(u i). Moreover, the resulting separation equations are ϕ′′i + f ′i fi ϕ′i + qiϕi = 0, i = 1, 2, 3. 4 A. Sym and A. Szereszewski 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 E3 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)) Hi,jk − 1 Hj Hi,jHj,k − 1 Hk Hi,kHk,j = 0, (1.13)( 1 Hi Hj,i ) ,i + ( 1 Hj Hi,j ) ,j + 1 H2 k Hi,kHj,k = 0, (1.14) 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 E3 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 E3 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 E3 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 E3, 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 E3, 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]. Definition 4. The metric (1.9) with Lamé coefficients (1.11) is called isothermic. On Darboux’s Approach to R-Separability of Variables 5 1.4 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 E3 to the case of the stationary Schrödinger equation on n-dimensional Riemann space Rn 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 3- dimensional 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 rep- resentative 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 n- elliptic metric (Subsection 3.2) (standard results) and 3) remarkable example of Kalnins–Miller revisited (Subsection 3.3), 4) fixed energy R-separation revisited (Subsection 3.4) (less standard examples). In all cases the approach offers alternative and simplified proofs or derivations. 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 pi coefficients in the separation equations. Secondly, we set up the correspon- ding R-equation which we treat as an equation for qi 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 qi 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 E3. 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 Dar- boux’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). 2 Isothermic metrics and R-equation 2.1 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. Theorem 3. A. The metric (1.4) and the potential V are R-separable in the Schrödinger equation (1.5) if and only if the following two conditions are satisfied • first condition of R-separability[ ln ( R2 h H2 i )] ,ij = 0, i 6= j, (2.1) 6 A. Sym and A. Szereszewski • second condition of R-separability (called R-equation) ∆R+ ( k2 − V − n∑ i=1 1 H2 i qi ) R = 0. (2.2) B. The metric (1.4) satisfies the first condition of R-separability if and only if it can be cast into the form ds2 = R4/(2−n)G 2/(n−2) (1) G 2/(n−2) (2) · · ·G2/(n−2) (n) n∑ i=1 G−2(i) 1 f2i ( dui )2 , (2.3) where G(i) does not depend on ui while fi depends only on ui. C. If conditions (2.3) and (2.2) are satisfied then the corresponding separation equations read ϕ′′i + f ′i fi ϕ′i + qiϕi = 0. (2.4) Proof. A. R-separability implies (2.1) and (2.2). Indeed, we insert ψ = R ∏ i ϕi into (1.5) and make use of (1.6). This results in ∑ i 1 H2 i [( lnR2 h H2 i ) ,i − pi ] ϕ′i ϕi +R−1∆R− ∑ i 1 H2 i qi + k2 − V = 0 (2.5) for an arbitrary choice of solutions ϕi. Let (ϕi1, ϕi2) be a basis in the solution space of the corresponding equation. We put ϕi = λiϕi1 + µiϕi2, λi, µi = const. Thus for each ϕi (λi 6= 0) we have ϕ′i ϕi = ϕ′i1 + αiϕ ′ i2 ϕi1 + αiϕi2 , (2.6) where αi = µi/λi = const. Since (2.5) with ϕ′ i ϕi replaced by r.h.s. of (2.6) is valid for arbitrary αi we have( lnR2 h H2 i ) ,i = pi, i = 1, 2, . . . , n (2.7) and from (2.7) both (2.1) and (2.2) follow. Conditions (2.1) and (2.2) imply R-separability. Indeed, we form the equations of (1.6) with pi = ( lnR2 h H2 i ) ,i and qi given by (2.2). Then the implication (1.6) in Definition 2 is satisfied. B. Indeed, the metric (1.4) satisfies (2.1) if and only if there exist 2n functions fi(u i) and F(i)(u 1, . . . , ui−1, ui+1, . . . , un) such that h H2 i = 1 R2 fiF(i). (2.8) Certainly, without loss of generality we can replace F(i) by ( ∏ k 6=i fk )−1 G2 (i), where G(i) = G(i)(u 1, . . . , ui−1, ui+1, . . . , un). Now (2.8) implies (2.3) and vice-versa. � On Darboux’s Approach to R-Separability of Variables 7 Remark 5. Willard Miller Jr. derived (2.1) in [23]. See (3.23) of [23] and notice that his R is our lnR. 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 2 ) functions Gij = Gij(u i, uj) (i < j). We select G(i) as follows G(i) = ∏ p 6=i 6=q Gpq. Then (2.3) assumes the form ds2 = R4/(2−n) n∑ i=1 ∏ i<q G2 iq ∏ p<i G2 pi f2i ( dui )2 . (2.9) Definition 6. The metric (2.9) is called binary. Example. The n-elliptic coordinates on En [20, 22, 17, 33]. We choose n real numbers bi such that b1 > b2 > · · · > bn. The n-elliptic coordinates λ = ( λ1, λ2, . . . , λn ) satisfy inequalities λ1 > b1 > λ2 > · · · > bn−1 > λn > bn. The following formulae give rise to a diffeomorphism onto any of 2n open n-hyper-octants of En equipped with the standard Cartesian coordinates x = (x1, x2, . . . , xn) ( xi )2 = n∏ j=1 ( λj − bi ) ∏ j 6=i (bj − bi) , i = 1, 2, . . . , n. The corresponding n-elliptic metric is ds2 = n∑ i=1 ∏ j 6=i (λi − λj) 4 n∏ k=1 (λi − bk) (dλi)2. (2.10) The n-elliptic metric is binary (R = 1, Gij = √ λi − λj and f2i = 4(−1)i−1 n∏ k=1 (λi − bk)) and thus isothermic. 2.2 R-equation Having found the general formulae (2.3) and (2.9) for isothermic metrics which – ex definitione – satisfy 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. Remark 6. (2.2) is not the Schrödinger equation since it is either a functional equation (when R is harmonic) or ∆ involves R. 8 A. Sym and A. Szereszewski Theorem 4. A. The metric (2.3) is R-separable in the Schrödinger equation if and only if n∑ i=1 G2 (i)f 2 i [( 1 R ) ,ii + f ′i fi ( 1 R ) ,i + qi 1 R ] = R(n+2)/(2−n)G 2/(n−2) (1) · · ·G2/(n−2) (n) ( k2 − V ) .(2.11) B. The binary metric (2.9) is R-separable in the Schrödinger equation if and only if n∑ i=1 f2i∏ i<q G 2 iq ∏ p<iG 2 pi [( 1 R ) ,ii + f ′i fi ( 1 R ) ,i + qi 1 R ] = R(n+2)/(2−n) (k2 − V ) . (2.12) 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). 2.3 Bôcher–Ushveridze identities Gaston Darboux was the first to discuss the so called triply conjugate coordinates in E3 [12]. These constitute a projective generalization of orthogonal coordinates in E3. In this context he introduced the following system of three equations for a single unknown M(x1, x2, x3) (x1 − x2)M,12 −M,1 +M,2 = 0, (x1 − x3)M,13 −M,1 +M,3 = 0, (2.13) (x2 − x3)M,23 −M,2 +M,3 = 0. and gave a general solution to it in the form M = m1(x1) (x1 − x2)(x1 − x3) + m2(x2) (x2 − x1)(x2 − x3) + m3(x3) (x3 − x1)(x3 − x2) , (2.14) where mi(xi) are arbitrary functions (see formulae (40), (41) and (42) in [12]). A generalization of (2.13) and (2.14) is straightforward. Consider in Rn the following system of ( n 2 ) PDEs for a single unknown M(x1, x2, . . . , xn) (xi − xj)M,ij −M,i +M,j = 0, i < j. (2.15) 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 M = n∑ i=1 mi(xi)∏ j 6=i (xi − xj) , where mi(xi) are arbitrary functions. On the other hand each single equation of the system (2.15) is a particular case of the Euler– Poisson–Darboux equation [14, p. 54] provided we ignore variables not explicitly involved in the equation. For simplicity (2.15) will be called the Euler–Poisson–Darboux system. Remark 8. Interestingly, in modern times the Euler–Poisson–Darboux system and its various modifications have been studied in the context of the so called integrable hydrodynamic type systems [31, 32, 25]. On Darboux’s Approach to R-Separability of Variables 9 Notice particular solutions to (2.15): M = 0, M = 1 and M = n∑ i=1 xi. The obvious question arises as to what functions mi(xi) correspond to them. Maxime Bôcher in his monograph on R-separability in the Laplace equation on En published without proof a series of remarkable algebraic identities [2, p. 250]. These can be written in a compact form as follows n∑ i=1 xp−1i∏ j 6=i (xi − xj) = δpn, p = 1, 2, . . . , n. (2.16) A.G. Ushveridze generalized the identities (2.16) [33]. We put m = 0, 1, 2, . . . , n = 2, 3, . . . , d = m+ 1− n and f (n) d (x1, x2, . . . , xn) = n∑ i=1 xmi∏ j 6=i (xi − xj) . (2.17) Then f (n) d =  0 for 0 ≤ m < n− 1, 1 for m = n− 1, homogeneous polynomial of degree and homogeneity = d for m ≥ n. (2.18) I.e. for m ≥ n f (n) d = ∑ 1l1+2l2+···+dld=d fl1l2...ld σ l1 1 σ l2 2 · · ·σ ld d , where σi are elementary symmetric polynomials: σ1 = n∑ i=1 xi, σ2 = ∑ i<j xixj , . . . and fl1l2...ld are constants defined uniquely by r.h.s. of (2.17). In particular f (n) 1 = σ1 = n∑ i=1 xi, f (n) 2 = σ21 − σ2, f (n) 3 = σ31 − 2σ1σ2 + σ3. (2.19) The identities (2.18) and in particular the identities (2.16) we call the Bôcher–Ushveridze identi- ties. Certainly, both sides of any Bôcher–Ushveridze identity is a particular solution to the Euler– Poisson–Darboux system. Notice also that functions mi(xi) 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. 3 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. 3.1 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. 10 A. Sym and A. Szereszewski In (1) of [27] we put k = 1 and replace E by k2. Notice that e.g. Robertson’s hi is our H−2i . The paper deals with the case R = 1. R-equation (2.2) is now the functional constraint which is bilinear in H−2i and qi n∑ i=1 1 H2 i qi = k2 − V. (3.1) We decompose qi as follows qi ( ui ) = k2qi1 ( ui ) − vi(ui) +Qi ( ui ) , (3.2) where vi are arbitrary. Inserting (3.2) into (3.1) gives n∑ i=1 1 H2 i qi1 = 1, (3.3) n∑ i=1 1 H2 i Qi = 0, (3.4) n∑ i=1 1 H2 i vi = V. (3.5) Formally (3.4) means that vector (Qi) belongs to (n − 1)-dimensional orthogonal complement of the vector (H−2i ). Select a basis (qij) (j = 2, 3, . . . , n) of the orthogonal complement n∑ i=1 1 H2 i qij ( ui ) = 0, j = 2, 3, . . . , n (3.6) and decompose (Qi) in this basis as follows Qi ( ui ) = n∑ j=2 kjqij ( ui ) , (3.7) where the coefficients of the decomposition are arbitrary constants. Remark 9. (3.3) and (3.6) introduce (non-uniquely!) an n×n matrix q = [qij(u i)]. We assume that q is non-singular everywhere. It is called the Stäckel matrix. Notice that the co-factor Qij of qij does not depend on ui. We collect (3.3) and (3.6) as n∑ i=1 1 H2 i qij = δ1j . (3.8) Inverting of (3.8) yields 1 H2 i = ( q−1 ) 1i = Qi1 det q . (3.9) It is clear that the metric ds2 = det q n∑ i=1 (dui)2 Qi1 (3.10) On Darboux’s Approach to R-Separability of Variables 11 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)( ln h H2 i ) ,ij = ( ln h det g Qi1 ) ,ij = ( ln h det q ) ,ij = 0, i 6= j, which implies h det q = n∏ i=1 fi ( ui ) (3.11) and thus (2.8) is h H2 i = fi ( ui ) Qi1 ∏ j 6=i fj ( uj ) , (3.12) 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 ϕ′′i + f ′i fi ϕ′i + k2qi1 + n∑ j=2 kjqij − vi ϕi = 0. (3.13) To conclude we arrive at the following identifications: (A), (B), (C) and (9) of [27] are now (3.9), (3.5), (3.11) and (3.13) respectively. Definition 7. (3.10) is called the Stäckel metric and (3.11) is called the Robertson condition. 3.2 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 po- tential function V (λ) = n∑ i=1 vi(λ i)∏ j 6=i (λi − λj) , where vi(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 ϕ′′i + 1 2 a′i ai ϕ′i + 1 ai [ n−2∑ m=0 km(λi)m + k2(λi)n−1 − vi(λi) ] ϕi = 0, i = 1, 2, . . . , n, where ai = 4 n∏ k=1 (λi − bk) and km are arbitrary constants (m = 0, 1, . . . , n− 2). 12 A. Sym and A. Szereszewski 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 n∑ i=1 ai(λ i)qi(λ i)∏ j 6=i (λi − λj) = k2 − V (λ). We put ai(λ i)qi(λ i) = n−2∑ m=0 km(λi)m + k2(λi)n−1 − vi(λi), i = 1, 2, . . . , n, where km = const and vi(λ i) are arbitrary functions. Now the Bôcher–Ushveridze identity (2.16) implies the statement. � 3.3 Remarkable example of Kalnins–Miller revisited Our setting can be easily extended to the pseudo-Riemannian case. Consider the following metric dσ2 = ( λ1 − λ2 ) ( λ1 − λ3 ) (dλ1)2 + ( λ2 − λ1 ) ( λ2 − λ3 ) (dλ2)2 + ( λ3 − λ1 ) ( λ3 − λ2 ) (dλ3)2, (3.14) where λ1 > λ2 > λ3 > 0. It is 3-dimensional Minkowski metric. Indeed, on replacing λi by t, x and y t = 1 9 ( λ1 + λ2 + λ3 ) − 9 16 ( λ1 + λ2 − λ3 ) ( λ1 − λ2 + λ3 ) ( λ1 − λ2 − λ3 ) , x = 1 9 ( λ1 + λ2 + λ3 ) + 9 16 ( λ1 + λ2 − λ3 ) ( λ1 − λ2 + λ3 ) ( λ1 − λ2 − λ3 ) , y = 1 4 ( λ1 + λ2 − λ3 )2 − λ1λ2, we arrive at dσ2 = −dt2 + dx2 + dy2. 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 ds2 = ( λ1 + λ2 + λ3 ) dσ2 (3.15) 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 (f21 = f23 = 1, f22 = −1) R(λ) = ( λ1 + λ2 + λ3 )−1/4 , (3.16) ϕ′′i + qiϕi = 0, i = 1, 2, 3. The point is that (3.16) is harmonic with respect of (3.15). Again R-equation is reducible to the functional constraint( λ1 + λ2 + λ3 ) k2 = q1(λ 1) (λ1 − λ2) (λ1 − λ3) + q2(λ 2) (λ2 − λ1) (λ2 − λ3) + q3(λ 3) (λ3 − λ1) (λ3 − λ2) . Then from the Bôcher–Ushveridze identities (2.16) and (2.19) we have immediately qi(λ i) = k2(λi)3 + k1(λ i) + k0, where k0, k1 are arbitrary constants. On Darboux’s Approach to R-Separability of Variables 13 3.4 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 R−1∆R− n∑ i=1 1 H2 i qi = 0. (3.17) We put V (u) = n∑ i=1 1 H2 i vi ( ui ) , (3.18) where vi(u i) are arbitrary functions and define q̄i = qi − vi. Then (3.17) can be rewritten as R−1∆R− n∑ i=1 1 H2 i vi − n∑ i=1 1 H2 i q̄i = 0, which is R-equation for n-dimensional Schrödinger equation (1.8) with the potential func- tion (3.18). � 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 R- separable 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 Hi Hi = 1 R2 G(1)G(2)G(3)G −1 (i) 1 fi , i = 1, 2, 3, or more explicitly H1 = G(2)G(3) R2f1 = G12G13 R2f1 , H2 = G(1)G(3) R2f2 = G12G23 R2f2 , H3 = G(1)G(2) R2f3 = G13G23 R2f3 . (4.1) Now see (2.9). � 14 A. Sym and A. Szereszewski To simplify notation we rewrite (4.1) as H1 = G2G3 Mf1 , H2 = G1G3 Mf2 , H3 = G1G2 Mf3 . In other words Gi does not depend on ui, fi depends on ui and R = √ M . Finally we arrive at the following general form of the isothermic metric in 3-dimensional case ds2 = 1 M2 [ G2 2G 2 3 f21 ( du1 )2 + G2 1G 2 3 f22 ( du2 )2 + G2 1G 2 2 f23 ( du3 )2] . (4.2) The procedure in question consists of three steps. Suppose we are given any 3-dimensional diagonal metric ds2 = H2 1 (u) ( du1 )2 +H2 2 (u) ( du2 )2 +H2 3 (u) ( du3 )2 . (4.3) In the first step we attempt to identify (4.3) as an isothermic metric (4.2). Suppose it is the case. This step provides us with (predicts) possible forms of R-factor and coefficients pi in the separation equations. In the second step we form the R-equation (2.2) for 3-dimensional Laplace equation either in terms of (4.3) as R−1∆R− 3∑ i=1 1 H2 i qi = 0, (4.4) or in terms of (4.2) as ∆R− ( s1 G2 2G 2 3 + s2 G2 1G 2 3 + s3 G2 1G 2 2 ) R5 = 0, (4.5) where si = f2i qi or as 3∑ i=1 G2 i f 2 i [ ∂2iR −1 + f ′i fi ∂iR −1 + qiR −1 ] = 0. (4.6) In the third step we treat (4.4) and (4.6) as equations for unknowns qi(u i) and (4.5) as equation for unknowns si(u i). Any solution to (4.4), (4.5) or (4.6) provides us with a coefficient qi in the separation equations. If the third step is successful, then the starting metric (4.3) is R-separable in 3-dimensional Laplace equation and the separation equations are constructed explicitly. If R is harmonic with respect of (4.2) (this case includes separability), then e.g. (4.4) becomes a linear in qi constraint 3∑ i=1 1 H2 i qi = 0 (4.7) and the corresponding solution space is at most 2-dimensional. If R is not harmonic with respect of (4.2) and if e.g. (4.4) admits a special solution qi0, then a general solution to (4.4) is qi = qi0 + qi1, where qi1 is a solution to (4.7). On Darboux’s Approach to R-Separability of Variables 15 4.2 Examples Here we present five examples proving efficiency of our procedure. 4.2.1 Spherical metric The spherical metric ds2 = dr2 + r2dθ2 + r2 sin2 θdφ2 is isothermic. It is easily seen that R = 1, G1 = sin θ, G2 = r, G3 = r, f1 = r2, f2 = sin θ, f3 = 1 in this case. Equation (4.7) reads q1 + 1 r2 q2 + 1 r2 sin2 θ q3 = 0 and can be easily solved q1 = − α r2 , q2 = α− β sin2 θ , q3 = β, α, β = const. The resulting separation equations read ϕ′′1 + 2 r ϕ′1 − α r2 ϕ1 = 0, ϕ′′2 + cot θ ϕ′2 + ( α− β sin2 θ ) ϕ2 = 0, ϕ′′3 + βϕ3 = 0. 4.2.2 Toroidal metric I The so called toroidal metric ds2 = (cosh η − cos θ)−2 ( dη2 + dθ2 + sinh2 η dφ2 ) , (4.8) discussed in e.g. [24, 7], is isothermic and R = √ cosh η − cos θ, G1 = G3 = 1, G2 = sinh η, f1 = sinh η, f2 = f3 = 1. We easily verify the equality ∆R− 1 4 R5 = 0. (4.9) Hence equation (4.5) is satisfied if and only if 1 sinh2 η s1 + s2 + 1 sinh2 η s3 = 1 4 . (4.10) A general solution to (4.10) is s1 = f21 q1 = ( 1 4 − α1 ) sinh2 η − α2, 16 A. Sym and A. Szereszewski s2 = f22 q2 = α1, s3 = f23 q3 = α2, α1, α2 = const. The resulting separation equations read ϕ′′1 + coth η ϕ′1 + ( 1 4 − α1 − 1 sinh2 η α2 ) ϕ1 = 0, ϕ′′2 + α1ϕ2 = 0, ϕ′′3 + α2ϕ3 = 0. 4.2.3 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 ds2 = sinh2 η (cosh η − cos θ)2 ( dη2 + dθ2 sinh2 η + dφ2 ) . (4.11) Metric (4.11) suggests the following identifications R = √ coth η − cos θ sinh η , G1 = G2 = 1, G3 = 1 sinh η , f1 = f2 = f3 = 1. Again (4.9) holds. Hence equation (4.5) is satisfied if and only if s1 sinh2 η + s2 sinh2 η + s3 = 1 4 . (4.12) A general solution to (4.12) is s1 = ( 1 4 − α2 ) 1 sinh2 η − α1, s2 = α1, s3 = α2, α1, α2 = const. The resulting separation equations read ϕ′′1 + [( 1 4 − α2 ) 1 sinh2 η − α1 ] ϕ1 = 0, ϕ′′2 + α1ϕ2 = 0, ϕ′′3 + α2ϕ3 = 0. 4.2.4 Cyclidic metric Consider the following metric ds2 = ( 1 + p √ λ1λ2λ3 )−2 [(λ1 − λ2)(λ1 − λ3)(dλ1)2 ϕ(λ1) + (λ2 − λ1)(λ2 − λ3)(dλ2)2 ϕ(λ2) + (λ3 − λ1)(λ3 − λ2)(dλ3)2 ϕ(λ3) ] , (4.13) where ϕ(x) = (x−a)(x− b)(x− c)(x−d) and p, a, b, c, d are constants. In general it is not flat. Proposition 3. The off-diagonal components of Ricci tensor of (4.13) vanish, i.e. part of Lamé equations (1.13) is satisfied. The diagonal components of Ricci tensor of (4.13) vanish, i.e. the other part of Lamé equations (1.14) is satisfied, if and only if pabcd = 0 and p2(abc+ abd+ acd+ bcd) = 1. On Darboux’s Approach to R-Separability of Variables 17 We select d = 0 and p = 1/ √ abc. Hence the metric ds2 = ( 1 + √ λ1λ2λ3 abc )−2 [ (λ1 − λ2)(λ1 − λ3)(dλ1)2 ϕ(λ1) + (λ2 − λ1)(λ2 − λ3)(dλ2)2 ϕ(λ2) + (λ3 − λ1)(λ3 − λ2)(dλ3)2 ϕ(λ3) ] (4.14) with ϕ(x) = x(x− a)(x− b)(x− c) is flat. It is isothermic and R2 = ( 1 + √ λ1λ2λ3 abc ) , G2 1 = λ2 − λ3, G2 2 = λ1 − λ3, G2 3 = λ1 − λ2, f21 = ϕ(λ1), f22 = −ϕ(λ2), f23 = ϕ(λ3). We readily check the equality ∆R− 3 16 R5 = 0. Hence equation (4.5) is satisfied if and only if ϕ(λ1)q1 (λ1 − λ2)(λ1 − λ3) + ϕ(λ2)q2 (λ2 − λ1)(λ2 − λ3) + ϕ(λ3)q3 (λ3 − λ1)(λ3 − λ2) = 3 16 . (4.15) From Bôcher–Ushveridze identities (2.16) (n = 3) we deduce immediately a general solution to (4.15) qi = 1 ϕ(λi) ( α1 + α2λ i + 3 16 ( λi )2) , i = 1, 2, 3, where α1, α2 = const. The resulting separation equations read ϕ′′i + 1 2 ϕ′(λi) ϕ(λi) ϕ′i + 1 ϕ(λi) [ α1 + α2λ i + 3 16 ( λi )2] ϕi = 0. Definition 8. A diagonal 3-dimensional flat metric all whose parametric surfaces are cyclides (Dupin cyclides) is called cyclidic (Dupin-cyclidic). General cyclides are discussed in [29]. For Dupin cyclides see Section 4.3 of the paper. Metric (4.14) is cyclidic but not Dupin-cyclidic. 4.2.5 Dupin-cyclidic metric The metric ds2 = b2(w − a cosh v)2 (a cosh v − c cosu)2 du2 + b2(w − c cosu)2 (a cosh v − c cosu)2 dv2 + dw2 (4.16) is Dupin-cyclidic [26]. It is R-separable in the Helmholtz equation (1.7) on E3 (see Theorem 1 in [26]). Here we give an alternative and remarkably simple proof of this result. Metric (4.16) is isothermic and R = (a cosh v − w)−1/2(w − c cosu)−1/2, G1 = (a cosh v − w)−1, G2 = (w − c cosu)−1, G3 = (a cosh v − c cosu)−1, f1 = f2 = b−1, f3 = 1. 18 A. Sym and A. Szereszewski It is easily to verify the equality R−1∆R− 1 4 ( H−21 −H −2 2 ) = 0, (4.17) which is equation (4.4) in this case. Taking into account H3 = 1 we rewrite (4.17) R−1∆R+ k2 − 1 4 ( H−21 −H −2 2 ) − k2H−23 = 0. (4.18) Certainly, (4.18) is R-equation (2.2) for 3-dimensional Helmholtz equation (1.7). The corre- sponding separation equations are ϕ′′1 + 1 4 ϕ1 = 0, ϕ′′2 − 1 4 ϕ2 = 0, ϕ′′3 + k2ϕ3 = 0. 4.3 Dupin-cyclidic metrics Gaston Darboux found a broad class of Dupin-cyclidic metrics which are R-separable in 3- dimensional Laplace equation [13, Section 162, p. 286]. Here we give an alternative and simplified proof of this remarkable result. The metric (4.16) belongs to this class. 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 E3 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 = (u1, u2, u3) be orthogonal coordinates in E3. Two arbitrary parametric surfaces ui = const and uj = const (i 6= j) intersect in a curvature line of each. Proposition 4. The metric (4.3) is Dupin-cyclidic (see Definition 8) if and only if it is flat and its Lamé coefficients satisfy the following six PDEs ∂ ∂uj H−1i ∂ ∂ui lnHj = 0 i, j = 1, 2, 3, i 6= j. (4.19) Proof. Indeed, kij = −H−1i ∂ ∂ui lnHj is a principal curvature on a parametric surface ui = const in the direction of a curvature line uj-variable [18, p. 608]. � A natural question arises as to when the isothermic metric (4.2) satisfies (4.19)? With no difficulty we prove the following result. Proposition 5. The isothermic metric (4.2) satisfies (4.19) if and only if( M G1 ) ,23 = ( M G2 ) ,13 = ( M G3 ) ,12 = 0. (4.20) Certainly, one solution to (4.20) is provided by the metric (4.16). On performing re-scaling in (4.16) u1 = c cosu, u2 = a cosh v, u3 = w On Darboux’s Approach to R-Separability of Variables 19 we arrive at ds2 = H2 1 (u) ( du1 )2 +H2 2 (u) ( du2 )2 +H2 3 (u) ( du3 )2 (4.21) with H1 = ( u3 − u1 )( u3 − u2 )( u1 − u2 )−1( u1 − u3 )−1 [−(u1)2 b2 + c2 b2 ]−1/2 , (4.22) H2 = ( u3 − u1 )( u3 − u2 )( u2 − u1 )−1( u2 − u3 )−1 [(u2)2 b2 − a2 b2 ]−1/2 , (4.23) H3 = ( u3 − u1 )( u3 − u2 )( u3 − u1 )−1( u3 − u2 )−1 . (4.24) Obviously, (4.21) is isothermic and from (4.22)–(4.24) we have G1 = ( u2 − u3 )−1 , G2 = ( u1 − u3 )−1 , G3 = ( u1 − u2 )−1 . (4.25) Let us insert (4.25) into (4.20). The resulting system of equations( ui − uj ) M,ij −M,i +M,j = 0 i, j = 1, 2, 3, i < j is exactly the Euler–Poisson–Darboux system (2.15) for n = 3. Hence M is of the form M = b1(u 1) (u1 − u2)(u1 − u3) + b2(u 2) (u2 − u1)(u2 − u3) + b3(u 3) (u3 − u1)(u3 − u2) , (4.26) where bi(u i) are arbitrary functions of a single variable. Thus we have the following result. Lemma 1. Any metric (4.21) with Hi = 1 M ( ui − uj )−1( ui − uk )−1 a −1/2 i , i, j, k are different, (4.27) where M is given by (4.26) while bi(u i) and ai(u i) are arbitrary functions of a single variable is isothermic and satisfies (4.19). Theorem 7. Suppose a 3-dimensional Riemann space R3 admits the metric described in Lem- ma 1. Then 1) the off-diagonal components of the Ricci tensor vanish, 2) R3 is conformally flat if and only if ai ( ui ) = mi ( ui )2 + 2niu i + pi, mi, ni, pi = const, (4.28) where 3∑ i=1 mi = 3∑ i=1 ni = 3∑ i=1 pi = 0, (4.29) 3) R3 is flat if and only if it is conformally flat and the following identities hold( n2i −mipi ) b2i + 2[(βimi − αini)u i + βini − αipi]bi + ( αiu i + βi )2 + γiai = 0, (4.30) where the constants αi, βi and γi satisfy identities 3∑ i=1 αi = 3∑ i=1 βi = 3∑ i=1 γi = 0. (4.31) 20 A. Sym and A. Szereszewski 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 Rij = 1 M ( ui − uj )−1 [( ui − uj ) M,ij +M,j −M,i ] , i < j. 2) The Cotton–York of the metric vanishes if and only if (4.28) and (4.29) hold. 3) Suppose (4.28), (4.29) are valid, then the diagonal components of the Ricci tensor vanish if and only if (4.30) and (4.31) hold. � Remark 10. The result 3) of Theorem 7 is essentially due to Gaston Darboux. See the formulae in [12, p. 335]. Lemma 2. Suppose E3 is equipped with the metric described in 3) of Theorem 7. Then R = √ M satisfies the equation( u2 − u3 )−2√ a1 [ ∂1 √ a1∂1R −1 − m1 4 √ a1 R−1 ] + ( u1 − u3 )−2√ a2 [ ∂2 √ a2∂2R −1 − m2 4 √ a2 R−1 ] + ( u1 − u2 )−2√ a3 [ ∂3 √ a3∂3R −1 − m3 4 √ a3 R−1 ] = 0. (4.32) We identify (4.32) as R-equation (4.6) for the Laplace equation on E3. Theorem 8. Any Dupin-cyclidic metric described in 3) of Theorem 7 is R-separable in the Laplace equation on E3. R = √ M and M is given by (4.26). 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Math. 126 (2011), 203–243. http://dx.doi.org/10.1016/0375-9601(95)00504-V http://arxiv.org/abs/solv-int/9502004 http://dx.doi.org/10.1215/S0012-7094-84-05114-7 http://dx.doi.org/10.2307/1968433 http://dx.doi.org/10.1007/BF01158149 http://dx.doi.org/10.1007/BF00943433 http://dx.doi.org/10.1070/RD2005v010n04ABEH000327 http://dx.doi.org/10.1007/3-540-12730-5_7 http://dx.doi.org/10.1063/1.1597946 http://arxiv.org/abs/nlin.SI/0301010 http://dx.doi.org/10.1016/j.physleta.2005.01.017 http://dx.doi.org/10.1007/BF01451624 http://dx.doi.org/10.1017/CBO9780511606359 http://dx.doi.org/10.1088/0264-9381/9/7/005 http://dx.doi.org/10.1088/0305-4470/21/7/024 http://dx.doi.org/10.1111/j.1467-9590.2010.00503.x 1 Introduction 1.1 R-separability setting 1.2 R-separability in the Schrödinger equation 1.3 Darboux's R-separability problem 1.4 Aims and results of the paper 2 Isothermic metrics and R-equation 2.1 The main result 2.2 R-equation 2.3 Bôcher-Ushveridze identities 3 Examples 3.1 Robertson paper revisited 3.2 The n-elliptic metric 3.3 Remarkable example of Kalnins-Miller revisited 3.4 Fixed energy R-separation revisited 4 R-separability in 3-dimensional case 4.1 Procedure to detect R-separable metrics 4.2 Examples 4.2.1 Spherical metric 4.2.2 Toroidal metric I 4.2.3 Toroidal metric II 4.2.4 Cyclidic metric 4.2.5 Dupin-cyclidic metric 4.3 Dupin-cyclidic metrics References

On Darboux's Approach to R-Separability of Variables

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