What does integrability of finite-gap or soliton potentials mean?

Yurii V Brezhnev

Abstract

In the example of the Schrödinger/KdV equation, we treat the theory as equivalence of two concepts of Liouvillian integrability: quadrature integrability of linear differential equations with a parameter (spectral problem) and Liouville's integrability of finite-dimensional Hamiltonian systems (stationary KdV equations). Three key objects in this field—new explicit Ψ-function, trace formula and the Jacobi problem—provide a complete solution. The Θ-function language is derivable from these objects and used for ultimate representation of a solution to the inversion problem. Relations with non-integrable equations are also discussed.

Keywords:

1. Introduction

As it is understood in contemporary language, the theory of algebraic–geometric (or finite-gap) integration is a theory of integration of integrable nonlinear (1+1)-soliton partial differential equations (PDEs), finite-dimensional Hamiltonian dynamical systems and spectral problems defined by ordinary differential equations (ODEs). A large number of papers devoted to these problems, which have appeared over the last three decades, show that the vague term ‘integration of integrable’ does not mean an automatic integrating procedure in some simple sense of the word. The contributions that are still being made to this theory testify to its vitality: all the evidence points to the continuance of its growth.

Although the term ‘completely integrable’ is quite an appropriate one, there is no commonly agreed answer to the questions about integrability and, in particular, to the question in the title of the present work. For example, in the theory of dynamical systems, this means the well-defined Liouvillian integrability, but the search for separability variables (followed by action-angle ones) is a subject of an independent theory (Flaschka & McLaughlin 1976; Vanhaecke 1996). In the case of (1+1)-integrable PDEs, the theory is, in fact, a treatment of these equations as infinite-dimensional analogues of Hamiltonian systems (Gardner 1971; Zakharov & Faddeev 1971) plus a set of non-trivial exact solutions in terms of elementary functions: solitons, positons and their relatives (Ablowitz & Segur 1981). There are natural multidimensional generalizations of the theory (the Kadomtsev–Petviashvili, KP, Davey–Stewartson equations, their hierarchies, etc.), which are rather well developed also. Integrability of spectral problems (direct and inverse) is usually associated with the inverse scattering method, soliton theory (Ablowitz & Segur 1981; Novikov 1984; Levitan 1987) and its Θ-generalizations (Dubrovin 1975; Flaschka 1975; McKean & van Moerbeke 1975; Matveev 1976; Novikov 1984; Belokolos et al. 1994; Gesztesy & Holden 2003). We must include here the deep links of this field to diverse areas of mathematics shedding light on the mechanism of integrability: Hirota's τ-function method, Painlevé analysis (Conte 1999), Abelian varieties (Dubrovin 1976; Vanhaecke 1996), Darboux–Bäcklund algebraic transformations (Matveev & Salle 1991), differential geometry (Darboux 1915), theory of commuting differential operators (Krichever 1978), etc. Each of these links is sufficient in itself to provide a complete development; combined, they exhibit an unusual wealth of ideas and furnish rich resources of new interrelations.

Nevertheless, there is a common object for all the approaches mentioned above: a Θ-representation for a wide class of solutions and fundamental Ψ-function as a solution to associated linear spectral problem. The key role of this object was discovered by Its & Matveev (1975a,b) in the example of the KdV equation. In the following years, Krichever (1976, 1977a) put such a construction into the basis of integration of soliton equations and it became clear that this link between the KdV equation and the Schrödinger operatorEmbedded Image(1.1)is not an exception but a common feature. The ideology is spread to the whole class of such problems. Presently, this is known as a concept of the Baker–Akhiezer function (Krichever 1978) and such an approach generates all the (1+1)-soliton equations and their hierarchies if an algebraic curve has been specified.

Originally, the term ‘finite-gap’ meant spectral problem for the smooth real periodic potential with finitely many lacunae at the spectrum of Schrödinger's operator. Later, in the 1970s, in works by Matveev, Its, Dubrovin, Krichever, and others that treatment was generalized to quasi-periodic complex valued potentials and related to methods of algebraic geometry and Θ-functions (Dubrovin 1976; Matveev 1976; Krichever 1977b). For this reason throughout the paper, we keep the traditional terminology ‘finite-gap’ but identify it with ‘algebraic–geometric’.

It is difficult to keep pace with the continuing growth of the literature, which is due to the activity of mathematicians. To become acquainted with the background of this field, as well as find a complete set of references, it is perhaps best to consult the surveys written by the initiators of the theory.

2. Solvable potentials for the Schrödinger equation

Until the pioneer work by Novikov (1974), the Schrödinger equation (1.1) was mostly an object of the spectral theory of operators (Akhiezer 1961; Levitan & Sargsjan 1975). By that time, few solvable (in different senses) examples were known.

  1. The constant potential (trivial case).

  2. Quantum harmonic oscillator u=x2.

  3. Linear potential u=x.

  4. Decaying potentials u=−n(n+1)cosh−2x and their rational degenerations.

  5. Reflectionless potentials of Bargmann (solitons) characterized by the property Embedded Image with a polynomial in λ function P (Darboux 1915, p. 212).

  6. The time isospectral deformations of these reflectionless potentials being governed by KdV-dynamics (Ablowitz & Segur 1981).

  7. Lamé's potentials Embedded Image (Whittaker & Watson 1927).

  8. Generalizations of the Lamé potentials (Darboux 1882, 1915, p. 228)Embedded Imagebeing called now the Treibich–Verdier potentials. The following year after Darboux, two comprehensive memoirs by Sparre (1883) appeared on further generalizations.

A considerable result of Matveev and Its is that they all, apart from the cases (ii) and (iii), lie within the framework of unified theory and are specifications or degenerations of a wider class, the class of algebraic–geometric potentials expressible by the following formula (Its & Matveev 1975a,b; Matveev 1976):Embedded Image(2.1)The same authors (Novikov 1974; Dubrovin 1975) proved that all these potentials are solutions of higher stationary ODEs which are presently named Novikov's equations. An important observation of Novikov (1974) was that these equations are representable as Hamiltonian finite-dimensional dynamical systems in x. Soon this property was completely clarified by Gel'fand & Dikii (1975, 1979).

The Hamiltonian treatment of Novikov's equations closely joined the examples mentioned above with the well-known Liouville's integrability. Moreover, Liouville's integrability of these equations has received a natural completion. Namely, all the solutions are given by the famous trace formula (Matveev 1976)Embedded Image(2.2)Note that the presence of Liouville's attributes itself does not automatically provide a procedure of the integration (this is a theorem of existence), but the trace formula (2.2) supplemented with the Jacobi inversion problem brings about such a procedure. In the language of dynamical systems this means, in fact, transformation to separability variables.

What could one say about solution Ψ corresponding to the potentials mentioned above? The answers are well known. Elementary functions in the cases (i) and (iv)–(vi), Airy's special functions in the case (iii), the Hermite polynomials (up to an exponential factor) for the case (ii) under λ=−2n−1 with integral n, Weber's (or parabolic cylinder) special functions for this case under arbitrary λ, and elliptic and related functions for the cases (vii) and (viii). For example, in the case (vii) with n=1, we haveEmbedded Image(2.3)The Θ-representation of the Ψ-function corresponding to the potentials (2.1), in traditional notations and terminology, is given by the formula of Alexandr ItsEmbedded Image(2.4)

On the other hand, the elementary solutions to the Ψ are representable by elementary functions, i.e. exactly solvable in terms of elementary indefinite integrals. The elliptic cases (vii) and (viii) and (2.3) are not exceptions: they are representable by indefinite elliptic integrals. A natural question arises: what about equations (2.1) and (2.4)? The answer is that these are not exceptions as well. Indefinite quadratures are a common property of the spectral problem under consideration. This means that the known Liouville's integrability of nonlinear Novikov's equations, in fact, turns out to be equivalent to the quadrature integrability of the ‘linear’ Ψ and conversely. In turn, the well-known effective solvability of direct/inverse spectral problems in the class of analytic non-singular decaying soliton potentials (Ablowitz & Segur 1981) turns out to be nothing else, but explicit solvability by elementary tools, i.e. indefinite integrations and their inversions. Moreover, such an opportunity is only one.

The next sections contain proof of the statements above and we suggest that this is a common feature of all spectral problems arising in the soliton theory. It should be emphasized here that the representation (2.4) (after crossing out λ and adding the KP-variables y, t1, t2, …) as a function of a point Embedded Image on arbitrary algebraic curve is a natural and fundamental object for KP-hierarchies of (2+1)-PDEs (Krichever 1977). We shall restrict our consideration only to spectral problems defined by ODEs as independent objects, so that their integrability, in the above sense, belongs to their intrinsic nature. Moreover, we will restrict ourselves only to the Schrödinger equation (1.1) that we view merely as a differential equation with a parameter (subject to explicit integration) rather than a spectral problem with any boundary conditions, commuting operator, etc.

3. Jules Drach

Perhaps the most surprising facet is the fact that the ideology mentioned above has not received mention in the modern literature in the context. It belongs to Drach (1919) and his name was revealed by Chudnovsky & Chudnovsky (1984) and Matveev (Belokolos et al. 1994, pp. 84–85), who drew attention to his remarkable results. The first sentence in Drach (1919) clearly indicates his motivations.1

The class of finite-gap (algebraic–geometric) potentials is the only one when the spectral problem (1.1) is integrable for the Ψ by quadratures under all values of the parameter λ. The solution has the formEmbedded Image(3.1)wherein the function R is a polynomial in λ solution of the equationEmbedded Image(3.2)

Drach himself does not give a proof; therefore to produce one, including step-by-step mechanism of integration, is not without interest. The proof will call to mind the classical result of Its & Matveev (1975a,b) about criteria of the potential to be finite-gap. However, we do not invoke any attributes of spectral theories: reality of the potential, periodicity, Weyl's bases, monodromies, squares of eigenfunctions Embedded Image, spectrums, resolvent, functional spaces, etc.Actually, we will be writing known and somewhat new formulae of the theory but keeping in mind only ideology of Drach. Before passing to the proof of the theorem, we will formulate a known statement that occurs in the literature in numerous contexts (Ermakov 1880; Marchenko 1974; Its & Matveev 1975a,b; Gel'fand & Dikii 1975; Al'ber 1979).

Integrability of equation (1.1) is equivalent to integrability of the third-order linear differential equationEmbedded Image(3.3)or compact nonlinear equation of Ermakov for quasi-amplitude Embedded Image of the Ψ:Embedded Image(3.4)

Though this proposition is a straight consequence of equations (3.1) and (3.2), we will give derivations to all the formulae (3.1)–(3.4). Arguments are as follows.

One sufficient test of quadrature integrability of some ODE is furnished by a solvable Lie point symmetry of this equation. This technique (group analysis of differential equations) is rather well developed (Eisenhart 1933; Ibragimov 1985), so that by applying these simple computations to the Schrödinger equation (1.1), we get necessary attributes of the theory in a natural way. Indeed, generator Embedded Image of the point symmetries Embedded Image for equation (1.1) has the following form:Embedded Image(3.5)wherein a=a(x; λ), b=b(x; λ) are arbitrary solutions of (1.1) and R=R(x; λ) satisfies equation (3.3). The functions a(x; λ), b(x; λ) do not help in further integrating (need to know solution Ψ itself) and we set them equal to zero. The remaining free constant in equation (3.5) indicates that the symmetry does not disappear and we have a solvable commutative two-parametric symmetry Embedded Image. Transformation to ‘integrable’ variables Embedded Image is provided by the standard Lie symmetry machinery. We get an explicit change (Eisenhart 1933, p. 91, case 3°)Embedded ImageIn new variables, as Lie's theory guarantees, the equation will be easily solvable. Making use of the onefold integrated form of the equation (3.3), i.e. equation (3.2) or (3.4), an equation for the function w=w(z) becomes Embedded Image, where the constant of Ermakov–Drach μ is a constant of integration. The equation is readily solved indeed and, after back transformations, we arrive at the formula (3.1). Point of departure of Ermakov (1880) was the same—integrability by quadratures.

4. Integrability of the Schrödinger equation: theorem 3.1

(a) Proof

Necessity. The function Ψ followed by the R is a function of two variables x and λ. Our aim is to find out the dependence of these functions upon both the variables. They are analytic entire functions of the parameter λ (Yakubovich & Starzhinskii 1975). This allows one to represent R in a form of analytical seriesEmbedded Image(4.1)The equation (3.3) is linear; hence, there exists a recurrence relation on the coefficients Rk derivable from the equation (3.3)Embedded Image(4.2)

The next step is to make use of Ermakov–Drach's equation (3.2), wherein the constant μ is, at the moment, arbitrary and independent of λ. We must require integrability under all values λ. This means that after substitution the series (4.1) into (3.2), coefficients in front of λk must be zeroes, independently of each other:Embedded ImageHence, the coefficients Rk(x) are determined by subsequent integration of this infinite system of chained equations. It is clear that quadrature integrability takes place not for arbitrary functions u=ϕ(x). By which restrictions are such functions distinguished from all possible ones?

Rewrite the series (4.1) as a power series in ζ=λ−1 (Gel'fand & Dikii 1975):Embedded Imagewhereupon the recurrence relation (4.2) acquires the well-known computable formEmbedded ImageThus, for arbitrary u=ϕ(x), the Ψ is expressed by the formula (3.1) with the differential polynomial R([u]; λ) of infinite order. Moreover, making use of this recurrence with subsequent collection in ζ, the equation (3.2) turns intoEmbedded Image(4.3)The dependence on u has gone at ∞: the first half (it is infinite!) contains only constants ck and the second one has differential polynomials of infinite order in u. In order to get conditions of finite order in derivatives u(k), we have the only possibility—to set R to be a finite polynomialEmbedded Image(4.4)Accordingly, splitted restrictions on the potential (4.3) become g constants Ik=Fk(c1, c2, c3, …, cg) and (g+1) differential conditions of finite orderEmbedded Image(4.5)Their compatibility means that such a class of potentials is not empty and the parameter μ must depend on the spectral one. This corresponds to a choice that the particular solution R among three linear independent R1,2,3 is defined by integration constants A1,2,3 such that the constant μ(A1,2,3) is algebraically related to λ: μ=μ(λ). These potentials admit such dependence and the condition (4.3), or which is the same (3.2), turns into a hyperelliptic algebraic curve of finite genusEmbedded Image(4.6)

The set of equations (4.5) is onefold integrated Novikov's equations (stationary KdV-equations) and a half of their integrals Ik=Fk([u]; c). Thus, quadrature integrability of the Ψ has become equivalent to integrability of these equations. If it has been established, a final answer is given by substitution u=ϕ(x) into the polynomial (4.4) and then into the formula (3.1). We emphasize that the well-known formula (3.1) does not mean any integrability without formula for ϕ(x). This is ansatz and such a form of solution to the ODE (1.1) does exist for arbitrary ϕ(x).

Sufficiency. The sufficiency is a procedure of explicit integration of the equations (4.5). It is widely known and we do not repeat details here (Drach 1919; Dubrovin 1975). The answer is the formula (2.2) plus integral representation for the functions γk(x). Nevertheless, we note that being a remarkable identity in the spectral theories (Levitan 1987), the trace formula (2.2) turns into a necessary and key object in the quadrature methodology.2 ▪

(b) Drach–Dubrovin equations and formula for the Ψ-function

Rather than mere identities, we consider consequences of the well-known definition of the polynomial R and new variables γk, for example Embedded Image, as algebraic transformations from the variables {u, ux, …} to the separability variables {γ, μ}. Namely, by writing the functionEmbedded Image(4.7)in factorized form R([u]; λ)=(λγ1(x))⋯(λγg(x)), we do define the first half of the change of variables as zeroes of this REmbedded Image(4.8)More precisely, the first part of the change (4.8) is as follows:Embedded Imageand not closed due to the odd derivatives of u. Missing ones and, therefore, the second half is extracted by involving the Drach–Dubrovin differential equationsEmbedded Image(4.9)where constants Ej are the functions of Novikov's constants cj and the integrals (4.5). We thus get the remaining part of the complete change (4.8):Embedded ImageThe equations (4.9) are readily rewritten into a promised integral form that coincides with the Jacobi inversion problem for the hyperelliptic algebraic curve (4.6)Embedded Image(4.10)With this equation, we obtain a very important formula for the Ψ-function (3.1), which is not found in the literature hitherto:Embedded Image(4.11)The variables of integration lie on the curve Embedded Image.

It may be remarked at once that the complete set of the transformations above to the separability variables {γ, μ} is not necessary for Novikov's equations themselves. This is an attribute of their Hamiltonian description (Gel'fand & Dikii 1979) derivable from the main trace formula (2.2) and the equations (4.9). It thus appears that the formula (3.1) or (4.11) supplemented with the two objects (2.2) and (4.9) and (4.10) explains the nature of integrability in question. Subsequent procedure of inversion for a symmetrical sum of γ's is necessary for the ultimate representation of the solution. In the degenerated cases (constant potential, solitons and the like), all the integrals (4.10) reduce to integrals of rational functions, so that the problem becomes trivial (inversion of logarithms) and leads to well-known exponents. In a general case, this transcendental problem is solved with the help of Riemann's Θ-functions (§7).

We note that even a ‘naive’ (not finite-gap) procedure of integration of the trivial potential Ψ″−const.Ψ=λΨ completely accords with the scheme described above. Holomorphic integrals do not appear but, instead, the following objects arise: integral definition of the logarithm, i.e. integration of a trivial rational function, and the necessity of inversion of the former (the exponent).

5. Some consequences

(a) Integrable λ-pencils

Quadrature integrability does not depend on a choice of dependent/independent variables. We could formally avoid the inversion of Abelian integrals (4.10) by rewriting the theory in the ‘inverse’ variable u. Indeed, after the change Embedded ImageEmbedded Image(5.1)we arrive at a second-order linear ODE ({Χ, u} is the standard Schwarzian)Embedded Image(5.2)which can be of interest in its own right as a new spectral problem (operator λ-pencil) for some ‘good’ functions Q(u; λ). The former depends on a chosen potential x=Χ(u). Apparently, the second equation in (5.2), written in the formEmbedded Image(5.3)and equation (6.4) play an important part in the theory,3 because the inversion of arbitrary finite-gap potential satisfies this integrable second-order nonlinear ODE with suitable function Q. Let us consider some examples (N=n(n+1)).

  1. Soliton potentials Embedded ImageEmbedded Image

  2. The Lamé potentials Embedded Image (Whittaker & Watson 1927):Embedded Image

  3. Arbitrary, even elliptic finite-gap potential Embedded Image, where U is a rational function of Embedded Image. A wide family of such potentials there provides the theory of elliptic solitons (Acta Appl. Math. 1994). The function Q is as follows:Embedded Imagewhere Embedded Image is another rational function of Embedded Image. Hence Embedded Image is a genus zero algebraic function of u or rational function of Embedded Image: Embedded Image.

  4. Arbitrary elliptic soliton u=ϕ(x). The equation (5.2) takes the formEmbedded Imagewhere Φ(u, v)=0 denotes a differential equation connecting the elliptic function u and its derivative ux=v (algebraic equation of genus unity).

By the previous constructions, all these equations and their relatives of the type (3.3) are of Fuchsian class and integrable by quadratures. We thus get a solution for all elliptic solitons generalizing the formulae of Hermite (Belokolos et al. 1994, p. 82):Embedded Image(5.4)wherein, owing to (4.7), Embedded Image becomes a rational function in (u, v). The last step is a standard problem to the representation of the elliptic Abelian integral (5.4), which is solved in terms of Jacobian θ-functions.

It should be remarked that the presence of explicit Ψ leads to explicit factorization of all of the linear operators (1.1), (3.3) and (5.2) with arbitrary λ's. For exampleEmbedded Imagewhere roots ±p(x; λ) have the quadrature formEmbedded ImageThe Θ-functional representation for this and other factorizations is readily written down using the formulae of §7.

(b) Liouvillian integrabilities

To all appearances, Liouville was the first (1833–1841) to recognize the significance of the object R=Ψ1Ψ2 and associated linear ODE of higher order in the context of integration of linear ODEs in closed form (Liouville 1839). The third-order equation (3.3) explicitly arose and was discussed on pp. 430–431 in Liouville (1839). Though Liouville was doing it in the spirit of algebraic solvability,4 the presence of a parameter in the equations turns the theory into the spectral one. The polynomials in λ arose also in the works by Darboux, and polynomial in Embedded Image was considered by Hermite in the case of Lamé's potentials Embedded Image (Whittaker & Watson 1927). This corresponds exactly to the ‘polynomial in u’ cases of Liouville (1839) for the equation (3.3).

V. Kuznetsov (2001, personal communication) pointed out a relationship of the theory with an algorithm of Kovacic (1986) and, as we have seen now, this key observation leads to the natural conclusion:

  1. Liouvillian integrability of linear ODEs (1830–1840s) with a parameter in finite terms is equivalent to algebraic integrability by Liouville (1840–1850s) of nonlinear Hamiltonian systems.5

The theory and examples above show that this is not a coincidence and, probably, the modern efficient computer-algorithmic theory, being applied to both equations (1.1) and (3.3), would provide the independent approaches to generation/classification of integrable linear operator pencils. For an excellent explanation of Liouville's ideas, see books by Mordukhai-Boltovskoi (1910), Ritt (1948) and ch. IX in Lützen (1990). Among other things, this book provides a full account of references for further study. The next section contains additional information, examples, and connections with non-integrable equations.

6. Integrable cases of Riccati and related equations

For simplicity and to avoid lengthening the terminology, we will refer to the second-order linear ODEs (or potential) and corresponding to them Riccati's equations of general form yz+a(z)y+b(z)y2=c(z) as one object. Well-known transformations between them have a quadrature characterization.

(a) Riccati's equations

The Q-functions corresponding to integrable Riccati's equations (5.2) can be rational/algebraic, elementary or transcendental. It is rather evident that non-integrable equations contain integrable subcases. For example, the quantum harmonic oscillator Yuu=(u2−2μ−1)Y shows that this potential, seemingly having nothing in common with the finite-gap ones, is integrable by quadratures if μ is an integer.

Let us reverse a view on the equations and change (5.1) and (5.2). Does there exist a transformation between non-finite-gap equation (with a parameter or no)Embedded Image(6.1)and a finite-gap one? Such a transformation depends on chosen equations and can be rather complicated. In contrast to the preceding (5.1) and (5.2), corresponding functional relation is given by the general change of variables Embedded ImageEmbedded Image(6.2)and depends on the potential u=ϕ(Χ).

Arbitrary equation (6.1) and the finite-gap one (1.1) are transformable into each other by the following functional relation:Embedded Image(6.3)where functions Ψ1,2(x; λ), Y1,2(z) are independent solutions of (1.1) and (6.1).

From (1.1) and (6.1) and (6.2) we haveEmbedded Image(6.4)Clearly, the sought for functional relation is an integral of this equation. The potential ϕ(Χ) is defined by the corresponding Ψ, which is known. From the second equality in (6.2), we haveEmbedded ImageIntegrating and supplementing with the propertyEmbedded Imagewe get the formula (6.3) and complete the proof. ▪

It thus appears that the product solution R=Ψ1Ψ2 is a fundamental object in the finite-gap theory and the ratio (6.3) is fundamental in transformations between Riccati's equations. Such arguments might seem to be trivial because every integrable equation is transformable to the trivial Yzz=0. But an example of Rawson (1883) shows non-trivial consequences: generating of finite-gap spectral problem (1.1) with ϕ(x)=n(n+1)x−2 from the oldest and classical equation of Riccati–Bernoulli with a parameter. The paper of Rawson is so short that we completely reproduce it in appendix A without any comments.6

The following extra examples exhibit a functional relation between the Lamé potentials and equidistance spectrum of harmonic oscillator Yzz=(z2−2μ−1)Y.

  1. ϕ(x)=n(n+1)x−2 and μ=0. We obtainEmbedded Image

  2. The Lamé 1-gap potential Embedded Image. The relation (6.3) has the formEmbedded Image

As we have seen, these formulae are representable by quadratures in both variables. The list of examples along these lines may be readily extended. For example to derive a generalization of Rawson's transformation (appendix A) from the classical equation of Riccati to the Lamé one. A counterexample of Liouville–Airy Yzz=(z+α)Y brings out the differential field independently of the parameter αEmbedded Imageas might be expected for the variable z.

In a broad sense, the variables u=ϕ(x) and Ψ(x; λ) may be thought of as ‘convenient’ variables for all integrable Riccati's equations with a non-trivial parameter because, in this case, the differential polynomial R([u]; λ) has a universal description as u=ϕ(x) is a solution of the equations of Novikov.

(b) Related equations

The equations of Riccati, Ermakov–Drach, Novikov's equations, and equations of the type (5.1)–(5.3) and (6.4) are not only hidden forms of integrability of one another. They generate other integrable linear and nonlinear equations. Besides them, the Ψ-function itself satisfies some nonlinear homogeneous autonomous differential equation of the third-order with a parameter(s). That equation and its relatives are obtained with the help of suitable elimination of the potential u. The following instances illustrate the remark above.

  1. Embedded Image:Embedded Imagewhere arbitrary parameters λk are restricted by the relation λ1+λ2+λ3=0.

  2. Arbitrary elliptic finite-gap potential u. Then the Ψ satisfies the equationEmbedded Imagewhere Φ(u, v)=0 is an algebraic relation between u and its derivative ux=v.

The result of elimination depends on a chosen independent variable x, u, … and all these integrable equations and their t-deformations can be of interest in their own right because they are closely related to the known third-order autonomous nonlinear ODE of Jacobi for the ϑ-constants in the framework of Fuchsian equations. We do not develop this topic here. It would appear reasonable that the widely known and universal Θ-description of the theory (Krichever 1977a,b) should be obtainable from the spectral problem itself. This is so indeed.

7. The Θ-representation

Here, we will obtain the Θ-representation (2.4) for the Ψ-function and, thereby, its properties as a function of Baker–Akhiezer. Insomuch as a regular derivation of this axiomatic representation is not described in the literature, to write up that procedure is, perhaps, not without interest.

As we mentioned above, pure spectral approaches were extended to the complex valued potentials (see the recent monograph by Gesztesy & Holden (2003) for a most exhaustive bibliography and new results in spectral treatment of the theory). Taking this into account, we will refer a general λ-dependence of the Ψ as its spectral property and its dependence upon x-variable is considered to be parametric. Such a spectral view arose in the paper by Akhiezer (1961) and was completed in full by Its & Matveev (1975a,b, §4). It seems helpful to compare this classical approach with a ‘reverse’ one, i.e. primary x-dependence of the Ψ. This would correspond to pure quadrature arguments with a parametrical λ-dependence made out in the previous sections.

The Θ-functional representation (2.4) to the Ψ-function is a consequence of the quadrature representations (3.1) and (4.11).

Since the spectral parameter λ is connected with the variable μ by the algebraic equation (4.6) of finite genus g, we view both these variables as meromorphic functions λ=λ(τ), μ=μ(τ) of a global parameter τ on the curve (4.6). Accordingly, we consider the Ψ-function (4.11) as a function of x and τEmbedded Image(7.1)where αk are arbitrary constants. It is a symmetrical function of the quantities γk and the formers, as functions of x, are defined from the inversion problem (4.10)Embedded Image(7.2)Let Ajk be a matrix of a-periods of the holomorphic Abelian integrals (7.2)Embedded ImageAll the terminology and notation in this section are standard and elucidated in any paper on the finite-gap integration. As usual, we normalize the integrals ωj(z), introducing the canonical base of normal holomorphic Abelian integrals Embedded ImageEmbedded Image(7.3)Jacobi's inversion problem (4.10), (7.2) acquires the formEmbedded Image(7.4)where we adopt the concise Baker's notation (Baker 1897) for Abelian integralsEmbedded ImageThe vector U depends on the curve Embedded Image, but αk and Cj are arbitrary constants. The function (7.1) depends on the point τ on the curve Embedded Image and on the variable x through γ's. Thus, it is a single-valued function (but not Abelian) of a point η(x)=−xU+C on the Jacobian Jac(Embedded Image) and, hence, has a Θ-functional representation. Indeed, the arisen sum (7.1) is nothing else but the fundamental T-function of Weierstrass (1856) and Clebsch & Gordan (1866)Embedded Imagewhere Embedded Image denotes the elementary normal Abelian differential of the third kind with first-order poles at the points z=ξ, η and residues +1, −1, respectively. Clebsch & Gordan (1866) devoted ch. 6–8 in their book to detail properties of this object and regular procedure of derivation of the Θ-representation for it (Baker 1897). We have from there (Clebsch & Gordan 1866, §§54 and 57; Baker 1897, §171, pp. 187–188)Embedded Imagewhere vector Embedded Image is a zero of the Θ-function. In our situation, we have α=λ, ν=∞, x=γk, z=αk. The representations for zeroes r and the Riemann constants Embedded Image are not unique because they depend on a lower bound of the holomorphic integrals Embedded Image. On the other hand, the point η on the Jacobian contains free constants Cj in (7.4) so that we may simplify the considerations putting m's equal to γj's apart from γk and setEmbedded Imagewith arbitrary points Embedded Image, say γ's. We thus arrived at Riemann's function Embedded Image. As a consequence of these arguments and explicit zeroes of the Ψ in (3.1), we get the following formula (Baker 1928, pp. 588–589):Embedded ImageThe integrals in (7.1) are elementary but not normal. Hence, we have an identityEmbedded Imagewith some normalizing constants hjk(τ). Taking into account Jacobi's problem (7.4), we arrive at the intermediate answerEmbedded Image(7.5)

Meromorphic part of the Ψ-function has been determined. In order to determine the function Embedded Image in (7.5), we involve the transformation properties of the Ψ as the function (7.1) on the curve Embedded Image. Let Embedded Image and Embedded Image denote linear-fractional transformations of the variable τ corresponding to the cycles ak and bk. From the primary representation (3.1), we conclude that Ψ(x; τ) saves its own form (3.1), up to a multiplier M(τ), when τ undergoes the transformations Embedded Image. Hence, the formula (7.5) must hold this property. Clearly, the function f(x; τ) is an entire function of x since Embedded Image's are everywhere finite. Further, the Embedded Image-transformations of the Θ's in (7.5) say that the function f has to be a linear function in x to compensate an exponential multiplier Embedded Image in (7.5):Embedded ImageInvariance of the Θ-functions in (7.5) with respect to Embedded Image-transformations impliesEmbedded ImageThe transformations Embedded Image implyEmbedded ImageLet τ approach the pole τ of the meromorphic function λ(τ). Such a pole is the only one and designating ξττ, we haveEmbedded ImageFrom these formulae and (3.1), we obtain (extracting terms independent of γ's)Embedded Imageand therefore we get one more propertyEmbedded Image(7.6)

A function with the properties (i)–(iii) does exist on Embedded Image. This is the normal elementary Abelian integral of the second kind Embedded Image with the first-order pole at the point τ. Its principal part and a constant in (7.6) are well defined so that this function is completely determined and unique. In turn, the three parameters {τ, A, B} may be freely chosen, say, {0, 1, 0}, respectively. The vector U in (7.4) and (7.5) becomes the vector of b-periods of the integral Embedded Image. Summarizing the arguments above and recovering a normalizing constantEmbedded Image(7.7)we get the spectral properties of the Ψ and complete the proof. ▪

From the last expansion of the Embedded Image, we obtain the formula (2.1). Indeed,Embedded Imagehence, by virtue of the property Embedded Image, the derivative d/dξ|ξ=0 may be replaced by −d/dx and we get the final formulaEmbedded Image(7.8)

We should conclude here that the spectral and quadrature considerations are mutually replaceable. The explicit transition between these approaches there provides the known Weierstrass's theorem on a permutation of arguments and parameters in the normal Abelian integrals of the third kindEmbedded ImageThis fact immediately leads to the equivalence of the ‘spectral’ formula for the Ψ (Its & Matveev 1975a,b, formulae (4.12) and (4.13)) and the quadrature one (4.11) or (7.1). The above-mentioned theorem, in our notation, has the formEmbedded Imagewhere Embedded Image. Since the Ψ contains all the information, we can obtain suitable expressions for all objects of the theory. In particular, Θ-functional representation for ‘finite-gap function’ of Ermakov (3.4) having numerous applications. Renormalizing it by the formula Embedded Image, we obtainEmbedded Image‘Finite-gap’ means that the constant μ in Ermakov's equation (3.4) is not independent of λ: μ=μ(λ). Note that the equation (3.3) itself exemplifies the integrable and factorizable linear operator pencil with Abelian coefficients (like soliton spectral problems), but its solution, as a counterexample, is not a function of Baker–Akhiezer.

Many of the constructions in §§3 and 5–7 can be carried over to arbitrary spectral problems although not so simply as in the case of the Schrödinger/KdV equation. Non-trivial examples of the Ψ were obtained in Ustinov & Brezhnev (2002) and modifications of Dubrovin's equations and trace formulae in Brezhnev (2002).

8. Conclusive comments and bibliographical remarks

Apparently Baker (1928, p. 587) was the first to realize the exponential property of the fundamental T-function in disguise; however, explicit meromorphic integrals like Embedded Image are, to all appearances, the result of the modern theory and became a universal property of all the integrable models (Krichever 1970s). Note that throughout the paper, we made no restrictions on the curve (4.6). It may be singular and the theory can be rewritten (with minor changes in §7) because the degenerated holomorphic integrals Embedded Image, in this case, turn into the integrals Embedded Image of the third kind. Clebsch & Gordan (1866, §43) call the corresponding problem the ‘extended inversion problem’, which is solved by theta as well. Such cases of degenerations were considered also by Baker (1897) and even by Abel (see his Œuvres 1881, I, pp. 170ff).

(a) On the Θ-series

It is to be noticed that contrary to the commonly accepted (?) viewpoint, the general Θ-series (B 1), as well as its argument Embedded Image, ought to be considered not as a special function or formal generalization of Jacobi's θ-function but regularly derivable fundamental object for explicit representation of all Abelian integrals, meromorphic/uniformizing functions, exponential Baker–Akhiezer functions, and the theory as a whole.7 Main ideology belongs to Weierstrass (1902, pp. 513–538) and was expounded in the books by Tikhomandritskii (1885, 1895, pp. 199–232) before the publication of Weierstrassian lectures on Abelian transcendents, but with explicit use of the Riemann surface. Primarily and naturally arising non-canonical form of the Θ-series is derived in these works with basic properties and identities. The canonical formula (B 1) is obtained after the normalization (7.3). See also the book by Clebsch & Gordan (1866, pp. 193–198). For lack of space, we do not pursue these important points here. This will be written up elsewhere. For the same reason, we have somewhat reduced the exposition between formulae (7.4) and (7.5).

(b) Algebraically integrable Hamiltonian systems

In what concerns an inverse transition in the thesis in §5b, i.e. transition from an algebraically integrable Hamiltonian system to some spectral problem, the answer is positive though somewhat non-effective. We assume that separability variables {γ, μ} sitting on a curve, say (4.9), exist. We can construct the Baker–Akhiezer function (7.7) and the Schrödinger operator (it certainly exists) for it. The computational part of the Liouville theorem is based on the Hamilton–Jacobi theory of canonical transformations. The variables γ's are always taken as poles of the Ψ:Embedded Imagesince they solve the problem (4.10), (7.4). Ostrogradskii's variables {u, ux, …} are constructed explicitly by the trace formulae. Isospectral t-deformations of finite-gap potentials are readily included in (7.2): the quantities dk become linear functions of xU+tV. Accordingly, there is no essential difference between x- and t-equations for the Ψ. Both of these equations (Lax's pair) may be thought of as quadrature integrable (factorizable) spectral problems/pencils: Ermakov–Drach's constant μ becomes an eigenvalue of the second commuting operator connected through the curve with the first oneEmbedded ImageThere are infinitely many Abelian functions built from {γ, μ} (followed by physical coordinates Embedded Image) and expressible through the Θ's. Non-constructiveness can appear because transformations between various dynamical systems are not necessary to be canonical. However, canonicity is not a necessary attribute of quadrature integrability. One fundamental Abelian function determines all the transformations. This is the potential u.

Acknowledgments

The author thanks Prof. J. C. Eilbeck and Prof. E. Previato for their numerous discussions and E. P. for hospitality in Boston University, where the work was carried out. The project was supported by NSF/NATO grant DGE-0209549.

Footnotes

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

  • …where h is an arbitrary parameter. The most interesting among them are those where the Riccati equation (and consequently the equation (d2y/dx2)=[φ(x)+h]y as well)Embedded Imagecan be integrated by quadratures; we will show how one determines the function φ in all these cases.

  • The independent role of these formulas for the KdV-equation, as well as their dynamical relatives, was pointed out by Matveev (1975). See also his appendix to Dubrovin et al. (1976).

  • See also a footnote on p. 97 in Gel'fand & Dikii (1975) apart from two misprints in one term.

  • See however p. 456 in Liouville (1839) about what is nowadays named Liouville's extension.

  • We speak here only about a general link and avoid discussion of the rigorous correspondence between finite-gap operators and Liouvillian solutions, differential algebra (Ritt 1948), algorithms of Singer (1981) and Kovacic (1986), Picard–Vessiot theory (van der Put & Singer 2003), etc. In particular, we do not touch an important question: when does isomorphism between these two Liouvillian integrabilities take place? We should mention here some comments about this analogy in Morales-Ruiz (1999, pp. 51–52) and van der Put & Singer (2003). However the main attributes of the theory (spectral curves, polynomial in λ, Θ-functions, etc.) are not discussed in these works.

  • See also Liouville's (1841, pp. 11–13) classical considerations on integrability of the potential ϕ(x)=Bx−2 with appearance B=n(n+1). Another example of ‘triviality’ is the class of potentials in elementary functions (solitons and the like) generated from the zero potential by the Darboux transformation (Matveev & Salle 1991).

  • Jacobi's case g=1 is not an exception (Tichomandritzky 1884). We note that neither Riemann nor Baker elucidate an origin of appearance of the Θ-series. Hyperelliptic case, based on Weierstrassian lectures, was considered in a dissertation by Tikhomandritskii (1885).

References

View Abstract