Lamé polynomials, hyperelliptic reductions and Lamé band structure

Robert S Maier

Abstract

The band structure of the Lamé equation, viewed as a one-dimensional Schrödinger equation with a periodic potential, is studied. At integer values of the degree parameter , the dispersion relation is reduced to the =1 dispersion relation, and a previously published =2 dispersion relation is shown to be partly incorrect. The Hermite–Krichever Ansatz, which expresses Lamé equation solutions in terms of =1 solutions, is the chief tool. It is based on a projection from a genus- hyperelliptic curve, which parametrizes solutions, to an elliptic curve. A general formula for this covering is derived, and is used to reduce certain hyperelliptic integrals to elliptic ones. Degeneracies between band edges, which can occur if the Lamé equation parameters take complex values, are investigated. If the Lamé equation is viewed as a differential equation on an elliptic curve, a formula is conjectured for the number of points in elliptic moduli space (elliptic curve parameter space) at which degeneracies occur. Tables of spectral polynomials and Lamé polynomials, i.e. band-edge solutions, are given. A table in the earlier literature is corrected.

Keywords:

1. Introduction

(a) Background

The term ‘Lamé equation’ refers to any of several closely related second-order ordinary differential equations (Whittaker & Watson 1927; Erdélyi 1953–1955; Arscott 1964). The first such equation was obtained by Lamé by applying the method of separation of variables to Laplace's equation in an ellipsoidal coordinate system. Lamé equations arise elsewhere in theoretical physics. Recent application areas include: (i) the analysis of preheating after inflation, arising from parametric amplification (Boyanovsky et al. 1996; Greene et al. 1997; Kaiser 1998; Ivanov 2001), (ii) the stability analysis of critical droplets in bounded spatial domains (Maier & Stein 2001), (iii) the stability analysis of static configurations in Josephson junctions (Caputo et al. 2000), (iv) the computation of the distance–redshift relation in inhomogeneous cosmologies (Kantowski & Thomas 2001) and (v) magnetostatic problems in triaxial ellipsoids (Dobner & Ritter 1998).

In some versions of the Lamé equation, elliptic functions appear explicitly, and in others (the algebraic versions) they appear implicitly. The version that appears most often in the physics literature is the Jacobi oneEmbedded Image(1.1)which is a one-dimensional Schrödinger equation with a doubly periodic potential, parametrized by m and . Here, sn(.|m) is the Jacobi elliptic function with modular parameter m. The parameter m is often restricted to (0, 1), though in general Embedded Image is allowed. When m∈(0, 1), the function sn2(.|m) has real period 2K≔2K(m) and imaginary period 2iK′≔2iK(1−m), with K(m) being the first complete elliptic integral.

If α is restricted to the real axis and m and are real, (1.1) becomes a real-domain Schrödinger equation with a periodic potential, i.e. a Hill's equation. Standard results on Hill's equation apply (Magnus & Winkler 1979; McKean & van Moerbeke 1979). Equipping (1.1) with a quasi-periodic boundary conditionEmbedded Image(1.2)where Embedded Image is fixed, defines a self-adjoint boundary value problem. For any real k (i.e. for any Floquet multiplier ξ with |ξ|=1), there will be an infinite discrete set of energies Embedded Image for which this problem has a solution, called a Bloch solution with crystal momentum k. Each such E will lie in one of the allowed zones, which are intervals delimited by energies corresponding to ξ=±1, i.e. to periodic and anti-periodic Bloch solutions. These form a sequence Embedded Image, where E0 is a ‘periodic’ eigenvalue, followed by alternating pairs of anti-periodic and periodic eigenvalues (each pair may be coincident). The allowed zones are the intervals Embedded Image. The complementary intervals Embedded Image are forbidden zones or lacunae. Any solution of Hill's equation with energy in a lacuna is unstable: its multiplier ξ will not satisfy |ξ|=1, and its crystal momentum k will not be real.

It is a celebrated result of Ince (1940b) that if the degree is an integer, which without loss of generality may be chosen to be non-negative, the Lamé equation (1.1) will have only a finite number of non-empty lacunae. A converse to this statement holds as well (Gesztesy & Weikard 1995a). If is an integer, the Bloch spectrum consists of the +1 bands Embedded Image, and Embedded Image is said to be a finite-band or algebro-geometric potential. The 2+1 band edges Embedded Image are algebraic functions of the parameter m. In other words, they are the roots of a certain polynomial, the coefficients of which are polynomial in m. The corresponding periodic and anti-periodic Bloch solutions are called Lamé polynomials: they are polynomials in the Jacobi elliptic functions sn(α|m), cn(α|m) and dn(α|m). The double eigenvalues embedded in the topmost band Embedded Image (the ‘conduction’ band), namely Embedded Image, j>, are loosely called transcendental eigenvalues. For =1 at least, they are known to be transcendental functions of m (Chudnovksy & Chudnovsky 1980).

There has been much work on algebraizing the integer- Lamé equation, to facilitate the computation of the band edges and the coefficients of the Lamé polynomials (Alhassid et al. 1983; Turbiner 1989; Li & Kusnezov 1999; Finkel et al. 2000; Li et al. 2000). Such schemes have been extended to the case when is a half-odd-integer, in which there are an infinite number of lacunæ. In this case, certain ‘mid-band’ Bloch functions, namely ones with ξ=±i and real period 8K, are algebraic functions of sn(α|m). Certain rational values of with Embedded Image also yield algebraic Bloch functions, provided the parameters m and E are chosen appropriately (Maier 2004).

An algebraic understanding of band edges is useful, but it is also desirable to have a closed-form expression for the dispersion relation: k as a function of E. The value of k is not unique, since it can be negated (equivalently Embedded Image) and any integer multiple of π/K can be added. However, each branch has the property that kE1/2 or k∼−E1/2 to leading order as Embedded Image. Also Embedded Image in each band.

The goal of this paper is the efficient computation of the dispersion relation when is an integer. The following are illustrations of why this is of importance in theoretical physics. In application (i) above (preheating after inflation), particle production is due to parametric amplification: a solution having a multiplier ξ with |ξ|>1. This corresponds to the energy E not being at a band edge, or even in a band, but in a lacuna. In application (ii) (the stability analysis of a critical droplet), the analysis includes an imposition of Dirichlet rather than quasi-periodic boundary conditions on an =2 Lamé equation (Maier & Stein 2001). The resulting Bloch solution is not a Lamé polynomial, but rather a mid-band solution.

When is an integer, the Lamé equation is integrable, and the general integral of (1.1) can be expressed in terms of Jacobi theta functions. The dispersion relations Embedded Image, ≥1, can in principle be computed in terms of elliptic integrals. The case =1 is by far the easiest. If =1, the solution space of (1.1), except when E is at a band edge, will be spanned by the pair of Hermite–Halphen solutionsEmbedded Image(1.3)Here H, Θ, Z are the Jacobi eta, theta and zeta functions with periods 4K, 2K, 2K, respectively, and Embedded Image is defined up to sign by Embedded Image. HenceEmbedded Image(1.4)up to multivaluedness. This is a parametric dispersion relation. It has been exploited in a study of Wannier–Stark resonances with non-real E and k (Grecchi & Sacchetti 1997; Sacchetti 1997). However, the extension to >1 is numerically non-trivial. Embedded Image turns out to equal Embedded Image, where Embedded Image satisfy coupled transcendental equations involving E and m (Whittaker & Watson 1927, §23.71). Li, Kusnezov & Iachello calculated and graphed k=k2(E|m) as well as k=k1(E|m) in the ‘lemniscatic’ case m=1/2 (Li & Kusnezov 1999; Li et al. 2000). Unfortunately, their graph of k=k2(E|1/2) is incorrect, as will be shown.

(b) Overview of results

When >1, we abandon the traditional Hermite–Halphen solutions, and examine instead the implications for the Lamé dispersion relations of what is now called the Hermite–Krichever Ansatz. This is an alternative way of generating closed-form solutions of the Lamé solution at arbitrary energy E; for small integer values of , at least. Until the 1980s, the only reference for the Ansatz was the classic work of Halphen (1888, ch. XII), who applied it to the cases =2, 3, 4, and in part to =5. Krichever (1980) revived it as an aid in the construction of elliptic solutions of the Korteweg–de Vries and other integrable evolution equations. Belokolos et al. (1986) and Belokolos & Enol'skii (2000) summarize early and recent developments.

The Hermite–Krichever Ansatz is easy to explain, even in the context of the Jacobi form of the Lamé equation, which is not the most convenient for symbolic manipulation. It asserts that for any integer ≥1, fundamental solutions of (1.1) can be constructed from the =1 solutions Embedded Image as finite series of the formEmbedded Image(1.5)where the parameter α0 is now computed from a reduced energy Embedded Image by the formula Embedded Image. The reduced energy Embedded Image, the exponent Embedded Image and the coefficients Embedded Image will depend on E and m. Embedded Image may be chosen to be rational in E and m, and Embedded Image to be of the form Embedded Image×Embedded Image, where Embedded Image is also rational and Embedded Image is the spectral polynomial Embedded Image, a degree-(2+1) polynomial in E the coefficients of which, as noted, are rational in m.

If the Lamé equation can be integrated in the framework of the Hermite–Krichever Ansatz, it follows from (1.5) that up to sign, etc.,Embedded Image(1.6)The dispersion relation for any integer ℓ can be expressed in terms of the ℓ=1 relation. To compute Embedded Image, only one transcendental function (i.e. Embedded Image) needs to be evaluated, since the other functions in (1.6) are elementary. The only difficult matter is choosing the relative sign of the two terms, since each is defined only up to sign.

The functions Embedded Image are rational with integer coefficients, but working them out when is large is a lengthy task. In principle, one can write down a recurrence relation for the coefficients Embedded Image and work out Embedded Image from the condition that the series terminate. However, their numerator and denominator degrees grow quadratically as increases. This explains why Halphen's treatment of the =5 case was only partial. In a series of papers, Kostov, Enol'skii and collaborators used computer algebra systems to perform a full analysis of the cases =2, 3, 4, 5 (Gerdt & Kostov 1989; Kostov & Enol'skii 1993; Eilbeck & Enol'skii 1994; Enol'skii & Kostov 1994). When =5, using Mathematica to compute the integer coefficients of rational functions equivalent to Embedded Image required 7 h of time on a Sparc-1, a Unix workstation of that era (Eilbeck & Enol'skii 1994). Until now, their analysis has not been extended to higher .

While performing extensive symbolic computations, we recently made a discovery, which is formalized in the central result of this paper, theorem 4.1 below. For all integer ≥2, the degree- Lamé equation can be integrated in the framework of the Hermite–Krichever Ansatz, and the rational functions that perform the reduction to the =1 case can be computed by simple formulae from certain spectral polynomials of the degree- equation, which are relatively easy to work out. These are the usual spectral polynomial Embedded Image associated with the band-edge solutions, and the spectral polynomials associated with two other types of closed-form solution that have not previously been studied in the literature. We call them twisted and theta-twisted Lamé polynomials. In the context of the Jacobi form, the former are polynomials in Embedded Image, Embedded Image and Embedded Image, multiplied by a factor Embedded Image. (If Embedded Image, ‘canted’ would be better than ‘twisted’). The latter contain a factor resembling (1.3).

Theorem 4.1 follows from modern finite-band integration theory: specifically, from the Baker–Akhiezer uniformization of the relation between the energy and the crystal momentum. This uniformization is closely tied to classical work on the Lamé equation (the parametrized Baker–Akhiezer solutions of the integer- Lamé equation are in fact equivalent to the Hermite–Halphen solutions). Theorem 4.1 greatly simplifies the computation of higher dispersion relations. It also has implications for the theory of hyperelliptic reduction: the reduction of hyperelliptic integrals to elliptic ones (Belokolos et al. 1986). This is on account of the following. For any ≥1 and m, the solutions of the Lamé equation, both the Hermite–Halphen solutions and those derived from the Hermite–Krichever Ansatz, are single-valued functions not of E, but rather of a point Embedded Image on the th Lamé spectral curve: a hyperelliptic curve comprising all Embedded Image satisfyingEmbedded Image(1.7)The fact that Embedded Image is two-valued (except at a band edge) is responsible for the two-valuedness of, e.g. the parameter Embedded Image of (1.3), and in general, for the uncertainty in the sign of k. The th spectral curve generically has genus and may be denoted Embedded Image. For any integer ≥2, the Embedded Image reduction map of the Hermite–Krichever Ansatz induces a covering Embedded Image. The first known covering of an elliptic curve by a higher genus hyperelliptic curve was constructed by Legendre and generalized by Jacobi (Belokolos et al. 1986, §2). However, it is difficult to enumerate such coverings or even work out explicit examples. Those generated by the Ansatz applied to the Lamé equation are a welcome exception.

The integral with respect to E of any rational function of E and Embedded Image, where Embedded Image is a polynomial, is a line integral on the algebraic curve defined by Embedded Image. The covering Embedded Image, a formula for which is provided by theorem 4.1, reduces certain such hyperelliptic integrals to elliptic ones. In modern language, the theorem specifies how certain holomorphic differentials on hyperelliptic curves can arise as pullbacks of holomorphic differentials on elliptic curves.

As part of our analysis, we investigate the degeneracies of the band edges Embedded Image that occur when the modular parameter m is non-real. Such level crossings were first considered by Cohn (1888) in a dissertation that seems not to have been followed up, though it was later cited by Whittaker & Watson (1927, §23.41). We conjecture a formula for the -dependence of the number of values of Embedded Image, or equivalently the number of values of the Klein invariant Embedded Image, at which two band edges coincide. When this occurs, the genus of the hyperelliptic spectral curve Embedded Image is reduced from to −1, though its arithmetic genus remains equal to . Band-edge degeneracies are responsible for a fact discovered by Turbiner (1989): if ≥2, the complex curve comprising all points Embedded Image in Embedded Image has only four, rather than 2+1, connected components.

This paper is organized as follows. Section 2 introduces the Lamé equation in its elliptic-curve form and relates the Hermite–Halphen solutions to the Baker–Akhiezer function. In §3, Lamé polynomials in the context of the elliptic-curve form are classified. In §4, the Hermite–Krichever Ansatz is introduced, and our key result, theorem 4.1, is stated and proved. The application to hyperelliptic reduction is covered in §5. In §§6 and 7, dispersion relations are worked out and the previously mentioned dispersion relation for the case =2 is corrected. The =3 dispersion relation is graphed as well. Finally, an area for future investigation is mentioned in §8.

2. The elliptic-curve algebraic form

In §§3–6, we use exclusively what we call the elliptic-curve algebraic form of the Lamé equation, which is the most convenient for symbolic computation. In this section, we derive it and also define a fundamental multivalued function Φ, which appears in the elliptic-curve version of both the Hermite–Halphen solutions and the Hermite–Krichever Ansatz.

(a) An elliptic-curve Schrödinger equation

Many algebraic forms can be obtained from (1.1) by changing to new independent variables that are elliptic functions of α, such as Embedded Image (e.g. Arscott & Khabaza 1962, §1.1; Arscott 1964, pp. 192–3). A form in which the domain of definition is explicitly a cubic algebraic curve of genus 1, i.e. a cubic elliptic curve, can be obtained as follows. First, the Lamé equation is restated in terms of the Weierstrassian function Embedded Image. This is the canonical elliptic function with a double pole at u=0, satisfying Embedded Image where Embedded Image. For ellipticity, the roots Embedded Image must be distinct, which is equivalent to the condition that the modular discriminant Embedded Image be non-zero. Either of Embedded Image may equal zero, but not both.

The relation between the Jacobi and Weierstrass elliptic functions is well known (Abramowitz & Stegun 1965, §18.9). Choose Embedded Image according toEmbedded Image(2.1)where Embedded Image is any convenient proportionality constant. ThenEmbedded Image(2.2)and the dimensionless (A-independent) Klein invariant Embedded Image will be given byEmbedded Image(2.3)The two sorts of elliptic function will be related by, e.g.Embedded Image(2.4)and the periods of Embedded Image, denoted Embedded Image, will be related to those of sn2 byEmbedded Image(2.5)The case when Embedded Image are real, or equivalently Embedded Image, Embedded Image (we assume Embedded Image), is the case when Embedded Image and Embedded Image (Abramowitz & Stegun 1965, §18.1).

Choosing for simplicity A=1, so that Embedded Image, and rewriting the Lamé equation (1.1) with the aid of (2.4), yields the Weierstrassian formEmbedded Image(2.6)where Embedded Image. (The translation of (1.1) by Embedded Image replaces msn2 by ns2.) Here, Embedded Image, i.e.Embedded Image(2.7)is a transformed energy parameter. Changing to the new independent variable Embedded Image converts (2.6) to the commonly encountered algebraic formEmbedded Image(2.8)This is a differential equation on the Riemann sphere Embedded Image with regular singular points at Embedded Image. Any solution of the original Lamé equation (1.1) or the Weierstrassian form (2.6), which is quasi-periodic in the sense that it is multiplied by Embedded Image when Embedded Image or Embedded Image, respectively (equivalently when Embedded Image or Embedded Image), will be a path-multiplicative function of x. In other words, it will be multiplied by Embedded Image when it is analytically continued around a cut joining the pair of points Embedded Image or Embedded Image, respectively. In the context of the algebraic form, the dispersion relation is still a relation between the energy and a multiplier ξ, but the multiplier is interpreted as specifying not quasi-periodicity on Embedded Image, but rather multivaluedness on Embedded Image.

The algebraic form (2.8) of the Lamé equation lifts naturally to the complex elliptic curve Embedded Image over Embedded Image, parametrized by Embedded Image. One may rewrite (2.8) in the elliptic-curve algebraic formEmbedded Image(2.9)This is an elliptic-curve Schrödinger equation of the formEmbedded Image(2.10)with the (rational) potential function q(x) taken to equal (+1)x. Equation (2.9) follows directly from (2.6), since Embedded Image. It is a differential equation on Embedded Image with a single singular point: a regular one at Embedded Image. Note that the two-to-one covering map Embedded Image defined by Embedded Image has Embedded Image and Embedded Image as simple critical points. One reason why (2.9) is more fundamental than (2.8) is that the singular points of (2.8) at Embedded Image can be regarded as artefacts: consequences of Embedded Image being critical points of π.

The complex analytic differential geometry of the elliptic curve Embedded Image takes a bit of getting used to. Both x and y are meromorphic Embedded Image-valued functions on Embedded Image, and the only pole that either has on Embedded Image is at the point Embedded Image. In a neighbourhood of any generic point Embedded Image other than O and the three points Embedded Image, either x or y will serve as a local coordinate. However, near each Embedded Image only y will be a good local coordinate, since dy/dx diverges at Embedded Image. In addition, x has a double and y has a triple pole at O, so the appropriate local coordinate near O is the quotient x/y. The 1-form dx/y is not merely meromorphic but holomorphic, with no poles on Embedded Image. Its dual is the vector field (or directional derivative) y d/dx.

Elliptic functions, i.e. doubly periodic functions, of the original variable Embedded Image correspond to single-valued functions on Embedded Image. These are rational functions of x and y and may be written as Embedded Image, i.e. Embedded Image. The formula Embedded Image allows such functions to be differentiated algebraically. In a similar way, quasi-doubly periodic functions of u (sometimes called elliptic functions of the second kind), which are multiplied by ξ when Embedded Image and by ξ′ when Embedded Image, correspond to multiplicatively multivalued functions on Embedded Image.

Embedded Image has genus 1 and is topologically a torus. A fundamental pair of loops that cannot be shrunk to a point may be chosen to be a loop that extends between Embedded Image and Embedded Image, and one that extends between Embedded Image and Embedded Image, with (if Embedded Image are real, at least) half of each loop passing through positive values of y, and the other half through negative values. One way of constructing an elliptic function of the second kind is to anti-differentiate a rational function R(x, y). Here, ‘anti-differentiate’ means to compute Embedded Image, its indefinite integral against the holomorphic 1-form dx/y. The resulting function will typically have a non-zero modulus of periodicity associated with each loop. If so, exponentiating it will yield a multiplicatively multivalued function on Embedded Image, with non-unit multipliers ξ, ξ′.

(b) The Hermite–Halphen solutions

The fundamental multivalued function Φ on the elliptic curve will now be defined. It is an elliptic-curve version of Halphen's l'élément simple (Halphen 1888).

On the elliptic curve Embedded Image, the multivalued meromorphic function Φ, parametrized by Embedded Image, is defined up to a constant factor by a formula containing an indefinite elliptic integral,Embedded Image(2.11)Its multivaluedness, which is multiplicative, arises from the path of integration winding around Embedded Image in any combination of the two directions. Each branch of Φ has a simple zero at Embedded Image and a simple pole at Embedded Image.

To motivate the definition of Φ, a brief sketch will now be given of the construction of the Hermite–Halphen solutions of the elliptic-curve algebraic Lamé equation (2.9) for integer ≥1. The standard published exposition is not fully algebraic, being framed largely in the context of the Weierstrassian form (Whittaker & Watson 1927, §23.7). This sketch will relate the Hermite–Halphen solutions to modern finite-band integration theory and the concept of a Baker–Akhiezer function (Treibich 2001; Gesztesy & Holden 2003). The starting point is the differential equationEmbedded Image(2.12)The differential operator in (2.12) is the ‘symmetric square’ of the elliptic-curve Schrödinger operator of (2.10), so the solutions of (2.12) include the product of any pair of solutions of (2.10). If the potential function q(x) is rational, it is known that the solution space of (2.12) contains a function Embedded Image which is (i) meromorphic in x (the only poles being at the poles of q(x)), and (ii) monic polynomial in B, if and only if (2.10) is a finite-band Schrödinger equation on Embedded Image, i.e. a finite-band Schrödinger equation with a doubly periodic potential (Its & Matveev 1974). This is an alternative to the characterization of Gesztesy & Weikard (1996), according to which (2.10) is finite-band if and only if for all Embedded Image, every solution Ψ is meromorphic on Embedded Image (multivaluedness being allowed).

For example, when Embedded Image, there is a polynomial solution Embedded Image of (2.12) that satisfies conditions (i) and (ii), of degree in both x and B. Hence, the integer- Lamé equation is a finite-band Schrödinger equation. Embedded Image is called the 'th Hermite–Halphen polynomial. It may be written Embedded Image when it is normalized to be monic in x, rather than in B (table 1).

View this table:
Table 1

Hermite–Halphen polynomials (van der Waall 2002, table A.2).

In the case of a general rational potential q(x) that is finite-band, let Embedded Image denote the specified solution of (2.12), and let its degree in B be denoted . It follows from manipulations parallel to those of Whittaker & Watson that the function Ψ on Embedded Image defined by a formula containing an indefinite integral,Embedded Image(2.13)will be a solution of the Schrödinger equation (2.10). Here, Embedded Image and ν is a B-dependent but position-independent quantity, determined only up to sign, that is computed by what Whittaker & Watson call an ‘interesting formula’,Embedded Image(2.14)(It is not obvious that the right-hand side is independent of the point Embedded Image.) It is widely known (Smirnov 2002) that ν is identical to the coordinate ν on the spectral curve Embedded Image defined by Embedded Image, where Embedded Image are the band edges of the Schrödinger operator; though no really simple proof of this fact seems to have been published. A consequence of this is that the formula (2.13) parametrizes solutions of the elliptic-curve Schrödinger equation (2.10) by Embedded Image. As defined, Ψ is called a Baker–Akhiezer function (Krichever 1990).

Consider now the special case of the integer- Lamé equation (2.9). In this case, the function Ψ computed by (2.13) from the Hermite–Halphen polynomial Embedded Image is in fact an Hermite–Halphen solution of the Lamé equation, re-expressed in terms of the elliptic curve coordinates (x, y). One can write Embedded Image, where the superscript ‘±’ refers to the ambiguity in the sign of Embedded Image. If Embedded Image, the two solutions Embedded Image are distinct. They are path-multiplicative, since they are exponentials of anti-derivatives of rational functions on Embedded Image.

It should be noted that the Hermite–Halphen polynomials are not merely a tool for generating the solutions Embedded Image of the Lamé equation. They are algebraically interesting in their own right. Klein (1892, figs 1 and 2) supplies a sketch of the real portion of the curve Embedded Image when =5, 6, showing how when ≥4, B is a band edge only if Embedded Image, regarded as a polynomial in x, has a double root.

The relevance of the fundamental multivalued function Φ can now be explained. It follows from (2.13), and the fact that Embedded Image (table 1), thatEmbedded Image(2.15)In other words, if Embedded Image is ‘above’ Embedded Image, then Embedded Image will be a solution of the =1 Lamé equation in the form (2.9). There are two such points, related by Embedded Image being negated, unless Embedded Image, i.e. unless Embedded Image, in which case Embedded Image is the only possibility. These are the three band-edge values of B for =1.

It is not difficult to show that the =1 Hermite–Halphen solutions (2.15) are identical to the solutions (1.3), though they are expressed as functions of the variable Embedded Image rather than the original independent variable Embedded Image. The parametrizing point Embedded Image corresponds to the parameter Embedded Image of (1.3). These solutions are clearly easier to formulate in the elliptic curve context.

For any integer , the Lamé dispersion relation can be computed numerically from (2.13) by calculating the multiplier arising from the path of integration winding around Embedded Image. However, (2.13) is not adapted to symbolic computation. By expanding the integrand in partial fractions, one can derive the remarkable formulaEmbedded Image(2.16)where Embedded Image are points on Embedded Image above Embedded Image, the B-dependent roots of the degree- polynomial Embedded Image (cf. Whittaker & Watson 1927, §23.7). Unfortunately, when ≥5, the roots Embedded Image cannot be computed in terms of radicals. This reduction to degree-1 solutions is less computationally tractable than the one that will be provided by the Hermite–Krichever Ansatz.

3. Finite families of Lamé equation solutions

The solutions of the integer- Lamé equation include the Lamé polynomials, which are the traditional band-edge solutions. In the Jacobi-form context, they are periodic or anti-periodic functions on [0, 2K], with Floquet multiplier ξ=±1, respectively. There are exactly 2+1 values of the spectral parameter Embedded Image, i.e. of the energy E, for which a Lamé polynomial may be constructed, the counting being up to multiplicity. By definition, these are the roots of the spectral polynomial Embedded Image.

As functions on the curve Embedded Image, the Lamé polynomials are single or double-valued and are essentially polynomials in the coordinates x, y. (In the Weierstrassian context, Embedded Image substitute for x, y.) However, no fully satisfactory table of the Lamé polynomials or the Lamé spectral polynomials has yet been published. Whittaker & Watson (1927, §23.42) refer to a list of Guerritore (1909) that covers ≤10. Sadly, although he produced it as a dissertazione di laurea at the University of Naples, most of his results on ≥5 are incorrect. This has long been known (Strutt 1967), but his paper is still occasionally cited for completeness (Gesztesy & Holden 2003). Arscott (1964, §9.3.2) gives a brief table of the Jacobi-form Lamé polynomials, covering only =1, 2, 3. His table is correct, with a single misprint (Fernández C. et al. 2000; Finkel et al. 2000). However, its brevity has been misinterpreted. An erroneous belief has arisen that when ≥4, the Lamé polynomial coefficients and band-edge energies cannot be expressed in terms of radicals. This sets in only when ≥8.

Owing to these confusions, we tabulate the Lamé polynomials and the spectral polynomials Embedded Image in this section. Both are computed from coefficient recurrence relations. We supply such relations and tables of spectral polynomials for the twisted and theta-twisted Lamé polynomials as well. The number of values of Embedded Image for which the latter two sorts of solution exist, i.e. the degrees of their spectral polynomials, will be given. All three sorts of solution will play a role in our key result, theorem 4.1. In fact, all will be special cases of the solutions constructed for arbitrary B by the Hermite–Krichever Ansatz.

When ≥2, many of the spectral polynomials will have degenerate roots if Embedded Image are appropriately chosen. This means that, for example, a pair of the 2+1 band-edge energies can be made to coincide by moving the modular parameter Embedded Image to one of a finite set of complex values. We indicate how to calculate these, or more precisely the corresponding values of Klein's absolute invariant Embedded Image.

J is the more fundamental parameter, in algebraic geometry at least, since two elliptic curves are isomorphic (birationally equivalent) if and only if they have the same value of J. The Embedded Image correspondence (2.3) maps Embedded Image onto Embedded Image, and it also maps Embedded Image (in fact Embedded Image) onto Embedded Image. Formally it is six-to-one. Each value of J corresponds to six values of m, with the exception of J=0 (i.e. Embedded Image), which corresponds to Embedded Image, and J=1 (i.e. Embedded Image), which corresponds to Embedded Image. Elliptic curves with J=0,1 are called equianharmonic and lemniscatic, respectively (Abramowitz & Stegun 1965, §§18.13 and 18.15). Any equianharmonic curve has a triangular period lattice, with Embedded Image, and any lemniscatic curve has a square period lattice, with Embedded Image.

(a) Lamé polynomials

The Lamé polynomials are classified into species 1, 2, 3, 4 (Whittaker & Watson 1927, §23.2). This is appropriate for some forms of the Lamé equation, but for the elliptic-curve algebraic form, a more structured classification scheme is better.

A solution of the Lamé equation (2.9) on the elliptic curve Embedded Image is said to be a Lamé polynomial of Type I if it is single-valued and of the form C(x) or D(x)y, where C, D are polynomials. A solution is said to be a Lamé polynomial of Type II, associated with the branch point Embedded Image of the curve (γ=1, 2, 3), if it is double-valued and of the form Embedded Image or Embedded Image, where E, F are polynomials. The subtypes of Types I and II are species 1, 4 and 2, 3, respectively.

To determine necessary conditions on and B for there to be a non-zero Lamé polynomial of each subtype, one may substitute the corresponding expression (C(x), etc.) into the Lamé equation (2.9), and work out a recurrence for the polynomial coefficients. This is similar to the approach of expanding in integer or half-integer powers of Embedded Image (Whittaker & Watson 1927, §23.41), though it leads to four-term rather than three-term recurrences. For the Type I solutions at least, the present approach seems more natural, since they are not associated with any singular point eγ.

If Embedded Image, Embedded Image, Embedded Image and Embedded Image, substituting the expression for each species of solution into (2.9) and equating the coefficients of powers of x leads to the recurrence relationsEmbedded Image(3.1)Embedded Image(3.2)Embedded Image(3.3)Embedded Image(3.4)It is easy to determine the integers for which C, D, E, F may be a polynomial.

If ℓ≥1 is odd, non-zero Type I Lamé polynomials of the fourth species and Type II ones of the second species can in principle be constructed from these recurrence relations with Embedded Image and Embedded Image, respectively. (The former assumes ℓ≥3.) If ℓ≥2 is even, non-zero Type I Lamé polynomials of the first species and Type II ones of the third species can be constructed similarly, with Embedded Image and Embedded Image, respectively.

The coefficients in each Lamé polynomial are computed from the appropriate recurrence relation by setting the coefficient of the highest power of x to unity and working downward. Unless B is specially chosen, the coefficients of negative powers of x may be non-zero. However, by examination, they will be zero if the coefficient of x−1 equals zero.

The Type I Lamé spectral polynomial Embedded Image is the polynomial monic in B which is proportional to the coefficient Embedded Image if is odd, and to c−1 if is even. (The former assumes ≥3; by convention Embedded Image.) The Type II Lamé spectral polynomial Embedded Image is similarly obtained from the coefficient e−1 if is odd and Embedded Image if is even. Each spectral polynomial may be regarded as Embedded Image, respectively, Embedded Image, where the roots Embedded Image are the values of B for which a Lamé equation solution of the indicated type exists, counted with multiplicity.

By examination, Embedded Image is (−1)/2 if is odd, and /2+1 if is even; and Embedded Image is (+1)/2 if is odd and /2 if is even. Hence, as expectedEmbedded Image(3.5)the full Lamé spectral polynomial, has degree Embedded Image in B.

It should be noted that Embedded Image, the full Type II Lamé spectral polynomial, is a function only of Embedded Image, since any symmetric polynomial in Embedded Image can be written in terms of Embedded Image. For example Embedded Image. This is why Embedded Image is absent on the left-hand side of (3.5). When using the recurrences (3.1)–(3.4), one should also note that Embedded Image, so Embedded Image. Any polynomial in Embedded Image can be reduced to one which is of degree at most 2 in Embedded Image, much as any polynomial in x, y can be reduced to one of degree at most 1 in y.

The Lamé polynomials of Types I and II are listed in table 2 and the corresponding spectral polynomials in table 3. They replace the table of Guerritore (1909), with its many unfortunate errors. The spectral polynomials with ≤7 were recently computed by a different technique (van der Waall 2002, table A.3). The table of van der Waall displays the full Type II spectral polynomials, rather than the more fundamental Embedded Image-dependent polynomials Embedded Image. The spectral polynomials can also be computed by the technique of Gesztesy & Weikard (1995a), which employs the Weierstrassian counterpart of the Ansatz used by Hermite in his solution of the Jacobi-form Lamé equation (Whittaker & Watson 1927, §23.71).

View this table:
Table 2

Lamé polynomials of Types I and II.

View this table:
Table 3

Lamé spectral polynomials of Types I and II. (Most of the ones with Graphic disagree with those published by Guerritore (1909).)

The roots of the spectral polynomials are the energies B for which the Lamé polynomials are solutions of the Lamé equation. It is clear that when ≥9, the Type II energies cannot be expressed in terms of radicals, since the degree of the spectral polynomial will be 5 or above. When =8 or ≥10, the Type I energies cannot be so expressed. These statements apply also to the coefficients of the Lamé polynomials, which depend on B. Hence when ≥10, the symbolic computation of the Lamé polynomials is impossible, and when =8 or 9, it is possible only in part. However, when Embedded Image take on special values, what would otherwise be impossible may become possible. For instance, when Embedded Image (the lemniscatic case, including m=1/2), the quintic spectral polynomial Embedded Image reduces to Embedded Image, the roots of which can obviously be expressed in terms of radicals.

In the context of the Jacobi form, the 2+1 values Embedded Image for which a Lamé polynomial solution exists can be thought of as the 2+1 branches of a spectral curve that lies over the triply punctured sphere Embedded Image, the space of values of the modular parameter m. Turbiner (1989) showed that if ≥2, this curve has only four connected components, not 2+1. The reason is clear now. These are the Type I component and the three Type II components, one associated with each point Embedded Image. Since each of the four is defined by a polynomial in E and m, each can be extended to an algebraic curve over Embedded Image. At the values Embedded Image, the four curves may touch one another. (Refer to Li et al. (2000, fig. 3), for the behaviour of the real portions of the =1, 2 curves as Embedded Image, and Alhassid et al. (1983) for =3.) These three values of m correspond to two of Embedded Image coinciding and the elliptic curve Embedded Image becoming rational rather than elliptic. Level crossings of this sort are perhaps less interesting than ‘intra-curve’ ones.

In the present context, E is replaced by the transformed energy B, and m by the pair Embedded Image or the Klein invariant J, with J=∞ corresponding to Embedded Image. It is easy to determine which finite values of J yield coincident values of B. One simply computes the discriminants of the Type I and full Type II spectral polynomials, Embedded Image and Embedded Image. Each discriminant is zero if and only if there is a double root. By using Embedded Image, one can eliminate Embedded Image and obtain a polynomial equation for J. For each of Types I and II, there are coincident values of B if and only if J is a root of what we shall call a Cohn polynomial.

In table 4, the Cohn polynomials are listed. Since the coefficients are rather large integers that may have number-theoretic significance, each is given in a fully factored form. An interesting feature of these polynomials is that none has a zero on the real half-line [1, ∞). Since Embedded Image corresponds to Embedded Image, the existence of such a zero would imply that for some Embedded Image, two of the 2+1 band edges become degenerate. That this cannot occur follows from a Sturmian argument (Whittaker & Watson 1927, §23.41). It also follows from the analysis of Gesztesy & Weikard (1995a, §3).

View this table:
Table 4

Cohn polynomials of Types I and II.

For any integer ℓ≥1, the degeneracies of the algebraic spectrum of the Lamé operator, which comprise the 2+1 roots (up to multiplicity) of the spectral polynomial Embedded Image, are fully captured by the Cohn polynomials of Types I and II. As the parameters Embedded Image are varied, a pair of roots will coincide, reducing the number of distinct roots from 2+1 to 2, if and only if the Klein invariant J is a root of one of the two Cohn polynomials; and there are no multiple coincidences.

This proposition will be proved in §4. The following conjecture is based on a close examination of the spectral and Cohn polynomials for all ≤25.

  1. As a polynomial in J with integer coefficients, no Cohn polynomial has a non-trivial factor, except for the Type I Cohn polynomials with Embedded Image, each of which is divisible by J. (These factors of J are visible in table 4.)

  2. If Embedded Image and Embedded Image denote the degrees of the spectral polynomials of Types I, II, which are given above, then the Cohn polynomials of Types I, II have degrees Embedded Image and Embedded Image, respectively. (In the first expression, […] is the integer part, or ‘floor’ function.)

The conjectured degree formulae constitute a conjecture as to the number of points in elliptic moduli space (elliptic curve parameter space), labelled by J, at which the 2+1 distinct energies in the algebraic spectrum are reduced to 2. For example, Embedded Image=1, 2, so when =3 the Cohn polynomials of Types I, II have degrees 0, 1, respectively. In other words, if =3 there is no Type I polynomial, and the Type II one is linear in the invariant J. According to table 4, it equals 4J+1. A non-degeneracy condition equivalent to the linear condition Embedded Image was previously worked out by Treibich (1994, §6.6), namely Embedded Image.

The remarks regarding extra J factors amount to a conjecture that in the equianharmonic case Embedded Image (i.e. J=0 or Embedded Image), there are only 2 distinct energies if and only if Embedded Image. For those values of , the double energy eigenvalue is evidently located at B=0. It should be mentioned that when J=0 and Embedded Image, there is also an eigenvalue at B=0, but it is a simple one.

A periodicity of length 3 in is present in the equianharmonic case of a third-order equation resembling (2.12), now called the Halphen equation (Halphen 1888, pp. 571–4). By the preceding, a similar periodicity appears to be present in the equianharmonic case of the Lamé equation. This was not previously realized.

(b) Twisted Lamé polynomials

The twisted Lamé polynomials are exponentially modified Lamé polynomials. They will play a major role in theorem 4.1 and in the hyperelliptic reductions following from the Hermite–Krichever Ansatz, but they are of independent interest.

A solution of the Lamé equation (2.9) on the elliptic curve Embedded Image is said to be a twisted Lamé polynomial, of Type I, or of Type II associated with the point Embedded Image, if it has the respective formEmbedded Imagewith Embedded Image non-zero. Here C, D, E, F are polynomials.

On the level of differential equation solutions, there is little to distinguish between twisted Lamé polynomials and ordinary Lamé polynomials, which are simply twisted polynomials with κ=0. A function Embedded Image will be a solution of the Lamé equation (2.9) if and only if Embedded Image satisfiesEmbedded Image(3.6)This is a differential equation on Embedded Image that generalizes but strongly resembles (2.9). It has a single singular point, Embedded Image, the characteristic exponents of which are Embedded Image. Like B, Embedded Image is an accessory parameter that does not affect these exponents. The values Embedded Image for which a twisted or conventional Lamé polynomial solution of (2.9) exists can be viewed as the points in a two-dimensional parameter space at which (3.6) has single- or double-valued solutions.

If Embedded Image, Embedded Image, Embedded Image and Embedded Image, substituting the expression for each type of twisted polynomial solution into (2.9) and equating the coefficients of powers of x yields the coupled pairs of recurrencesEmbedded Image(3.7)Embedded Image(3.8)Embedded Image(3.9)Embedded Image(3.10)It is easy to determine the maximum value of the exponent j in each of C, D, E, F.

Non-zero twisted Lamé polynomials of Type I (if ≥3) and of Type II (if ≥2) can in principle be constructed from these recurrence relations. If ℓ is odd, respectively, even, then Embedded Image are Embedded Image, respectively, Embedded Image. The coefficients are computed by setting the coefficient of the highest power of x to unity in D and E, respectively, C and F, and working downward.

Unless Embedded Image are specially chosen, the coefficients of negative powers of x may be non-zero. However, by examination, they will be zero if the coefficients of Embedded Image in C and D (for Type I) or E and F (for Type II), equal zero. Embedded Image, Embedded Image and Embedded Image, Embedded Image, are coupled polynomial equations in Embedded Image, and their solutions may be computed by polynomial elimination, e.g. by computing resultants. A minor problem is the proper handling of the case Embedded Image, in which (3.7)–(3.10) reduce to (3.1)–(3.4). If Embedded Image is odd, respectively, even, then Embedded Image and Embedded Image, respectively, Embedded Image and Embedded Image, turn out to be divisible by Embedded Image. By dividing the appropriate equations by Embedded Image before solving each pair of coupled equations, the spurious Embedded Image solutions can be eliminated.

The Type I twisted Lamé spectral polynomial Embedded Image is the polynomial monic in B which is proportional to the resultant of Embedded Image with respect to Embedded Image, with Embedded Image factors removed as indicated. (This assumes Embedded Image; by convention Embedded Image.) The Type II twisted Lamé spectral polynomial Embedded Image is similarly obtained from Embedded Image. (This assumes Embedded Image; by convention Embedded Image.) Each twisted spectral polynomial may be regarded as Embedded Image, respectively, Embedded Image, where the roots Embedded Image are the values of B for which a Lamé equation solution of the specified type exists, counted with multiplicity.

The twisted Lamé spectral polynomials for ≤8 are listed in table 5. The polynomials Embedded Image and Embedded Image are omitted on account of lack of space (their respective degrees are 12 and 16). The following proposition will be proved in §4.

View this table:
Table 5

Twisted Lamé spectral polynomials of Types I and II.

  1. For any integer ℓ≥3, respectively, ≥2, there is a non-trivial twisted spectral polynomial of Type I, respectively, II.

  2. For any integer Embedded Image, Embedded Image is Embedded Image if Embedded Image is odd and Embedded Image if Embedded Image is even; and Embedded Image is Embedded Image if Embedded Image is odd and Embedded Image if Embedded Image is even.

Hence, the full twisted Lamé spectral polynomial Embedded Image, which by definition is Embedded Image, will be of degree Embedded Image in the spectral parameter B. Like the ordinary degree-(2+1) spectral polynomial Embedded Image, Embedded Image is a function of Embedded Image only, because any symmetric polynomial in Embedded Image can be written in terms of the invariants Embedded Image.

(c) Theta-twisted Lamé polynomials

Lamé equation solutions of a third sort can be constructed for certain values of the spectral parameter B. These are linear combinations, over polynomials in the coordinate x, of (i) the multivalued meromorphic function Embedded Image parametrized by the point Embedded Image, and (ii) its derivativeEmbedded Image(3.11)One way of seeing that Embedded Image are a natural basis is to note that when Embedded Image, they reduce to Embedded Image. Hence, the class of functions constructed from them will include the Lamé polynomials of Type II.

A solution of the Lamé equation (2.9) on the elliptic curve Embedded Image is said to be a theta-twisted Lamé polynomial, if it is of the form Embedded Image, with Embedded Image for γ=1, 2, 3. Here, Embedded Image, Embedded Image are polynomials, and the innocuous ‘2’ factor compensates for the ‘1/2’ factor of (3.11).

If Embedded Image and Embedded Image, substituting this expression into (2.9) and equating the coefficients of powers of x yields the coupled pair of recurrencesEmbedded Image(3.12)

Embedded Image(3.13)

If ℓ≥4, non-zero theta-twisted polynomials can in principle be computed from these recurrences. If ℓ is odd, respectively, even, then Embedded Image are Embedded Image respectively, Embedded Image. The coefficients are computed by setting the coefficient of the highest power of x in Embedded Image, respectively, Embedded Image, to unity, and working downward.

Unless B and the point Embedded Image are specially chosen, the coefficients of negative powers of x may be non-zero. However, by examination, they will be zero if the coefficients of Embedded Image in Embedded Image and Embedded Image are both zero. Embedded Image, Embedded Image are equations in Embedded Image. Together with the identity Embedded Image, they make up a set of three polynomial equations for these three unknowns. This system may be solved by polynomial elimination. For example, to obtain a single polynomial equation for B (involving Embedded Image of course), one may eliminate Embedded Image from Embedded Image, Embedded Image by computing their resultants against the third equation, and then eliminate Embedded Image. Alternatively, a Gröbner basis calculation may be performed (Brezhnev 2004).

Irrespective of which procedure is followed, there is a minor problem: the handling of the improper case Embedded Image, in which (3.12) and (3.13) reduce to (3.3) and (3.4). If Embedded Image is odd, respectively, even, then the left-hand side of the equation Embedded Image, respectively, Embedded Image, turns out to be divisible by Embedded Image. By dividing the appropriate equation by Embedded Image before eliminating Embedded Image, the spurious solutions with Embedded Image can be eliminated.

The theta-twisted Lamé spectral polynomial Embedded Image is the polynomial monic in B which is obtained by eliminating Embedded Image from the equations Embedded Image, Embedded Image, with Embedded Image factors removed as indicated. (This assumes Embedded Image; by convention Embedded Image.) Each theta-twisted spectral polynomial may be regarded as Embedded Image, where the roots Embedded Image are the values of B for which a theta-twisted Lamé polynomial exists, counted with multiplicity.

The theta-twisted Lamé spectral polynomials for Embedded Image are listed in table 6. The following proposition will be proved in §4.

View this table:
Table 6

Theta-twisted Lamé spectral polynomials.

  1. For any integer Embedded Image, there is a non-trivial theta-twisted spectral polynomial Embedded Image.

  2. For any integer Embedded Image, Embedded Image is Embedded Image, if Embedded Image is odd, and Embedded Image if Embedded Image is even.

4. The Hermite–Krichever Ansatz

The Hermite–Krichever Ansatz is a tool for solving any Schrödinger-like differential equation, not necessarily of second order, with coefficient functions that are elliptic. Such an equation should ideally have one or more independent solutions that, according to the Ansatz, are expressible as finite series in the derivatives of an elliptic Baker–Akhiezer function, including an exponential factor (cf. (1.5)).

In the context of (2.9), the elliptic-curve algebraic form of the Lamé equation, this means that one hopes to be able to construct a solution on the curve Embedded Image, except at a finite number of values of the spectral parameter Embedded Image, as a finite series in the functions Embedded Image, Embedded Image, multiplied by a factor Embedded Image. Here, Embedded Image is the fundamental multivalued meromorphic function on Embedded Image introduced in §2. Actually, a different but equivalent sort of series is easier to manipulate symbolically. By examination Embedded Image, from which it follows by induction on j that any finite series in Embedded Image, Embedded Image, is a combination (over polynomials in x) of the basis functions Embedded Image. This motivates the following definition.

A solution of the Lamé equation (2.9) on the elliptic curve Embedded Image is said to be an Hermite–Krichever solution if it is of the formEmbedded Image(4.1)for some Embedded Image and Embedded Image. Here Embedded Image, Embedded Image are polynomials.

As defined, Hermite–Krichever solutions subsume most of the solutions explored in §3. If Embedded Image, they reduce to theta-twisted Lamé polynomials. If Embedded Image for γ=1, 2, 3, in which case Embedded Image degenerate to Embedded Image, they reduce to twisted Lamé polynomials of Type II. If both specializations are applied, they reduce to ordinary Lamé polynomials of Type II. The Lamé polynomials of Type I, both ordinary and twisted, are not of the Hermite–Krichever form, but they can be viewed as arising from a passage to the Embedded Image limit.

If Embedded Image and Embedded Image, substituting (4.1) into (2.9) and equating the coefficients of powers of x yields the coupled pair of recurrencesEmbedded Image(4.2)Embedded Image(4.3)If Embedded Image, (4.2) and (4.3) reduce to (3.12) and (3.13), and if Embedded Image, they reduce to (3.9) and (3.10). If both specializations are applied, they reduce to (3.3) and (3.4).

For all Embedded Image, HermiteKrichever solutions can in principle be computed from these recurrences. If ℓ is odd, respectively, even, then Embedded Image are Embedded Image, respectively, Embedded Image. The coefficients are computed by setting the coefficient of the highest power of x in Embedded Image, Embedded Image, to unity, and working downward.

Unless Embedded Image and the point Embedded Image are specially chosen, the coefficients of negative powers of x may be non-zero. However, by examination, they will be zero if the coefficients of Embedded Image in Embedded Image and Embedded Image are both zero. Embedded Image, Embedded Image are equations in Embedded Image. They are ‘compatibility conditions’ similar to those that appear in other applications of the Hermite–Krichever Ansatz. Together with the identity Embedded Image, they make up a set of three equations for these four unknowns.

Informally, one may eliminate any two of the four unknowns Embedded Image, thereby deriving an algebraic relation between the remaining two (involving g2, g3 of course). A rigorous investigation must be more careful. For example, if the ideal generated by the three equations contained a polynomial involving only B (and g2, g3), then a solution of the Hermite–Krichever form would exist for very few values of B (Brezhnev 2004). In practice, this problem does not arise: except at a finite number of values of B (at most), the Ansatz can be employed (Gesztesy & Weikard 1998). In fact, in previous work, an algebraic curve in Embedded Image has been derived for each ≤5. The Embedded Image-curve is the one with the most physical significance, since B is a transformed energy and κ is related to the crystal momentum.

Solving for Embedded Image as functions of Embedded Image reveals that x0 is a rational function of B, and that if κ is not identically zero, y0 is a rational function of B, times κ. These facts can be interpreted in terms of the following seemingly different curve.

The 'th Lamé spectral curve Embedded Image is the hyperelliptic curve over Embedded Image comprising all Embedded Image satisfying Embedded Image, where Embedded Image is the full Lamé spectral polynomial, of degree 2+1 in B. (Embedded Image was informally introduced as Embedded Image in §1, where the original energy parameter E was used.) Embedded Image will have genus unless two roots of Embedded Image coincide, i.e. unless the Klein invariant J is a root of one of the two Cohn polynomials of table 4, in which case the genus equals −1.

For each , there must exist a parametrization of Lamé equation solutions by a point Embedded Image on the punctured curve Embedded Image, by the general theory of Hill's equation on Embedded Image (McKean & van Moerbeke 1979). For any finite-band Schrödinger equation on an elliptic curve, including the integer- Lamé equation, the Baker–Akhiezer function (2.13) provides such a parametrization of solutions. In the general theory, the parametrizing hyperelliptic curve Γ for any finite-band Hill's equation arises from differential–difference bispectrality: as a uniformization of the relation between the energy parameter B and the crystal momentum k (Treibich 2001). This curve Embedded Image is defined by an irrationality of the form Embedded Image, and B and k are meromorphic functions on it; the former single valued, and the latter additively multivalued. The energy is computed from the degree-2 map Embedded Image given by Embedded Image, and the crystal momentum from the formulaEmbedded Image(4.4)in which the line integrals on Embedded Image are taken over the appropriate fundamental loop. Here, Embedded Image is a certain projection Embedded Image, and Embedded Image is a certain auxiliary meromorphic function. These two morphisms of complex manifolds are ‘odd’ under the involution Embedded Image, i.e. Embedded Image and Embedded Image, and Embedded Image. Hence, x0 must be a rational function of B, and each of y0 and κ must be a rational function of B, times ν.

In the general theory of finite-band equations, the Baker–Akhiezer uniformization is viewed as more fundamental than the Hermite–Krichever Ansatz. However, in the case of the integer- Lamé equation, one can immediately identify the curve in Embedded Image, derived from the Ansatz as explained previously, with the th Lamé spectral curve Embedded Image. It is isomorphic to it by a birational equivalence of a simple kind: the ratio Embedded Image is a rational function of B.

This interpretation makes possible a geometrical understanding of each of the types of Lamé spectral polynomial worked out in §3. Owing to oddness, each of the finite Weierstrass points Embedded Image on Embedded Image, which correspond to band edges, must be mapped by the projection Embedded Image to one of the finite Weierstrass points Embedded Image or to (∞, ∞). Bearing in mind that Type I Lamé polynomials, both ordinary and twisted, are not of the Hermite–Krichever form, on account of Embedded Image formally equalling (∞, ∞) and κ equalling ∞, one has the following proposition.

  1. The roots of the Type I Lamé spectral polynomial Embedded Image are the B-values of the finite Weierstrass points Embedded Image that are projected by Embedded Image to the infinite Weierstrass point Embedded Image. Moreover, the roots of the Type I twisted Lamé spectral polynomial Embedded Image include the B-values of the finite non-Weierstrass points that are projected to Embedded Image and do not include the B-value of any finite point that is not projected to Embedded Image.

  2. For Embedded Image, the roots of the Type II Lamé spectral polynomial Embedded Image are the B-values of the finite Weierstrass points Embedded Image that are projected by Embedded Image to the finite Weierstrass point Embedded Image. Moreover, the roots of the Type II twisted Lamé spectral polynomial Embedded Image include the B-values of the finite non-Weierstrass points that are projected to Embedded Image and do not include the B-value of any finite point that is not projected to Embedded Image.

  3. The roots of the theta-twisted Lamé spectral polynomial Embedded Image include the B-values of the finite non-Weierstrass points that are zeroes of Embedded Image and do not include the B-value of any finite point that is not a zero.

The phrasing of the proposition leaves open the possibilities that (i) a root of Embedded Image may be a root of Embedded Image, (ii) a root of Embedded Image may be a root of Embedded Image and (iii) a root of Embedded Image may be the B-value of a finite Weierstrass point, i.e. a band edge. Generically, these three types of coincidence do not occur, but instances are not difficult to find. One is the case Embedded Image and Embedded Image (i.e. J=0), in which by examination Embedded Image and Embedded Image have the common root B=0.

To proceed beyond the proposition, a significant result from finite-band integration theory is needed. For the integer- Lamé equation, the covering Embedded Image and the auxiliary map Embedded Image are both of degree (+1)/2, irrespective of the choice of elliptic curve Embedded Image. From this fact, supplemented by proposition 4.2, the two propositions 3.4 and 3.6, which were left unproved in §3, immediately follow.

The fibre over any point Embedded Image must comprise (+1)/2 points of Embedded Image, the counting being up to multiplicity. Consider the fibre over (∞, ∞), which by examination includes with unit multiplicity the point Embedded Image. It also includes each finite Weierstrass point of Embedded Image that corresponds to a Type I Lamé polynomial. As was shown in §3, the number of these up to multiplicity, Embedded Image, equals (−1)/2 if is odd and /2+1 if is even. Hence, the number of additional points above Embedded Image is Embedded Image if is odd and Embedded Image if is even. Since the projection Embedded Image is odd under the involution Embedded Image, these occur in pairs. Hence, Embedded Image must equal Embedded Image if is odd and Embedded Image if is even; which is the formula for Embedded Image as given in proposition 3.4. A similar computation applied to the fibre above any finite Weierstrass point Embedded Image yields the formula for Embedded Image given in that proposition.

The formula for Embedded Image stated in proposition 3.6 can also be derived with the aid of proposition 4.2. Since the map Embedded Image has degree Embedded Image, the fibre above 0 comprises that number of points, up to multiplicity. It includes each finite Weierstrass point of Embedded Image that corresponds to a Type II Lamé polynomial (but not the Weierstrass points corresponding to Type I Lamé polynomials, since those are not of the Hermite–Krichever form and formally have Embedded Image). The number of these up to multiplicity is three times Embedded Image, which equals 3(+1)/2 if is odd and 3/2 if is even. Hence, the number of additional points above 0 is Embedded Image if is odd and Embedded Image if is even. They come in pairs, and division by two yields the formula for Embedded Image given in proposition 3.6.

A geometrized version of proposition 3.2 can also be proved, with the aid of an additional result that goes beyond the Hermite–Krichever Ansatz. The Lamé spectral curve Embedded Image is non-singular with genus Embedded Image for generic values of the Klein invariant Embedded Image, and when it degenerates to a singular curve Embedded Image, the singular curve has genus Embedded Image, with singularities that are limits as Embedded Image of Weierstrass points of Embedded Image. It follows that the th Types I and II Cohn polynomials, which characterize the pairs (g2, g3) for which the th Lamé operator on Embedded Image has degenerate algebraic spectrum of the specified type, i.e. for which Embedded Image has a pair of degenerate finite Weierstrass points of the specified type, in fact do more: they characterize the (g2, g3) for which Embedded Image is singular. Owing to the reduction of the genus by at most unity, there can be, as proposition 3.2 states, no multiple coincidences of the algebraic spectrum.

The problem of explicitly constructing a Hermite–Krichever solution of the integer- Lamé equation, of the form (4.1), will now be considered. What are needed are the quantities Embedded Image, or equivalently Embedded Image. Each of the latter is a rational function of the spectral parameter B.

One way of deriving these functions is to eliminate variables from the system of three polynomial equations in Embedded Image, as previously explained. Coupled with the spectral equation Embedded Image, this yields explicit expressions for the three desired functions. By the standards of polynomial elimination algorithms, this procedure is not time consuming. It is much more efficient than the manipulations of compatibility conditions that other authors have employed. Beginning with Halphen (1888), it has been the universal practice to apply the Hermite–Krichever Ansatz to the Weierstrassian form of the Lamé equation, i.e. to (2.6), rather than to the elliptic curve form (2.9). It is now clear that this Weierstrassian approach is far from optimal. For example, the computation of the covering map Embedded Image in the case =5, by Eilbeck & Enol'skii (1994), required 7 h of computer time. The just-sketched elliptic curve approach requires only a fraction of a second.

It turns out that for constructing Hermite–Krichever solutions, this revised elimination scheme is also not optimal. Remarkably, at this point no elimination needs to be performed at all, since the covering Embedded Image and auxiliary function Embedded Image can be computed directly from the spectral polynomials of §3.

For all integer ℓ≥1, the covering map Embedded Image appearing in the Ansatz maps Embedded Image to Embedded Image according toEmbedded Image(4.5)Embedded Image(4.6)with γ in (4.5) being any of 1, 2, 3. The auxiliary function Embedded Image is given byEmbedded Image(4.7)

With the exception of the three -dependent prefactors, such as Embedded Image, the formulae (4.5)–(4.7) follow uniquely from proposition 4.2, regarded as a list of properties that Embedded Image and Embedded Image must satisfy.

Embedded Image must map each point Embedded Image, where Bs is a root of Embedded Image, singly to (∞, ∞), and each point Embedded Image, where Bs is a root of Embedded Image, singly to Embedded Image. It must also map each point Embedded Image, where Bt is a root of Embedded Image, singly to (∞, ∞), and each point Embedded Image, where Bt is a root of Embedded Image, singly to Embedded Image. In all these statements, the counting is up to multiplicity.

In addition, Embedded Image must map each point Embedded Image, where B′ is a root of Embedded Image, singly to zero, and must map each point (Bs, 0), where Bs is a root of Embedded Image, and each point Embedded Image, where Embedded Image is a root of Embedded Image, singly to ∞. In these statements as well, the counting is up to multiplicity.

The -dependent prefactors in (4.5)–(4.7) can be deduced from the leading-order asymptotic behaviour of x0 and Embedded Image as Embedded Image. ▪

The remarkably simple formulae of the theorem permit Hermite–Krichever solutions of the form (4.1) to be constructed for very large values of , since the Lamé spectral polynomials L, Lt, (ordinary, twisted and theta-twisted) are relatively easy to work out, as §3 made clear. Tables 3, 5 and 6 may be consulted.

It should be stressed that in the formula (4.5) for x0, the same right-hand side results, irrespective of which of the three values of γ is chosen. All terms explicitly involving eγ will cancel. Of course, all powers of eγ higher than the second must first be rewritten in terms of g2, g3 by using the identity Embedded Image. In the same way, it is understood that the numerator of the right-hand side of (4.6), the terms of which are symmetric in e1, e2, e3, should be rewritten in terms of g2, g3. (This can always be done: for example, the symmetric polynomial Embedded Image equals Embedded Image).

The application of theorem 4.1 to the cases =1, 2, 3 may be illuminating.

  1. If =1, then Embedded Image and κ=0. The map Embedded Image is a mere change of normalization, since Γ1 is isomorphic to Embedded Image; cf. (2.15).

  2. If =2, thenEmbedded Image(4.8)Embedded Image(4.9)and Embedded Image.

  3. If Embedded Image, thenEmbedded Image(4.10)Embedded Image(4.11)and Embedded Image.

The formulae for =2, 3 were essentially known to Hermite. Setting =4, 5 in theorem 4.1 yields the less familiar and more complicated formulae that Enol'skii & Kostov (1994) and Eilbeck & Enol'skii (1994) derived by eliminating variables from compatibility conditions. Theorem 4.1 readily yields the covering map Embedded Image and auxiliary function Embedded Image for far larger .

5. Hyperelliptic reductions

The cover Embedded Image introduced as part of the Hermite–Krichever Ansatz, i.e. the map Embedded Image, is of independent interest, since explicit examples of coverings of elliptic curves by higher genus algebraic curves are few, and the problem of determining which curves can cover Embedded Image, for either specified or arbitrary values of the invariants Embedded Image, remains unsolved. Embedded Image generically has genus Embedded Image, as noted, and the cover will always be of degree Embedded Image. The formula for Embedded Image given in theorem 4.1 is consistent with this, since N equals Embedded Image, the maximum of the degrees in B of the numerator and denominator of Embedded Image. The degrees Embedded Image, as well as the twisted degrees Embedded Image, were computed in §3 (for the latter, see proposition 3.4).

Since Embedded Image is hyperelliptic (defined by the irrationality Embedded Image) and Embedded Image is elliptic (defined by the irrationality Embedded Image), the map Embedded Image enables certain hyperelliptic integrals to be reduced to elliptic ones. Just as Embedded Image is equipped with the canonical holomorphic 1-form Embedded Image, so can Embedded Image be equipped with the holomorphic 1-form Embedded Image. Any integral of a function in the function field of a hyperelliptic curve (here, any rational function Embedded Image) against its canonical 1-form is called a hyperelliptic integral. Hyperelliptic integrals are classified as follows (Belokolos et al. 1986). The linear space of meromorphic 1-forms of the form Embedded Image, i.e. of Abelian differentials, is generated by 1-forms of the first, second and third kinds. These are (i) holomorphic 1-forms, with no poles; (ii) 1-forms with one multiple pole and (iii) 1-forms with a pair of simple poles, the residues of which are opposite in sign. The indefinite integrals of (i)–(iii) are called hyperelliptic integrals of the first, second and third kinds. They generalize the three kinds of elliptic integral (Abramowitz & Stegun 1965, ch. 17).

Hyperelliptic integrals of the first kind are the easiest to study, since the linear space of holomorphic 1-forms is finite-dimensional and is spanned by Embedded Image, Embedded Image, where g is the genus. Hence, there are only g independent integrals of the first kind. A consequence of the map Embedded Image is that on any hyperelliptic curve of the form Embedded Image, there are really only g−1 independent integrals of the first kind, modulo elliptic integrals (considered trivial by comparison). Changing variables in Embedded Image, the elliptic integral of the first kind, yieldsEmbedded Image(5.1)The quantity in square brackets is rational in B and, in fact, is guaranteed to be a polynomial in B of degree less than or equal to g−1, since the left-hand integrand is a pulled-back version of the right-hand one and must be a holomorphic 1-form. Equation (5.1) is a linear constraint relation on the g basic hyperelliptic integrals of the first kind. It reduces the number of independent integrals from g to g−1.

The cases =2, 3 of (5.1) may be instructive. The maps Embedded Image were given in (4.8), (4.10), and the degree-(2+1) spectral polynomials Embedded Image follow from table 3. If =2, one obtains the hyperelliptic-to-elliptic reductionEmbedded Image(5.2)where the change of variables is performed by (4.8). If =3, one obtainsEmbedded Image(5.3)where the full spectral polynomial Embedded Image isEmbedded Imageand the change of variables is performed by (4.10). These reductions were known to Hermite (Königsberger 1878; Belokolos et al. 1986). More recently, the reductions induced by the =4, 5 coverings have been worked out (Eilbeck & Enol'skii 1994; Enol'skii & Kostov 1994). However, the reductions with >5 proved to be too difficult to compute. Theorem 4.1 makes possible the computation of many such higher reductions.

The following proposition specifies the normalization of the pulled-back 1-form. It follows from the known leading order asymptotic behaviour of Embedded Image as Embedded Image.

For all integer Embedded Image, the polynomial function Embedded Image in the hyperelliptic-to-elliptic reduction formulaEmbedded Imagewhere the change of variables Embedded Image is given by theorem 4.1, equals Embedded Image times a polynomial Embedded Image which is monic and of degree Embedded Image in B.

The polynomials Embedded Image are listed in table 7. Embedded Image agree with those found by Enol'skii et al. if allowance is made for a difference in normalization conventions.

View this table:
Table 7

Polynomials specifying the holomorphic 1-forms pulled back from Graphic.

A complete analysis of Lamé-derived elliptic covers will need to consider exceptional cases of several kinds. The covering curve Embedded Image generically has genus g=, but if the Klein invariant Embedded Image is a root of one of the two Cohn polynomials of table 4, the genus will be reduced to −1. According to conjecture 3.1, this will happen, for instance, if Embedded Image and g2=0 (i.e. J=0), so that the base curve Embedded Image is equianharmonic. When g is reduced to −1 in this way, the linear space of holomorphic 1-forms will be spanned by Embedded Image, Embedded Image, where B0 is the degenerate root of the spectral polynomial; but (5.1) will still provide a linear constraint on the associated hyperelliptic integrals.

Another sort of degeneracy takes place when the modular discriminant Embedded Image equals zero, i.e. when J=∞. In this case, Embedded Image will degenerate to a rational curve due to two or more of e1, e2, e3 being coincident. Lamé-derived reduction formulae such as (5.2) and (5.3) will continue to apply. (They are valid though trivial even in the case e1=e2=e3, in which g2=g3=0.) Hence, these formulae include as special cases certain hyperelliptic-to-rational reductions.

Subtle degeneracies of the covering map Embedded Image can occur, even in the generic case when Embedded Image has genus and Embedded Image has genus 1. The branching structure of Embedded Image is determined by the polynomial Embedded Image of table 7, which is proportional to Embedded Image. If Embedded Image has distinct roots Embedded Image, then Embedded Image will normally have 2−2 simple critical points on Embedded Image, of the form Embedded Image. However, if any Embedded Image is located at a band edge, i.e. at a branch point of the hyperelliptic Embedded Image-curve, then Embedded Image will be a double critical point. This appears to happen when Embedded Image and the base curve Embedded Image is equianharmonic; the double critical point being located at Embedded Image. Even if no root of Embedded Image is located at a band edge, it is possible for it to have a double root, in which case each of a pair of points Embedded Image will be a double critical point. By examination, this happens when =4 and Embedded Image.

A few hyperelliptic-to-elliptic reductions, similar to the quadratic (N=2) reduction of Legendre and Jacobi, can be found in the handbooks of elliptic integrals (Byrd & Friedman 1954, §§575 and 576). The Lamé-derived reductions, indexed by , should certainly be included in any future handbook. It is natural to wonder whether they can be generalized in some straightforward way. The problem of finding the genus-2 covers of an elliptic curve was intensively studied in the nineteenth century, by Weierstrass and Poincaré among many others, and one may reason by analogy with results on =2. One expects that for all ≥2 and for arbitrarily large N, a generic Embedded Image can be covered by some genus- curve via a covering map of degree N. Each Lamé-derived covering Embedded Image has degree Embedded Image and may be only a low-lying member of an infinite family. Generalizing the Lamé-derived coverings may be possible even if one confines oneself to Embedded Image. One can of course pre-compose with an automorphism of Embedded Image and post-compose with an automorphism of Embedded Image (a modular transformation). However, when =2, a rather different covering map with the same degree is known to exist (Belokolos et al. 1986). π2 has two simple critical points on Γ2, but the other degree-3 covering map has a single double critical point on its analogue of Γ2. Both can be generalized to include a free parameter (Burnside 1892; Belokolos & Enol'skii 2000). It seems possible that when >2, similar alternatives to the Lamé-derived coverings may exist, with degree (+1)/2 but different branching structures.

6. Dispersion relations

It is now possible to introduce dispersion relations and determine the way in which the Hermite–Krichever Ansatz reduces higher- to =1 dispersion relations. The starting point is the fundamental multivalued function Φ introduced in §2. As noted, if the parametrization point Embedded Image on the punctured elliptic curve Embedded Image is over Embedded Image, then Embedded Image will be a solution of the =1 case of the Lamé equation (2.9). Embedded Image is defined by Embedded Image, so the hypothesis here is that Embedded Image should equal Embedded Image.

In the Jacobi form (with independent variable α), respectively the Weierstrassian form (with independent variable u), the crystal momentum k characterizes the behaviour of a solution of the Lamé equation under Embedded Image, respectively Embedded Image. Both shifts correspond to motion around Embedded Image, along a fundamental loop that passes between Embedded Image and (∞, ∞), and cannot be shrunk to a point. (If e1, e2, e3 are defined by (2.1), this will be because y is positive on one-half of the loop, and negative on the other.) By definition, the solution will be multiplied by Embedded Image. It follows from the definition (2.11) of Φ that when =1,Embedded Image(6.1)In other words, when =1, the crystal momentum is given by a complete elliptic integral. In the context of finite-band integration theory, this is a special case of (4.4).

It was pointed out in §4 that the spectral curve Γ1 that parametrizes =1 solutions can be identified with Embedded Image itself, via the identification Embedded Image. This suggests a subtle but important reinterpretation of k. In §1, it was introduced as a function of the energy parameter, here B, which is determined only up to integer multiples of Embedded Image, and which is also undetermined as to sign. If the presence of y0=2ν in (6.1) is taken into account, it is clear that the =1 crystal momentum, called k1 henceforth, should be regarded as a function not on Embedded Image, but rather on Embedded Image. In this interpretation, the indeterminacy of sign disappears. The additive indeterminacy, on account of which k1 is an elliptic function of the second kind, remains but can be viewed as an artefact: it is due to k1∝log ξ, where ξ is the Floquet multiplier. The behaviour of k1 near the puncture Embedded Image is easily determined. It follows from (6.1) that as Embedded Image, i.e. Embedded Image, each branch of k1 is asymptotic to Embedded Image to leading order. Since Embedded Image have double and triple poles there, respectively, it follows that each branch of k1 has a simple pole at the puncture.

The multiplier ξ is a true single-valued function on the punctured spectral curve Embedded Image, and moreover is entire. One can write Embedded Image, since the multiplier is never zero. Like k1, this function is not algebraic: it necessarily has an essential singularity at the puncture. The Embedded Image-curve over Embedded Image, which is a single cover, and the (B, ξ)-curve over Embedded Image, which is a double cover, are both transcendental curves.

The crystal momentum for each integer ≥1 may similarly be viewed as an additively multivalued function on the punctured spectral curve Embedded Image. It will be written as Embedded Image, with the understanding that for this to be well defined, Embedded Image must be related by the spectral curve relation Embedded Image. The quantity Embedded Image will not be undetermined as to sign. Suppose now that the projections Embedded Image of the Hermite–Krichever Ansatz are regarded as maps Embedded Image. In other words, Embedded Image maps Embedded Image to the point Embedded Image. The reductions Embedded Image for Embedded Image, for example, follow from (4.8)–(4.11).

If the integration of the Lamé equation on the elliptic curve Embedded Image, for integer Embedded Image, can be accomplished in the framework of the HermiteKrichever Ansatz by maps Embedded Image and Embedded Image, where Embedded Image and Embedded Image map the point Embedded Image to Embedded Image and Embedded Image, respectively, then the dispersion relation for the HermiteKrichever solutions will be Embedded Image, in which Embedded Image can be expressed in terms of k1 byEmbedded Image(6.2)

This proposition follows immediately from the form of the Hermite–Krichever solutions (4.1). The first term in (6.2) arises from the Φ, Φ′ factors, and the second from the exponential. The factors Embedded Image, Embedded Image in (4.1) do not contribute to the crystal momentum. Equation (6.2) could also be derived from the general theory of finite-band integration, specifically from the formula (4.4). However, a derivation from the Hermite–Krichever Ansatz seems more natural in the present context.

The effort expended in replacing two-valuedness by single-valuedness is justified by the following observation. As a function of B alone, rather than of the pair Embedded Image, each of the two terms of (6.2) would be undetermined as to sign. This ambiguity could cause confusion and errors. The present formulation, though a bit pedantic, facilitates the determination of the correct relative sign.

The form of the Embedded Image map assumed in the proposition is of course the form supplied by theorem 4.1. Substituting the formulae of the theorem into (6.2) yields an explicit expression for Embedded Image. It follows readily from this expression that for all integer ≥1, each branch of Embedded Image on the hyperelliptic Embedded Image-curve Embedded Image satisfies Embedded Image, Embedded Image. Since Embedded Image have double and order-(2+1) poles at the puncture (∞, ∞), respectively, this implies that irrespective of , each branch of the crystal momentum has a simple pole at the puncture.

7. Band structure of the Jacobi form

The results of §§3–6 were framed in terms of the elliptic-curve algebraic form of the Lamé equation. Most work on Lamé dispersion relations has used the Jacobi form instead, and has led accordingly to expressions involving Jacobi theta functions. To derive dispersion relations that can be compared with previous work, the formulation of §6 must be converted to the language of the Jacobi form.

The relationships among the several forms were sketched in §2. In the Weierstrassian and Jacobi forms, the Lamé equation is an equation on Embedded Image with doubly periodic coefficients, rather than an equation on the curve Embedded Image. In the conversion to the Jacobi form, the invariants Embedded Image are expressed in terms of the modular parameter m by (2.2), with A=1 by convention. The coordinate x on Embedded Image is interpreted as the function Embedded Image, i.e. as Embedded Image, where Embedded Image and Embedded Image are the respective independent variables of the Weierstrassian and Jacobi forms. The holomorphic differential dx/y corresponds to du or dα, and the derivative yd/dx to d/du or d/dα. The coordinate y=(yd/dx)x on Embedded Image is interpreted as Embedded Image, i.e. as the doubly periodic function Embedded Image on the complex α-plane. The functions Embedded Image, γ=1, 2, 3, correspond to Embedded Image, Embedded Image and Embedded Image. The equationEmbedded Image(7.1)relates the accessory parameters B, E of the different forms.

With these formulae, it is easy to convert the Lamé polynomials of table 2 to polynomials in Embedded Image, and the spectral polynomials of table 3 to polynomials in E, for comparison with the list given by Arscott (1964, §9.3.2).

The Jacobi-form spectral polynomial Embedded Image is the negative of the spectral polynomial Embedded Image, when Embedded Image are expressed in terms of E and m. It is a monic degree-(2+1) polynomial in E with coefficients polynomial in m, and can be regarded as Embedded Image, where the roots {Es} are the values of the energy E for which there exists a Lamé polynomial solution of the Lamé equation, counted with multiplicity. (The negation is due to the relative minus sign in the Embedded Image correspondence (7.1)).

The th Jacobi-form spectral curve Embedded Image is the hyperelliptic curve over Embedded Image comprising all Embedded Image satisfying Embedded Image. In the usual case when m is real, Embedded Image will be real if E is in a band, and non-real if E is in a lacuna. In both cases, it is determined only up to negation. By convention, the correspondence between the curve Embedded Image and the previously introduced curve Embedded Image, which was defined by Embedded Image, is given by Embedded Image.

The following cases are examples. When =1, 2, 3, the spectral polynomial factors over the integers into polynomials at most quadratic in E. In full,Embedded Image(7.2)Embedded Image(7.3)Embedded Image(7.4)In (7.3) and (7.4), the first factor arises from Embedded Image and the remaining three from the factors Embedded Image, γ=1, 2, 3. In (7.2), there is no Type I factor. The polynomials (7.2)–(7.4) agree with those obtained by Arscott.

The derivation of the Jacobi-form spectral polynomial Embedded Image from Embedded Image sheds light on a regularity noted by Ince (1940a, §7), which arises in the lemniscatic case m=1/2. Ince observed that if ≤6, at least, then Embedded Image has an integer root, namely Embedded Image. In fact, this is the case for all integer . By (7.1), the presence of this root is equivalent to the full spectral polynomial Embedded Image having B=0 as a root. But if m=1/2, it follows from (2.1) that e2=0. A glance at the pattern of coefficients in table 3 reveals that if g3=0 and a singular point eγ also equals zero, then either the Type I spectral polynomial Embedded Image (if Embedded Image) or one of the three Type II spectral polynomials Embedded Image (if Embedded Image) will necessarily have B=0 as a root.

Dispersion relations in their Jacobi form can now be investigated. Recall that if =1, the Jacobi-form Lamé equation (1.1) has Embedded Image as a solution, where the theta quotient Embedded Image (the Jacobi-form version of Halphen's l'élément simple) is defined in (1.3), and the multivalued parameter α0 is defined by Embedded Image. This solution has crystal momentum Embedded Image equal to Embedded Image, which is undetermined as to sign, and is also determined only up to integer multiples of Embedded Image. The sign indeterminacy is due to Embedded Image being even. This causes α0 to be undetermined as to sign, and k1 as well, since the function Embedded Image is odd.

The parametrization point α0, or the equivalent point Embedded Image of the Weierstrassian form, corresponds to the parametrization point Embedded Image of the fundamental function Φ on the elliptic curve Embedded Image. The correspondence is the usual one: Embedded Image, Embedded Image. The first of these two equations says that Embedded Image, and the latter that Embedded Image. The formula which computes α0 from E, namely Embedded Image, is readily seen to be a translation to the Jacobi form of the familiar condition Embedded Image, which simply says that the parametrization point Embedded Image must be ‘over’ Embedded Image.

The correspondence between the Jacobi and elliptic-curve forms motivates the following reinterpretation of the crystal momentum of the fundamental solution Embedded Image, which is modelled on the reinterpretation of the last section. k1 should be viewed as a function not of the energy Embedded Image, but rather of a point Embedded Image on the punctured Jacobi-form spectral curve Embedded Image. There are two such points for each energy E, except when E is a band edge. This is the source of the sign ambiguity in the parameter α0. Since Embedded Image, the equationEmbedded Image(7.5)determines a unique sign for α0, provided that Embedded Image is specified in addition to E. k1 will be written as Embedded Image, with the understanding that for this to be well defined, the pair Embedded Image must be related by the spectral curve relation Embedded Image. The additively undetermined quantity Embedded Image will not be undetermined as to sign. It is easily checked that on each branch, Embedded Image as Embedded Image.

A solution of the Jacobi-form Lamé equation (1.1) is said to be a Hermite–Krichever solution if it is of the formEmbedded Image(7.6)for some Embedded Image and Embedded Image. Here Embedded Image are polynomials, and Embedded Image.

The expression (7.6) is a replacement for the original Jacobi-form expression (1.5), to which it is equivalent. Regardless of which is used, it is easy to compute the crystal momentum of an Hermite–Krichever solution. The momentum computed from (7.6) will be Embedded Image, up to additive multivaluedness. The first term arises from the Embedded Image factors, and the second from the exponential. The factors Embedded Image do not contribute, since Embedded Image is periodic in α with period 2K(m).

The Jacobi form of the Hermite–Krichever Ansatz asserts that for all integer and Embedded Image, there is an Hermite–Krichever solution for all but a finite number of values of the energy E. On the elliptic curve Embedded Image, these solutions were constructed from two maps: a projection Embedded Image and an auxiliary function Embedded Image. However, Embedded Image should really be regarded as a map from Embedded Image to Γ1, on account of the correspondence between Embedded Image and Γ1 provided by Embedded Image. The following is the Jacobi-form version of proposition 6.1.

Suppose that the integration of the Lamé equation on the elliptic curve Embedded Image, for integer ℓ≥1, can be accomplished in the framework of the Hermite–Krichever Ansatz by the maps Embedded Image and Embedded Image, where Embedded Image and Embedded Image map the point (B, ν) to Embedded Image and Embedded Image, respectively. Then the dispersion relation for the solutions of the Jacobi form of the Lamé equation will be Embedded Image, where up to additive multivaluednessEmbedded Image(7.7)in whichEmbedded Image(7.8)Embedded Image(7.9)Embedded Image(7.10)

The formula (7.7) follows by inspection. The projection Embedded Image reduces the integration of the Lamé equation to the integration of an =1 equation, the ‘B’ parameter of which equals Embedded Image. By (7.1), the ‘E’ parameter of the =1 equation will be the right-hand side of (7.8). The two terms of (7.7) are simply the two terms of Embedded Image. The equality Embedded Image has been used.

It is straightforward to apply proposition 7.1 to the cases =2, 3, since the coverings π2, π3 and auxiliary functions κ2, κ3 were worked out in §4. A brief discussion of the =2 case should suffice. After some algebra, one findsEmbedded Image(7.11)Embedded Image(7.12)Embedded Image(7.13)from which Embedded Image may be computed by (7.7). Like k1, k2 is determined only up to integer multiples of Embedded Image. Each branch of k2 has the property that Embedded Image, Embedded Image, with ‘±’ determined by the sign of Embedded Image. This is a special case of a general fact: for all integer ≥1, Embedded Image, Embedded Image, since each branch of Embedded Image is asymptotic to Embedded Image as Embedded Image.

The real portions of the dispersion relations Embedded Image, Embedded Image and Embedded Image are graphed in figure 1. For ease of viewing, each crystal momentum is regarded as lying in the interval Embedded Image; which is equivalent to choosing the sign of Embedded Image in an E-dependent way. As (7.2)–(7.4) imply, the two =1 bands are Embedded Image, the three =2 bands are Embedded Image, and the four =3 bands are Embedded Image.

Figure 1

Dispersion relations for =1, 2, 3 in the lemniscatic case m=1/2.

The =1 graph agrees with that of Li et al. (2000, fig. 6), and for confirmation, with that of Sutherland (1973, fig. 1). Unfortunately, the =2 graph disagrees with that of Li et al. in the placement or direction of curvature of each of the two upper bands. The algorithm they used for reducing =2 to =1, which was based on Hermite's solution of the Jacobi-form Lamé equation (Whittaker & Watson 1927, §23.71), evidently yielded incorrect results for these bands. It appears that for the middle band, at least, the discrepancy can be traced to an incorrect choice of relative sign for the two terms of k=k2. The reinterpretation of the crystal momentum as a function on the spectral curve, rather than a function of the energy, eliminates such sign ambiguities.

8. Summary and final remarks

A new approach to the closed-form solution of the Lamé equation has been introduced. Our key result, theorem 4.1, provides a formula for the covering map of the Hermite–Krichever Ansatz in terms of certain polynomials which are of independent interest, namely twisted spectral polynomials. The theorem permits an efficient computation of Lamé dispersion relations, of a mixed symbolic–numerical kind. Cohn polynomials, which are a new concept, have also been introduced. The roots of such a polynomial are the points in elliptic moduli space at which a Lamé spectral polynomial has a double root, so that the Lamé spectral curve becomes singular. Twisted and theta-twisted Cohn polynomials could be defined, as well.

The approach of this paper can be extended from the Lamé equation to the Heun equation, which as a differential equation on the elliptic curve Embedded Image has up to four regular singular points, positioned at the finite Weierstrass points Embedded Image as well as at (∞, ∞). Its Weierstrassian form is called the Treibich–Verdier equation (Smirnov 2002), and its Jacobi form, at least when only two of the Weierstrass points are singular points, the associated Lamé equation (Magnus & Winkler 1979, §7.3).

The ‘four triangular numbers’ condition for the Heun equation to have the finite-band property, due to Treibich & Verdier (1992) and Gesztesy & Weikard (1995b), is now well known. In the finite-band case, the number of points in the algebraic spectrum has been computed up to multiplicity (Gesztesy & Weikard 1995b). The corresponding band-edge solutions are Heun polynomials. Applying the Hermite–Krichever Ansatz to the finite-band Heun equation leads to a greater variety of coverings of Embedded Image than arise in the solution of the integer- Lamé equation; for example, coverings by a genus-2 hyperelliptic curve that have degrees 3, 4, 5 (Belokolos & Enol'skii 2000). These coverings play a role in the construction of elliptic soliton solutions of certain nonlinear evolution equations that occur in fibre optics (Christiansen et al. 2000). A treatment of the Heun equation along the lines of this paper will appear elsewhere.

Acknowledgments

This work was partially supported by NSF grant PHY-0099484. The symbolic computations were performed with the aid of the Macsyma computer algebra system. The helpful comments of an anonymous referee are gratefully acknowledged.

Footnotes

  • One contribution of 15 to a Theme Issue ‘30 years of finite-gap integration’.

References

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