## Abstract

We collect known results about the Schrödinger operators *L*=−Δ+*u*(*x*), generalizing to higher dimension those algebro-geometric operators *L*=−d^{2}/d*x*^{2}+*u*(*x*) with rational, trigonometric and elliptic potential which appear in the finite-gap theory.

## 1. Introduction

The present paper is a survey of the results of Chalykh & Veselov (1990), Veselov *et al*. (1993), Chalykh (1998) and Chalykh *et al*. (1999, 2003) on higher-dimensional analogues of the so-called *algebro-geometric* operators of the form(1.1)which appeared in the framework of the finite-gap theory in the 1970s. A great achievement of that theory was constructing explicit solutions for various nonlinear integrable equations, for instance KdV, and the spectral theory of the operator (1.1) was an important tool of the theory (see Dubrovin *et al*. (1990) and references therein).

Apart from their relation to the KdV equation, the algebro-geometric Sturm–Liouville operators (1.1) are interesting due to many other reasons (e.g. commutative rings of differential operators, exact solutions, finite-gapness, interesting moduli spaces). And while there is no completely satisfactory analogue of the finite-gap theory in many dimensions, what we shall explain below should convince the reader that there is a nice class of multidimensional Schrödinger operators exhibiting many, if not all, of the characteristic properties of the one-dimensional algebro-geometric operators.

The approach that we follow here has been developed starting from the work of Chalykh & Veselov (1990). It was motivated by the results of Dubrovin *et al*. (1976) and Veselov & Novikov (1984*a*,*b*), who proposed a nice theory of the operators in two variables which are algebro-geometric on *one energy level*. This made natural the following question posed by Novikov: find those of these operators which are algebro-geometric on *every* energy level. The theory developed by Dubrovin *et al*. (1976) and Veselov & Novikov (1984*a*,*b*) predicted that such operators must be rare and exceptional, which made a task of finding them rather tricky. A starting point for us was the idea that, perhaps, such examples might be related to other exceptional mathematical objects like simple Lie algebras or root systems. And while this has been partly confirmed in the follow-up papers of Chalykh & Veselov (1993) and Veselov *et al*. (1993) (see also a recent paper of Sergeev & Veselov (2004)), the whole story turned out to be more complex than we initially thought. Before proceeding, let us also mention the papers of Nakayashiki (1991, 1994), Feldman *et al*. (1992), Rothstein (1996, 2002), Parshin (1999, 2001) and Osipov (2001), devoted to other approaches for generalizing the finite-gap theory to higher dimensions.

We start by recalling that one of the characteristic features of the algebro-geometric operators is the existence of non-trivial operators commuting with *L*. Namely, such *L* are completely characterized by the property that they admit a commuting operator *A* of odd order,(1.2)In other words, *L* is a member of a non-trivial commutative ring of ordinary differential operators. An elementary lemma due to Burchnall and Chaundy says that *L* and *A* are related by an algebraic relation *P*(*L*, *A*)=0, so the commutative ring generated by *L* and *A* is isomorphic to , the coordinate ring of an affine algebraic curve. This is a part of the famous Burchnall–Chaundy–Krichever correspondence between the commutative rings of ordinary differential operators and algebraic curves (Krichever 1977). This theory is the most complete in the case of rings of *rank one* when the common eigenspace of the operators from the ring is one-dimensional. This suggests that similarly one should be interested in commutative rings *R* of partial differential operators in *n* variables such that *SpecR* is a *n*-dimensional affine variety and that the common eigenspace of the operators from *R* is one-dimensional (see §3*b* for a precise definition). A Schrödinger operator *L*=−Δ+*u* is called *algebraically integrable* if it is a member of such a ring.

One can construct trivial examples of algebraically integrable operators by taking direct sums *L*=*L*_{1}(*x*, ∂_{x})+*L*_{2}(*y*, ∂_{y}) of two one-dimensional algebro-geometric *L*_{1} and *L*_{2}. The first non-trivial examples were found in Chalykh & Veselov (1990, 1993). They are related to the quantum Calogero–Moser system and have the form(1.3)where *m* is an integer. One can associate a version of the Calogero–Moser system to any root system or Coxeter group (Olshanetsky & Perelomov 1983), and this gives more examples of algebraically integrable Schrödinger operators (Veselov *et al*. 1993).

Note that the Calogero–Moser operator (1.3) can be viewed as a multivariable version of the operatorThe latter is a member of a nice family related to the rational solutions of KdV. They all have the form(1.4)and can be characterized by the property (Duistermaat & Grünbaum 1986) that the solutions of the equationare single-valued in the complex domain for any *λ*. The latter property is equivalent to the following overdetermined system of algebraic relations on the position of the poles *x*_{i} and their ‘multiplicities’ *m*_{i}:(1.5)

A surprising fact, first observed by Chalykh (1998) and further explored by Chalykh *et al*. (1999), was that the very similar conditions are responsible for the algebraic integrability in higher dimension. To illustrate this parallel, let us formulate one result. We will assume that the potential *u*(*x*) has the following form, generalizing (1.4) in a natural way:(1.6)Here, is a finite collection of non-parallel, non-isotropic vectors *α*∈ with prescribed multiplicities *m*_{α}∈, and (., .) stands for the standard Euclidean product in .

Impose next the following conditions (‘locus relations’) onto the set and the multiplicities *m*_{α}, for each *α*∈ and every *s*=1, …, *m*_{α}(1.7)The parallel between these conditions and (1.5) is transparent. Such configurations {*α*, *m*_{α}} were called ‘locus configurations’ by Chalykh *et al*. (1999).

Before formulating the theorem, let us introduce the ring of *quasi-invariants* , as the ring of polynomials *f*(*k*), *k*∈, satisfying the following conditions for every *α*∈:Here, *s*_{α} is the orthogonal reflection with respect to the hyperplane (*α*, *k*)=0.

The following theorem follows from the results of Chalykh *et al*. (1999) (see theorems 3.5 and 3.7 below).

*For any locus configuration, the corresponding operator L*=−Δ+*u*(*x*) *is algebraically integrable. Moreover, L is a member of a commutative ring of partial differential operators, which is isomorphic to the ring* *of quasi-invariants*.

As we will explain below (theorem 3.7), this ring is a *maximal* commutative ring. Furthermore, the commuting operators *L*_{f} and their common eigenfunction can be given by rather explicit formulae. Note that the ring replaces the coordinate ring of the *spectral curve P*(*λ*, *μ*)=0, which we have in dimension one. Thus, one should think of as the ‘spectral variety’ for *L*. Informally speaking, *X* looks like with cusps along hyperplanes (*α*, *x*)=0. These varieties have been studied in many important cases (Etingof & Ginzburg 2002; Feigin & Veselov 2002, 2003; Berest *et al*. 2003).

Theorem 1.1 and its generalizations are the main subject of the present survey which is organized as follows. In §2, we review the one-dimensional algebro-geometric operators (1.1) for the cases when *u* is rational, trigonometric and elliptic, highlighting their features which extend to higher dimension. In §3, we consider the case of *L*=−Δ+*u* with rational potential. In the process, we define algebraic integrability and prove theorem 1.1. Trigonometric and elliptic cases are treated in §§4 and 5, respectively. Note that as a by-product, we obtain a proof of a conjecture due to Gesztesy & Weikard (1996) (theorem 5.3). In §6, we list known irreducible examples of the operators studied in §§3–5. Section 7 contains some concluding remarks.

## 2. One-dimensional case

In this section, we recall known facts about the algebro-geometric operators in dimension one. Let be a one-dimensional Schrödinger operator with (locally) analytic potential *u*. It is called algebro-geometric, or ‘finite-gap’, if there is an ordinary differential operator *A* of odd order, commuting with *L*. This terminology is motivated by a remarkable fact, observed by Novikov (1974), that in the case when *u* is real-valued smooth and periodic, the existence of *A* implies that the spectrum of *L* in has a finite number of instability intervals (‘gaps’). Further, according to the celebrated Its & Matveev (1975) formula, *u* can be expressed via multivariable *θ*-function on the Jacobian of an algebraic curve, thus ‘algebro-geometric’. The following three families of potentials have a rich intrinsic algebraic structure associated with them and are of particular interest: rational, trigonometric and elliptic *u*. These and their higher-dimensional analogues will be the main subject of this survey.

Let us begin with the class of rational *u*(*x*). In this case, one can give the following characterization of the algebro-geometric operators.

*Assume that* *has a rational potential with* .*Then, the following statements about L are equivalent:*

*L is algebro-geometric*;*L can be obtained from**by applying classical Darboux transformations at the zero level*;*there exists a differential operator D with rational coefficients such that LD=DL*_{0},*where*;*(exact solvability) L has an eigenfunction ψ(k, x) of the form ψ=P(k, x*)^{kx}*such that Lψ=−k*^{2}*ψ, where P(k, x) is polynomial in k and rational in x; and**(trivial local monodromy) for any λ∈*,*all solutions f(x) of the equation Lf=λf are meromorphic in the complex domain*.

Equivalence of (i) and (ii) is well known from the theory of the KdV equation (Airault *et al*. 1977; Adler & Moser 1978); (iii) follows from (ii) since *D* factors as a product of successive first-order Darboux transformations; equivalence of (iii) and (iv) is expressed by the formula *ψ*=*D*(e^{kx}); an implication from (iv) to (v) is almost obvious; and finally, implication from (v) to (ii) is the result of Duistermaat & Grünbaum (1986).

Following Duistermaat & Grünbaum (1986), one can give an explicit formulation of the property (v) in the theorem. First, (v) implies that *L* has regular singularities. This, together with the decay at infinity, leads to the formula (1.4). After that the local analysis at each pole leads to the conditions (1.5). In generic situation, all *m*_{i}=1 and the number of the poles is *N*=*m*(*m*+1)/2 for some *m*∈. In that case, and the system of equations (1.5) reduces to(2.1)This is the so-called *rational locus* introduced by Airault *et al*. (1977). It describes the equilibrium points of the classical rational Calogero–Moser system of *N* particles. The set of solutions to (2.1) is empty unless *N*=*m*(*m*+1)/2, while for each *N*=*m*(*m*+1)/2 the solutions depend on *m* complex parameters and can be given explicitly (Adler & Moser 1978).

Theorem 2.1 has a nice generalization (Oblomkov 1999) to the case when *u*−*x*^{2}→0 as (see also Chalykh & Oblomkov (2000) for the multidimensional version).

Let us now turn to the trigonometric case, i.e. when *u* is a rational function of *z*=e^{2ix}. We can formulate the following analogue of the previous result.

*Assume that* *has a trigonometric potential u(x), such that* . *Then, the following statements about L are equivalent:*

*L is algebro-geometric;**L can be obtained from**by applying classical Darboux transformations at the levels of the form λ=−n*^{2}*with n*∈;*there exists a differential operator D with trigonometric coefficients such that LD=DL*_{0},*where*;*(exact solvability) L has an eigenfunction ψ(k, x) of the form ψ=P(k, x*)^{kx}*such that Lψ=−k*^{2}*ψ, where P(k, x) is polynomial in k and trigonometric in x; and**(trivial local monodromy) for any λ∈*,*all solutions f(x) of the equation Lf=λf are meromorphic in the complex domain*.

Essentially, this is a result of Weikard (1999). An implication from (v) to (iv) is less straightforward (it does not follow from the standard facts of the KdV theory). An alternative proof of this follows from theorem 4.1 below.

Again, one can reformulate (v), using the local analysis of the equation *Lf*=*λf*. We will make use of the following definition.

Given a point *x*_{0}∈ and *m*∈, let us say that a function *f*(*x*) is *m*-quasi-invariant or simply quasi-invariant at *x*_{0} if *f*(*sx*)−*f*(*x*)=*o*((*x*−*x*_{0})^{2m}) near *x*=*x*_{0}, where *s* denotes the symmetry across *x*_{0}, i.e. *sx*=2*x*_{0}−*x*.

This property means the absence of odd terms, up to (*x*−*x*_{0})^{2m+1}, in the Laurent expansion of *f* at *x*_{0}. Under such terminology, the conditions (1.5) express the quasi-invariance of *u* at each of the poles.

The assumptions of the theorem together with (v) imply that *u* has the form(2.2)with *x*_{i} distinct modulo *π*, and similar to (1.5), we conclude that *u* must be *m*_{i}-quasi-invariant at each of the poles *x*_{i}.

Since the algebro-geometric operators of the form (2.2) can be obtained from by a sequence of Darboux transformations, one gets an explicit formula for the potential *u*(*x*). As a result, the set of such operators breaks into separate families, each labelled by a finite sequence of integers 0<*k*_{1}<⋯<*k*_{n}. Such a family depends on *n* extra non-zero complex parameters (‘phases’), and the generic *u*(*x*) in this family has *N*=*k*_{1}+⋯+*k*_{n} poles *x*_{i} with all *m*_{i}=1 (in general, ). For these poles, we have the following system of equations (‘trigonometric locus’ from Airault *et al*. (1977)):(2.3)Thus, in contrast with the rational case, for a fixed *N* the variety (2.3) has several components, corresponding to different partitions of *N* (each being topologically), with the KdV flows acting transitively on every component.

Finally, let us consider the elliptic case, i.e. when *u* is an elliptic function. The following elegant characterization of the algebro-geometric operators with elliptic potential is the result of Gesztesy & Weikard (1996).

*An operator* *with an elliptic potential is algebro-geometric if and only if it has trivial local monodromy, i.e. for any λ∈* *all solutions f(x) of the equation Lf=λf are meromorphic in the complex domain*.

Again, using the local analysis, one shows that such *u* must be of the form(2.4)with *x*_{i} distinct modulo periods, and similar to (1.5), *u* must be *m*_{i}-quasi-invariant at each of its poles *x*_{i}.

Generically, all *m*_{i}=1 and the quasi-invariance of *u* reduces to the following system (‘elliptic locus’ from Airault *et al*. (1977)):(2.5)The structure of the components of the elliptic locus is much more complicated (Airault *et al*. 1977; Krichever 1980; Treibich & Verdier 1990; Treibich 2001).

Summing up, we see that in all three cases (rational, trigonometric and elliptic), triviality of the local monodromy singles out the algebro-geometric operators. Note that this happens only for specific analytic classes of potentials, otherwise this would not be very restrictive; for instance, any operator with holomorphic *u* has trivial local monodromy (it has no singular points at all).

In the following sections, we will generalize these results to higher dimension, following Chalykh (1998) and Chalykh *et al*. (1999, 2003). In particular, this will provide alternative proofs in dimension one.

## 3. Rational case

### (a) Exact solvability

Let us consider a Schrödinger operator in *L*=−Δ+*u*(*x*) with rational potential *decaying at infinity*, i.e. *u*=*p*(*x*)/*q*(*x*) where deg *q*>deg *p*. For convenience and in analogy with the one-dimensional case, we will denote this symbolically as *u*(∞)=0.

Let us call *L*=−Δ+*u* with meromorphic potential *u exactly solvable* if it admits an eigenfunction *ψ*(*k*, *x*)=*P*(*k*, *x*)e^{(k, x)}, *k*,*x*∈, with *Lψ*=−*k*^{2}*ψ*, where *P* is polynomial in *k* and meromorphic in *x*. Equivalently, there exists a PDO *D*(*x*, ∂/∂_{x}), such that .

Following the work of Chalykh (1998) and Chalykh *et al*. (1999), we are going to characterize all exactly solvable operators *L* in the spirit of theorem 2.1. The first interesting fact is that the highest symbol −Δ of *L* puts serious restrictions onto geometry of singularities of *u*. The following result is due to Berest & Veselov (1998, 2000).

*If L*=−Δ+*u*(*x*) *with meromorphic u is exactly solvable, then the singularities of u form a union of hyperplanes in* .

Next, we need the following analytic result (see theorem 2.2 of Chalykh *et al*. (1999)).

*Let L*=−Δ+*u be exactly solvable and Π be one of the hyperplanes where u has poles. Then*,

*Π is non-isotropic, and thus in suitable Cartesian coordinates x*_{1}, …,*x*_{n}*it is given by the equation x*_{1}=0;*u has the form*,*where m is an integer and u*_{0}*is regular along Π; and**the first m odd derivatives of u*_{0}*with respect to x*_{1}, i.e. ,*vanish along Π*.

Properties (ii) and (iii) mean that the potential *u* is *m*-quasi-invariant with respect to the hyperplane *Π*: *x*_{1}=0, in the sense of definition 3.2.

Given a non-isotropic hyperplane *Π*⊂ and an integer *m*∈, we say that a function *f* of *x*∈ is *m*-quasi-invariant with respect to *Π* if *f*(*x*)−*f*(*sx*) has zero of order greater than or equal to 2*m*+1 along *Π*. Here, *s* is the orthogonal reflection with respect to *Π*.

In dimension one, the quasi-invariance of *u* guarantees the triviality of the local monodromy. A similar interpretation exists in higher dimension (Chalykh *et al*. 1999).

Suppose now that *u*(*x*) is rational with and such that *L*=−Δ+*u* is exactly solvable. Then, according to the previous two results, its only singularities are the second-order poles along a finite collection of hyperplanes. Each of these hyperplanes in standard coordinates can be given by a linear equation where . Here and . Thus, we arrive at a finite set of affine–linear functions encoding the poles of the potential. Further, due to lemma 3.1, the singular part of *u* along *Π*_{α}:*α*(*x*)=0 must be of the form , for some integer *m*_{α}. Together with the decay at infinity, this allows us to conclude that the potential *u*(*x*) of any exactly solvable operator must be of the form(3.1)Additionally, part (iii) of the lemma says that *u* must be *m*_{α}-quasi-invariant with respect to *Π*_{α}:*α*(*x*)=0.

These are necessary conditions for exact integrability of *L*. Remarkably, they turn out to be sufficient.

*If the potential u of a Schrödinger operator L has the form* (3.1) *and is m*_{α}-*quasi-invariant with respect to each Π*_{α}, *then L is exactly integrable*.

For the proof see Chalykh (1998). The proof is effective and gives an explicit formula for the eigenfunction *ψ*(*k*, *x*) such that *L*_{ψ}*=*−*k*^{2}*ψ*; namely, one has(3.2)In this formula, the differential operator (*L*+*k*^{2})^{M} is applied to the quasi-polynomial in the brackets. Such formula was first discovered for the Calogero–Moser systems in Berest (1998), and this inspired the results of Chalykh (1998).

From (3.2), it is easy to deduce that *ψ* has a form *ψ*=*P*(*k*, *x*)e^{(k, x)}, where *P* is rational in *x* and polynomial in *k* with the highest term . Equivalently, one can say that *ψ* has the form *ψ*=*D*e^{(k, x)}, where *D* is differential operator in *x* with the highest symbol .

If all the hyperplanes pass through the origin, they are determined by the set of their normal vectors. In this case, *u*(*x*) is given by the formula (1.6) and the quasi-invariance reduces to (1.7).

Combining all these results together we get the following multidimensional generalization of theorem 2.1.

*Let L*=−Δ+*u be a Schrödinger operator with rational potential u(x) such that* . *Then, L is exactly solvable if and only if u has a form* (3.1) *and is m*_{α}-*quasi-invariant with respect to each Π*_{α}:*α*(*x*)=0.

### (b) Algebraic integrability

In this section, we discuss the notion of algebraic integrability for the Schrödinger operators following Chalykh & Veselov (1990, 1993), Veselov *et al*. (1993) and Braverman *et al*. (1997). In particular, we demonstrate that all operators from theorem 3.3 are algebraically integrable, which strengthens the analogy between the above results and the one-dimensional case.

We start by recalling that a *quantum completely integrable* (or Liouville integrable) *system* (QCIS) with *n* degrees of freedom is given by a set of *n* algebraically independent, pairwise commuting partial differential operators in *n* variables *L*_{1}, …, *L*_{n}: [*L*_{i}, *L*_{j}]=0. Many examples of such systems are obtained as a quantization of classical completely integrable Hamiltonian systems. In many interesting cases, the quantized Hamiltonian takes the form of a Schrödinger operator *L*_{1}=−Δ+*u*(*x*), and *L*_{2}, …, *L*_{n} should be thought of as *quantum integrals* of the corresponding quantum system. The main problem usually is to compute the spectrum and the eigenstates of the system, i.e. the eigenfunctions *ψ* in an appropriate Hilbert space such that *L*_{i}*ψ*=*λ*_{i}*ψ*.

However, from the algebraic point of view, we can think of *L*_{i} as operators with analytic coefficients in some open chart of or, more generally, some smooth algebraic variety *X*. From that point of view, a QCIS is an embeddingof the ring of polynomials into the ring of differential operators on *X*. More generally, one can consider an embeddingwhere *Λ* is an affine variety of dim(*Λ*)=*n* and (*Λ*) denotes the coordinate ring of *Λ*. (The discussion above corresponds to the case of *Λ* being an affine space .)

Given a quantum system like this, one considers for any point *λ*∈*Λ* the corresponding ‘eigenvalue problem’ *θ*(*g*)*ψ*=*g*(*λ*)*ψ*, . The dimension of the local solution space of this system at generic point of *X* for generic *λ* is called the *rank* of a QCIS. Recall further that a QCIS *S*=(*Λ*, *θ*) is *algebraically integrable* if it is dominated by another QCIS *S*′=(*Λ*′, *θ*′) of *rank one* (*S* is dominated by *S*′ if there is an algebra homomorphism such that ). This is the definition from Braverman *et al*. (1997); the original definition in Chalykh & Veselov (1990) and Veselov *et al*. (1993) was slightly different, though essentially equivalent to it for the case when the operators have constant highest symbols. If the rank of the original system is greater than one (as it usually is), then its algebraic integrability implies that, apart from the operators *L*_{1}, …, *L*_{n}, we have additional commuting operators which are not algebraic combinations of *L*_{i} (though they are algebraically dependent on *L*_{i}). In dimension *n*=1, this is equivalent to saying that *L* is a member of a rank one commutative subring of *D*(*X*); this is known to coincide with the class of algebro-geometric operators.

Let us turn to the exactly integrable operators *L* from theorem 3.3 in order to prove their algebraic integrability.

First, using the idea of Chalykh (1997), we will construct sufficiently many operators commuting with *L*. Recall that *L* has an eigenfunction *ψ*(*k*, *x*) of the form *ψ*=*D*e^{(k, x)}, where *D* is a differential operator in *x* with the highest symbol . In proposition 3.1, *D*^{*} will denote the *formal adjoint* of *D*, i.e. the result of applying an anti-involution *x*→*x*, ∂→−∂ to *D*. For example, *L*^{*}=*L*.

**(****Chalykh 1997****).** (i) *For any constant coefficient operator p(∂), the operator A*_{p}=*D*_{p}(∂)*D*^{*} *commutes with L. Moreover, all these operators for various polynomials p pairwise commute and A*_{p}*ψ=λ*_{p}*ψ, where* . (ii) *D*^{*}*D is a constant coefficient operator*: .

In Chalykh (1997), the proof is given for the Calogero–Moser system, but it applies to our situation as well. Note also that any algebraic relation between the eigenvalues *λ*_{p}(*k*) of the operators forces the same relation between *A*_{p}'s. This follows from a simple fact that any partial differential operator *A*(*x*, ∂) annihilating a quasi-polynomial family *ψ*(*k*, *x*) must be zero.

Now take *n*+1 operators *A*_{0}, *A*_{1}, …, *A*_{n} corresponding to *p*_{0}=1 and *p*_{i}=∂_{i}. Let us denote the polynomial *P*_{0}(*k*)*P*_{0}(−*k*) which appears in theorem 3.4 as *R*(*k*),(3.3)It is homogeneous of degree . According to the theorem 3.4, we have *A*_{i}*ψ*=*λ*_{i}*ψ*, where(3.4)The eigenvalues *λ*_{0}, …, *λ*_{n} are subject to the relation(3.5)and we have the same relation between *A*_{0} and *A*_{1}, …, *A*_{n}. It is clear that *λ*_{1}, …, *λ*_{n} and the operators *A*_{1}, …, *A*_{n} are algebraically independent, so they define a QCIS. Note also that the map(3.6)from onto the hypersurface (3.5) in is bijective outside the locus *R*(*k*)=0.

We will now prove that the holonomic system of equations *A*_{i}*f*=*λ*_{i}*f* (*i*=0, …, *n*) for generic point *λ* of the hypersurface (3.5) has a unique (up to a factor) locally analytic solution, which is *f*=*ψ*(*x*, *k*) with *k*=*π*^{−1}(*λ*). Clearly, this would imply the algebraic integrability of the QCIS defined by *A*_{1}, …, *A*_{n}, and the algebraic integrability of *L* would follow from that.

So, let us consider, for a fixed *k*∈, the system(3.7)and an associated system with *constant coefficients*(3.8)

*The partial differential operator D*^{*} *takes the (locally analytic) solutions f(x) of* (3.7) *to the solutions g=D*^{*}*f of the system* (3.8). *If the spectral parameter k is such that λ*_{0}*=R(k)≠0, then the inverse map is given by f=(R(k))*^{−1}*Dg*.

To prove the proposition, we use the fact that the constant coefficient operators in the left-hand side of (3.8) have the form *λ*_{0}(∂)=*D*^{*}*D* and *λ*_{i}(∂)=*D*^{*}*D∂*_{i} for *i*>0. This is due to the second part of theorem 3.4. It follows that . Similarly, for *i*>0, we get . This proves the first claim. Now, since *Dg*=*DD*^{*}*f*=*A*_{0}*f*, which is *λ*_{0}(*k*)*f* by the first equation in (3.7), the second claim follows.

*Any (locally analytic) solution f(x) to* (3.7) *with R(k)≠0 is a constant multiple of ψ=D*e^{(k, x)}.

To prove this, take any solution *f*(*x*) analytic in some open chart of and apply *D*^{*} to get a solution *g*(*x*)=*D*^{*}*f* to (3.8). But this constant coefficient system has the only solution e in the case . Indeed, by the Nullstellensatz applied to the commutative ring , the system of equations implies that for all , as soon as the system of equations has as its only solution. And this is the case if . Thus, we get for all , which implies that *g* is a constant multiple of .

Summing up, we get theorem 3.5.

*Any exactly solvable Schrödinger operator* *from* theorem 3.3 *is algebraically integrable*.

### (c) Baker–Akhiezer function and quasi-invariant polynomials

The commutative ring constructed in theorem 3.4 is not a maximal one and can be extended in a way we are going to describe now. An important tool is a multidimensional Baker–Akhiezer function, which plays a central role in the approach of Chalykh & Veselov (1990), Veselov *et al*. (1993) and Chalykh *et al*. (1999).

For simplicity, let us restrict to the case when the configuration of the poles of *u* is central, i.e. when has the form (1.6)(3.9)Here, is a finite collection of non-parallel vectors with prescribed multiplicities , which satisfies the locus conditions (1.7).

According to theorem 3.2, the Schrödinger operator has an eigenfunction , quasi-polynomial in the spectral parameter *k*. This function is a common eigenfunction of the commutative ring constructed in theorem 3.4 and, as such, it is unique up to a factor depending on *k*. A non-trivial fact (Chalykh *et al*. 1999) is that there is a distinguished way to normalize to achieve a symmetry between *x* and *k*. The resulting function has the following form:(3.10)where and is polynomial in *k, x* with the highest term .

**(**Chalykh *et al*. 1999**).** *The (properly normalized) common eigenfunction* *of the commutative ring constructed in* *theorem 3.4* *has the form* (3.10) *and is symmetric in k, x in the sense that* . *As a result*, *satisfies the same equations in the spectral parameter*.

The last property is called *bispectrality* following the fundamental paper of Duistermaat & Grünbaum (1986). In Chalykh *et al*. (1999), it was also shown that in the spirit of the finite-gap theory, can be uniquely characterized by (3.10) together with certain analytic properties. Namely, for each of the hyperplanes , the function must be -quasi-invariant with respect to this hyperplane, in the sense of definition 3.2. This will allow us to extend the commutative ring from theorem 3.4. Before formulating the result, let us make one remark which is as follows.

Note that . From this equation and from the type of the singularity that *u* has along a hyperplane , one easily derives that should either have a pole of order or a zero of order along this hyperplane. But the latter clearly contradicts (3.10). Thus, we conclude that must have a pole of order . By the symmetry between *k* and *x,* we get the same properties in *k*. In particular, this implies that the polynomial in (3.10) is irreducible as polynomial in *k*.

Given a configuration , define the ring of quasi-invariants as the ring of polynomials which are -quasi-invariant with respect to the hyperplane for all .

(i) *For each homogeneous quasi-invariant polynomial* , *there exists a partial differential operator* *with the highest symbol* *such that* . *All these operators pairwise commute*.

(ii) *This commutative ring is a maximal commutative subring of the ring of analytic differential operators on any open chart* *outside the singular locus* .

Part (i) is the result of Chalykh *et al*. (1999). Note that there is the following explicit formula for due to Berest: , where and *C* is an appropriate constant (Berest 1998; Chalykh *et al*. 1999).

To prove the second claim, let *A* be a differential operator on *U* that commutes with all of from theorem 3.4. Then, must be proportional to since it is the common eigenfunction of , which is unique according to corollary. Thus, for some polynomial in *k*. We want to prove that *f* is quasi-invariant polynomial, thus . To this end, observe that since does not involve *k*, the function has the same analytic properties in *k* as . Namely, must be -quasi-invariant as a function of *k* with respect to the hyperplane . From the remark above, we conclude that the function is not zero along this hyperplane. And we know that multiplying it by gives another -quasi-invariant function. This immediately implies that *f* itself must be -quasi-invariant with respect to the hyperplane . Since this must be true for any hyperplane, we conclude that and we are done.

In papers of Feigin & Veselov (2002, 2003), a result close to part (ii) of theorem 3.7 is proved in some important special cases.

The question whether or not the ring is finitely generated is not quite trivial; it seems to be finitely generated in most cases, except some ‘degenerate’ ones (see Kasman & Previato (2001) for an example of an infinitely generated maximal commutative ring of partial differential operators). Now, assuming that is finitely generated, we may think of it as the coordinate ring of an affine algebraic variety; this is analogous to the spectral curve in dimension one. In many important cases, these varieties were shown to have a number of very interesting properties (Etingof & Ginzburg 2002; Feigin & Veselov 2002, 2003; Berest *et al*. 2003).

## 4. Trigonometric case

Let be an *n*-dimensional Schrödinger operator with trigonometric potential. By this we mean that *u* is a rational function of , where generate a rank *n* lattice in . Since we will not be using the real structure, there will be no difference between the trigonometric and hyperbolic settings. Clearly, the periods of form the reciprocal lattice generated by with . We will impose the following assumption on *u* which is analogous to the assumption in theorem 2.2. Namely, we will assume that *u* decays at infinity along any ray , , with generic . For example, satisfies this assumption, as does for any . In analogy with *n*=1 case, let us denote this property symbolically as .

We want to characterize those *L* that are exactly solvable. Similar to the rational case, one uses theorem 3.1 and lemma 3.1 to conclude that the potential *u* of such an operator must have second-order poles along the collection of (pairwise distinct) hyperplanes with , and for a finite set of affine–linear functions of the form with non-isotropic . Since the configuration of the poles must be periodic, we assume that all the vectors belong to the lattice . As a result, we get that the singularities of *u* are encoded in a finite set with the assumptions as above, with prescribed multiplicities . Owing to the condition , the potential *u* is completely determined by its singularities and is described by the formula(4.1)

As an example, note that a one-dimensional with does not meet the requirements above, since and have common zeros. Instead, one should consider

Now, lemma 3.1 tells us that the potential (4.1) of an exactly solvable operator must be -quasi-invariant with respect to each of the hyperplanes , . These are necessary conditions, which turn out to be sufficient according to the next theorem.

(Chalykh 1998). *If the potential* *of a Schrödinger operator* *has the form* (4.1) *with the assumptions as above, and u is quasi-invariant with respect to each of its pole hyperplanes, then L is exactly solvable*.

We outline the proof since it was omitted in Chalykh (1998). Suppose has a quasi-invariant potential of the form (4.1). We will effectively construct an eigenfunction of *L* of the form . Let *V* denote a subspace of meromorphic functions with the following properties:

the poles of

*f*must be along some of the hyperplanes , , with the order of the pole andfor each , the function must be -quasi-invariant with respect to .

The central observation (Chalykh 1998) is that the operator *L* preserves the space *V*: . This follows directly from the quasi-invariance of *u*. As a result, *L* will preserve the subspace of *V* consisting of those *f* having the form , where is a trigonometric polynomial in *x* and polynomial in *k*, and is as follows:(4.2)

Now, for a trigonometric polynomial , by its *support* we will mean the convex hull of those with . Then, one can show that an application of *L* to does not increase the support of *Q*, i.e. if , then has the form , where . Moreover, one can compute how an application of *L* affects the coefficients at the vertices of . More precisely, let be from and be a vertex of the support of the trigonometric polynomial *Q*, with being the corresponding coefficient. Then, the coefficient in is given by the formula . Here, stands for and is one of the vertices of the support of . (One can say exactly which vertex should be taken, but we will not need this.)

Now, we can construct as follows. Take , so with . Clearly, belongs to . Now, let us start applying operators of the form successively to decrease the support of *Q* at each step. To achieve this, we just choose any vertex of the support of *Q* and take , in accordance with the paragraph above. Since the support is finite, repeating this sufficiently many times will produce zero. The last non-zero function *f* in this process will be a quasi-polynomial in *k*, such that for some . Shifting *k* by will give us , quasi-polynomial in *k*, such that . This finishes the proof. ▪

One can refine the arguments used in the proof above, and get an explicit formula for the eigenfunction , generalizing the Berest's formula (3.2) (see §5.6 of Chalykh (2002) for the Calogero–Moser case).

Combining this theorem with the previous discussion, we obtain the following result.

*Let* *be a Schrödinger operator with trigonometric potential such that* . *Then, L is exactly solvable if and only if the potential u has the form* (4.1) *and is* *-quasi-invariant with respect to each of the hyperplanes* .

Finally, by the arguments similar to those used in section b, we obtain the algebraic integrability of all these operators.

*All exactly solvable operators* *from* *theorem 4.2* *are algebraically integrable*.

The structure of the corresponding commutative ring of partial differential operators is described below. For simplicity, we will restrict to the case when *u* has the form(4.3)

Here, is a finite collection of non-parallel vectors with prescribed multiplicities . We keep all the assumptions of theorem 4.1, namely, the vectors are assumed to belong to a lattice of rank *n*, and *u* must be -quasi-invariant with respect to each of the hyperplanes .

For , define the ring as the ring of polynomials such that for all and any

For a given , all rings are isomorphic, say, to . The reason we introduce is that then this definition can be viewed as a deformation of the definition 3.3, with the latter corresponding to the limit . In that sense, we may think of the ring in theorem 3.7 as . This suggests that coincides with the associated graded ring for . It is easy to prove the inclusion , but we do not know a general proof of the fact that they coincide.

Now, according to theorem 4.1, has an eigenfunction , which is quasi-polynomial in the spectral parameter *k*. More precisely, , where is given by (4.2) and *P* is polynomial in *k* and trigonometric polynomial in *x*. Theorem 4.4 is a trigonometric version of theorem 3.7.

(i) *For any polynomial* , *there exists a partial differential operator* *such that* . *All these operators commute, and the operator corresponding to* *coincides with* .

(ii) *The commutative ring consisting of all* *with* *is a maximal commutative subring* (*of rank one*) *inside the ring of analytic operators on any open chart* *outside the singular locus* . *(Here* *is the function* (4.2)*.)*

The proof is similar to the proof of theorem 3.7. The main idea is to characterize in a unique way by its analytic properties in *x* and *k*. This can be done by exploiting the technique developed by Chalykh (2000, 2002). A similar result is true for any from theorem 4.1, not necessarily for those of the form (4.3). Details will appear elsewhere.

As a by-product, one can conclude that satisfies certain difference equations in the spectral parameter *k*; this is a deformation of the symmetry between *k* and *x* that we had in the rational case. As an example, for the Sutherland–Calogero–Moser operator *L*, these dual difference operators are nothing but (a rational version of) the Macdonald–Riujsenaars operators from Ruijsenaars (1987) and Macdonald (1988). For other root systems, one gets similarly the Macdonald difference operators (see Chalykh 2000). For the deformed root systems, this bispectral duality gives a natural way to deform the Macdonald–Ruijsenaars operators. For , this was done by Chalykh (2000) and for other cases see Feigin (2005).

## 5. Elliptic case

In this section, we consider the operators with elliptic potential, following the work of Chalykh *et al*. (2003). Our results are parallel to those for the rational and trigonometric cases, though they are less complete.

Let us first describe the class of potentials we will be considering. They are the following elliptic analogues of the trigonometric potentials (4.1):(5.1)Here, is the Weierstrass -function with the periods and . As in the trigonometric case, stands for a finite set of affine–linear functions on , each given by its equation , and the multiplicities are non-negative integers. We assume that the vectors are non-isotropic and belong to a rank *n* lattice . In this case, the period lattice for *u* will be of rank and can be described as , where is the reciprocal of , i.e. . Thus, *u* may be viewed as a meromorphic function on a (compact) torus , which is isomorphic to the product of *n* copies of the elliptic curve . Singularities of are the second-order poles along the following set of the hyperplanes (or hypertori):Similarly to the trigonometric case, we have to assume that all the hyperplanes are *pairwise distinct*. Next, motivated by the results above, let us require the quasi-invariance of *u*.

Let us call a Schrödinger operator a generalized Lamé operator if *u* has the form (5.1) and is -quasi-invariant with respect to each of the hyperplanes .

As we already know, in dimension one this definition singles out all elliptic algebro-geometric operators , with the Lamé operator being one of them. This, together with the previous results for the rational and trigonometric cases, makes conjecture 5.1 very natural.

(Chalykh *et al*. 2003). *Any generalized Lamé operator is algebraically integrable*.

While this conjecture remains open in general, in Chalykh *et al*. (2003) it was proved under additional assumption of the complete integrability of *L* (in a slightly stronger sense, see below). This is an important result since the complete integrability in many cases is easier to establish. In particular, as we will explain, this allows one to prove the conjecture of Chalykh & Veselov (1990) about the algebraic integrability of the quantum elliptic Calogero–Moser systems for all root systems.

Let us formulate the main result of Chalykh *et al*. (2003) precisely. Let *L* be a generalized Lamé operator. Recall that *L* is *completely integrable* if it is a member of a commutative family of differential operators which are algebraically independent. We assume that the *L*_{i}'s have meromorphic coefficients and are periodic with respect to the same lattice, which makes them (singular) differential operators on the torus .

Let us say that a completely integrable Schrödinger operator in is *strongly integrable* if the commuting operators have algebraically independent homogeneous constant highest symbols such that is finitely generated as a module over the ring generated by or, equivalently, that the system has a unique solution .

For example, in dimension *n*=1, any is strongly integrable. On the other hand, consider the QCIS defined by constructed in §3*b*. Then, this system is not strongly integrable since the highest symbols are all divisible by .

(Chalykh *et al*. 2003). *Any generalized Lamé operator L which is strongly integrable is algebraically integrable*.

Let us explain the main idea behind the proof of this theorem. It uses the following important result of Braverman *et al*. (1997), which links the algebraic integrability to the differential Galois theory. Namely, for a given QCIS system defined by , consider the eigenvalue problem(5.2)with . Then, for generic *λ* this system is holonomic (Braverman *et al*. 1997). Next, for a fixed *λ* one defines the differential Galois group of (5.2) (e.g. Kaplansky 1957). The main result of Braverman *et al*. (1997) is the following characterization of algebraically integrable systems.

(Braverman *et al*. 1997). *A quantum completely integrable system is algebraically integrable if and only if the differential Galois group of* (5.2) *is commutative for generic λ*.

Now, let *L* be a strongly integrable generalized Lamé operator, so we have the commuting operators with meromorphic coefficients on the torus . Then, one can show that our assumptions on the highest symbols of imply that the holonomic system (5.2) has regular singularities (see Chalykh *et al*. 2003). Next, fix a point and consider the finite-dimensional space of locally analytic solutions. Then, the monodromy group of this system gives us a representation of the fundamental group of in the local solution space. Now, from the quasi-invariance of the potential one derives that the solutions to (5.2) have no branching along the hypertori from , i.e. they are *locally meromorphic* functions on *T*. As a result, the monodromy representation factors through the fundamental group of *T*, which is Abelian. Thus, the monodromy group of the system (5.2) is Abelian. However, a general result due to Deligne (1970) says that for a regular holonomic system on a smooth projective variety the differential Galois group is the Zariski closure of the monodromy group (see Chalykh *et al*. (2003) for a simple proof of this fact in our situation). As a result, we conclude that the differential Galois group of (5.2) is Abelian for any *λ*, and theorem 5.1 follows.

Moreover, according to Braverman *et al*. (1997) and Chalykh *et al*. (2003), in that situation the solution space of (5.2) for generic *λ* will be generated by the *quasi-periodic* solutions.

*There exist meromorphic* 1-*forms* *on the torus T, with first-order poles and depending analytically on λ, such that the functions* *give a basis of the solution space of* (5.2) *for generic λ*.

Each of these functions will be a *double-Bloch solution*, in the following sense (cf. Krichever & Zabrodin 1995):(5.3)for appropriate , where is the basis of the lattice (see §5 of Chalykh *et al*. (2003) for a general procedure of computing these double-Bloch solutions).

Our proof applies to a more general situation when one has commuting operators on an Abelian variety, such that the system , , is regular holonomic. Then, the triviality of its local monodromy around singularities implies that this system is algebraically integrable.

Let us finish this section by considering some examples when theorem 5.1 applies.

### (a) One-dimensional case

We consider with elliptic potential *u*. In this case, *u* must have a form (2.4)with distinct (modulo the periods) and . The quasi-invariance of *u* boils down to the following conditions:(5.4)By doing the local analysis (cf. Duistermaat & Grünbaum 1986), one can see that in the one-dimensional case our definition of a generalized Lamé operator is equivalent to the property that all solutions of the equation are meromorphic in the complex domain.

Thus, we get an alternative proof of theorem 2.3. Moreover, it generalizes easily to the higher-order case. Indeed, let(5.5)be an ordinary differential operator of order *n*. It is called algebro-geometric if it is a member of a rank one maximal commutative ring of differential operators. The rank is defined as the dimension of the common eigenspace or, equivalently, as the greatest common divisor of the orders of the operators from the ring. In other words, *L* is algebro-geometric if there exists another operator *A* of order coprime to *n* such that (Krichever 1977). The above results imply immediately theorem 5.3, conjectured by Gesztesy & Weikard (1996).

(Chalykh *et al*. 2003). *Suppose that L given by* (5.5) *has elliptic coefficients and all solutions of the equation* *are meromorphic in the complex domain for any* . *Then, L is algebro-geometric*.

Converse is also true, owing to the general formula expressing the Baker function via -function for the KP hierarchy (Segal & Wilson 1985; Gesztesy & Weikard 1996).

### (b) Calogero–Moser problem

A natural example of a generalized Lamé operator in dimension greater than one is given by the quantum Calogero–Moser problem with integer coupling constants. Recall that the quantum elliptic Calogero–Moser problem is described by the following *n*-dimensional Schrödinger operator:(5.6)Here, *m* is the coupling constant, which is an arbitrary complex number. In physical literature, *m* is usually assumed to be real; also, to ensure that *u* is real-valued, the period of is often taken to be purely imaginary. The singularities of *L* are the second-order poles along the hyperplanes , and clearly *L* is symmetric with respect to any of these hyperplanes. As a result, for any *integer m*, the Calogero–Moser operator (5.6) gives a particular example of a generalized Lamé operator. In the same manner, we may consider a generalization of the Calogero–Moser problem for any root system *R* due to Olshanetsky & Perelomov (1983),(5.7)Here, is a positive half of a root system , and the coupling constants are chosen in a *W*-invariant way, where *W* is the Weyl group of *R*. Specializing the parameters to integers, we get more examples of the generalized Lamé operators. The standard Calogero–Moser problem corresponds to .

In Chalykh & Veselov (1990), it was conjectured that the quantum Calogero–Moser problems related to the root systems are algebraically integrable if the coupling constants are integers. This conjecture, clearly, is a special case of a more general conjecture. For the -case, it was proved by Braverman *et al*. (1997). To prove it in full generality, one uses the fact due to Cherednik (1995) that the operator (5.7) is completely integrable (see also Oshima & Sekiguchi (1995), where the set of commuting operators was given explicitly for all classical root systems). One easily checks then that the highest symbols of satisfy our condition of strong integrability. Thus, the following theorem results.

(Chalykh *et al*. 2003). *For any root system R and any integer W-invariant coupling constants* , *the quantum elliptic Calogero–Moser operator* (5.7) *is algebraically integrable*.

A similar result is true for the Inozemtsev operator, which is a version of (5.7) (§6). Also, the algebraic integrability has been checked by Chalykh *et al*. (2003) for all known generalized Lamé operators in dimension two. This provides an extra evidence in support of conjecture 5.1.

## 6. Examples

In §§3–5, we considered algebraically integrable Schrödinger operators with three classes of potential: rational, trigonometric and elliptic. Known (irreducible) examples of these operators in dimension greater than two fall into three categories: (i) Calogero–Moser systems related to Coxeter groups and root systems (Olshanetsky & Perelomov 1983); (ii) Berest–Lutsenko family (Berest & Lutsenko 1997), this is an interesting series of two-dimensional examples with rational *u*; and (iii) generalized Calogero–Moser systems related to the deformed root systems, the first examples of which appeared in the works of Veselov *et al*. (1996) and Chalykh *et al*. (1999).

### (a) Calogero–Moser systems

These examples are related to the Calogero–Moser systems with integer coupling parameters . The corresponding operator *L* is given by the formula (5.7) with being a positive half of a root system , and *u* being or the Weierstrass' , respectively.

In , the corresponding operator in the trigonometric case involves three parameters and looks as follows (Olshanetsky & Perelomov 1983):(6.1)

Its elliptic version due to Inozemtsev (1989) involves five parameters *m*, *g*_{0}, *g*_{1}, *g*_{2}, and *g*_{3} and looks as follows:(6.2)with denoting the half-periods of . According to Chalykh *et al*. (2003), this operator is algebraically integrable for any integer .

In the rational case, we may consider any finite reflection group *W* in and take *R* to be the set of the unit normals to its reflection hyperplanes, choosing in a *W*-invariant way. The corresponding Schrödinger operator is given by the same formula (5.7) with . Thus, in the rational case, apart from cases, we have also (dihedral groups), and .

### (b) Berest–Lutsenko family

This family was discovered by Berest & Lutsenko (1997) in relation to the Huygens' principle and Hadamard problem. It is in dimension two, so *L* has the form , where the potential *U* is rational homogeneous of degree −2. The Berest–Lutsenko operators *L* are in one-to-one correspondence with *-periodic soliton potentials* (as discussed in §2). Namely, for any such , we define *U* in polar coordinates as . The singularities of form a collection of lines passing through the origin, and the positions of these lines correspond to the poles of . For instance, taking we get the dihedral case of the Calogero–Moser problem. An interesting feature of this family is that it has continuous parameters, all other known irreducible examples in higher dimension appear in discrete families. Note that the notion of irreducibility we refer to is not quite straightforward and involves *isotropic reduction* (see Berest & Lutsenko (1997) and Chalykh *et al*. (1999) for more details and examples).

### (c) Deformed root systems

Other known irreducible examples of the generalized Lamé operators in dimension greater than one are related to the deformed root systems. Below we describe the set of linear functions and the corresponding multiplicities . For convenience, we will allow to be negative, with *L* given by the same formula as above,(6.3)where , or , respectively.

system (Veselov

*et al*. 1996). It consists of the following linear functions in :Here,*m*is an integer parameter.system (Chalykh

*et al*. 1999). It consists of the following linear functions in :Here, are integer parameters related by . In two-dimensional case*n*=1, the first group of roots is vacuous and there is no restriction for*k*to be an integer.Here is a -type generalization of the previous example for the trigonometric and elliptic cases (Chalykh

*et al*. 2003). For the elliptic case, we denote the half-periods by , as in (6.2). The set of linear functions and the corresponding multiplicities look as follows:Here,*k*,*l*and_{s}*m*are nine integer parameters related through for all . The previous case corresponds to and . Again, in the case_{s}*n*=1, the first group of roots is not present and*k*does not have to be an integer.In the trigonometric case, we have the same formulae but with just two half-periods and . Respectively, we will have five integer parameters with for .

Hietarinta operator (Hietarinta 1998). In this case, we have three covectors in ,with . Here, are arbitrary complex parameters such that . Note that the system is essentially two-dimensional since .

system (Chalykh & Veselov 2001). It consists of the following linear functions in :Here,

*m*is an integer. Note that for*m*=1, this system coincides with the system above.

## 7. Conclusion

We have described three classes of multidimensional Schrödinger operators which generalize the algebro-geometric operators with rational, trigonometric and elliptic potential . Here, we would like to mention some open questions related to that approach.

One of the main questions is to describe completely all operators in those classes, for instance, by proving that the list of examples we gave exhausts all irreducible cases. This would be very important for solving the Hadamard problem (see Chalykh *et al*. 1999).

Secondly, it would be very desirable to find more explicit description of the spectral variety in the elliptic case. For the -case, Felder & Varchenko (1997) constructed the Hermite–Bloch variety parametrizing the double-Bloch eigenfunctions, but its relation to the spectral variety has yet to be clarified. In Chalykh *et al*. (2003), the Hermite–Bloch variety was also (implicitly) constructed for the generalized Lamé operators, and we have the same question in that case. Also, for the -case, the double-Bloch solutions were constructed by Felder & Varchenko (1997) in a rather effective way, via a version of Bethe ansatz. Thus, it would be very interesting to generalize the Bethe ansatz of Felder–Varchenko to other root systems and their deformations.

Finally, it is conceptually clear that there is a *q*-analogue of this story, related to the commutative rings of partial difference operators, but it is yet to be worked out in detail. For the root systems, this is related to the operators of Macdonald–Ruijsennars type, and their algebraic integrability (in trigonometric case) was established by Etingof & Styrkas (1998) for -case and by Chalykh (2000) in general. This has nice applications to the Macdonald theory. The deformed case was treated by Chalykh (2000), and the methods used there are applicable to all deformed root systems. However, the elliptic case is more difficult and very little is proven in dimension greater than one. Let us mention here the paper of Billey (1998) where the Bethe ansatz of Felder–Varchenko was extended to the elliptic Ruijsenaars system (); also see some very explicit formulae in Chalykh *et al*. (2003) for a particular case related to . Perhaps, a first natural step in that direction would be to try to generalize the results of Braverman *et al*. (1997) to the difference setting.

## Acknowledgments

I am grateful to Yu. Berest, P. Etingof, M. Feigin and A. P. Veselov for many useful discussions. This work was partially supported by Russian President grant MK-2050.2003.01.

## Footnotes

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

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