aa r X i v : . [ m a t h . SP ] J a n BURCHNALL-CHAUNDY THEORY
SEBASTIAN KLEIN, EVA L ¨UBCKE, MARTIN ULRICH SCHMIDT,AND TOBIAS SIMON
Abstract.
The Burchnall-Chaundy theory concerns the classifi-cation of all pairs of commuting ordinary differential operators. Wephrase this theory in the language of spectral data for integrablesystems.In particular, we define spectral data for rank 1 commutativealgebras A of ordinary differential operators. We solve the inverseproblem for such data, i.e. we prove that the algebra A is (essen-tially) uniquely determined by its spectral data. The isomorphytype of A is uniquely determined by the underlying spectral curve. Introduction
The Burchnall-Chaundy theory ([B-C-I, B-C-II, B-C-III]) concernsthe classification of all pairs (
P, Q ) of commuting ordinary differen-tial operators P and Q of order m and n , respectively. Burchnall andChaundy carried out their work before the relationship between inte-grable systems and Riemann surfaces or complex curves, i.e. the spec-tral theory for integrable systems, was discovered in the course of theinvestigation of the integrable system defined by the Korteweg-de Vriesequation.The main purpose of the present article is to rephrase the Burchnall-Chaundy theory in terms of the theory of spectral data for integrablesystems. For this the Krichever construction and the theory of Baker-Akhiezer functions will play important roles. Because the spectralcurve can have singularities, we will use the description of these con-cepts for analytic singular curves in [K-L-S-S]. Our construction canalso serve as an explanation of the relation of these two concepts todifferential operators.We will consider algebras A that are generated by pairs ( P, Q ) ofcommuting differential operators of order m , n . In Section 3 we willassociate to any such algebra a holomorphic matrix-valued function M : C → C m × m , λ M ( λ ), and thereby spectral data which are com-posed of a generally singular complex curve X ′ describing the eigen-values of ( M ( λ )) λ ∈ C and a second datum describing the corresponding eigenvector bundle. In contrast to parts of Burchnall’s and Chaundy’soriginal theory, we will here restrict ourselves to the case where theorders m and n of the generating differential operators are relativelyprime. This restriction corresponds to the algebra A being of rank 1,which means by definition that the eigenspaces of M ( λ ) are generically1-dimensional. In this situation the eigenvectors of M ( λ ) comprise aholomorphic line bundle Λ ′ on X ′ .Whenever X ′ has no singularities and is therefore a Riemann surface,the well-known 1–1 correspondence between line bundles and divisorson Riemann surfaces is often used to define spectral data as the pair( X ′ , D ) of the eigenvalue curve X ′ and the divisor D corresponding tothe eigenline bundle Λ ′ . In order to define spectral data in a similarmanner also for complex curves X ′ with singularities, the concept of adivisor needs to be generalised. The proper generalisation to use in thiscontext is the concept of generalised divisors introduced by Hartshorne[Ha-86], at first for Gorenstein curves. In this sense, a generalised di-visor on X ′ is a subsheaf of the sheaf of meromorphic functions on X ′ which is locally finitely generated over the sheaf of holomorphicfunctions on X ′ . The usefulness of this concept for our purposes isexpressed by the fact that one again has a 1–1 correspondence betweenline bundles and generalised divisors on complex curves X ′ . By virtueof this fact, we will define the spectral data corresponding to a rank 1commutative algebra A (essentially) as the pair ( X ′ , S ′ ) comprising theeigenvalue curve X ′ and the generalised divisor S ′ on X ′ that corre-sponds to the eigenline bundle Λ ′ on X ′ . It is also one of the purposes ofthis paper to explore the extent of the usefulness of generalised divisorsin Hartshorne’s sense, also for non-Gorenstein curves, and to convincethe reader of their manifest value.We do not consider the case of commutative algebras of rank higherthan 1 (which are generated by differential operators whose degreesare not relatively prime), because in this case the eigenvector bundleof M ( λ ) is a vector bundle of rank higher than 1. Such vector bundlesdo not correspond to generalised divisors, because the sheaf of theirsections is not contained in the sheaf of meromorphic functions on X ′ .For this reason the construction and investigation of spectral data inthis case would require a very different theory.In Section 4 we will solve the inverse problem for the spectral datathus defined for rank 1 algebras A of commuting differential operators.This means that we will prove that A is essentially uniquely determinedby its spectral data (in fact, the domain of definition of the differentialoperators is extended to a certain maximum), and we will also see howto reconstruct A from its spectral data (see Theorems 4.2 and 4.4). URCHNALL-CHAUNDY THEORY 3
This constitutes the rephrasing of Burchnall’s and Chaundy’s mainclassification result in terms of the present, modern concepts.We will additionally show that two rank 1 algebras of commutingdifferential operators are isomorphic to each other as algebras if andonly if the corresponding spectral curves X ′ are biholomorphic (seeTheorem 4.5). In other words, the family of all rank 1 algebras ofcommutative differential operators that are isomorphic to a given one A can be generated by taking the spectral data ( X ′ , S ′ ) corresponding to A and then varying the spectral divisor S ′ throughout its connectedcomponent in the space of generalised divisors on X ′ . To the bestof our knowledge, this statement is new in the sense that it has nocounterpart in Burchnall’s and Chaundy’s classical work.We know from the discussion in [K-L-S-S, Section 4] that the pairs( X ′ , S ′ ) occur in families where the complex curves X ′ are partial nor-malisations, i.e. branched one-fold coverings, of one another, and thegeneralised divisors S ′ are direct images under the corresponding cov-ering maps. These spectral data obtained by partial normalisation cor-respond to commutative algebras which contain A as subalgebra. Suchfamilies always contain one member ( X ′′ , S ′′ ) of minimal δ -invariant,i.e. minimal singularity. X ′′ was called the S ′ -halfway normalisationof X ′ in [K-L-S-S]. This minimal member corresponds to the maximalcommutative rank 1 algebra which contains A , i.e. to the centraliserof A in the algebra of all ordinary differential operators. This con-struction also permits to find pairs ( X ′′′ , S ′′′ ) which are below ( X ′ , S ′ )by a branched one-fold covering and so that X ′′′ has arbitrarily large δ -invariant. Such pairs correspond to rank 1 subalgebras of the givencommutative rank 1 algebra A .We now begin our work by deriving a certain standard form for pairsof commuting differential operators in Section 2, which will facilitatethe construction of the spectral data.2. The algebra of differential operators
Let us first introduce the algebra of ordinary differential operators.We consider three different algebras of differential operators:Case 1: The domain is an open interval I = ( a, b ) and the algebra is A ( I ) := C ∞ ( I, R )[ D ] where D = ddt with parameter t ∈ I .Case 2: The domain I is a real 1-dimensional non-compact submanifoldof C which is simply connected and the algebra is A ( I ) := C ∞ ( I, C )[ D ] where D = ddz with parameter z ∈ I .Case 3: The domain is an open, connected subset I of C and the algebrais A ( I ) := O I [ D ] where D = ddz with parameter z ∈ I . S. KLEIN, E. L ¨UBCKE, M. SCHMIDT, AND T. SIMON
Along with the domain I we fix a point t ∈ I in any of these cases.We write these polynomials as P = α m D m + . . . + α . We denote the operator of multiplication with a function f also by f . The algebra results as a subalgebra of the linear endomorphismsof the coefficient functions defined on I . Due to the Leibniz rule thecommutator of the differential operator D and the operator of multi-plication with a coefficient function f acting on functions on I is givenby the multiplication with f ′ , the derivative of f . Therefore, we set Df − f D = f ′ . Lemma 2.1.
For any coefficient function f on I and any n ∈ N thecomposition of the operator D n with f acting on the functions on I satisfies the identity D n f = X ≤ i ≤ n (cid:18) ni (cid:19) f ( i ) D ( n − i ) . Proof.
Let ad( D ) denote the operator A [ D, A ] acting on the linearoperators on the space of smooth functions. Then the Leibniz rulemay be written as ad( D ) f = f ′ . Hence we have the identity Df = f D + ad( D ) f = (ad ( D ) f ) D + (ad ( D ) f ) D . This implies D n f = X ≤ i ≤ n (cid:18) ni (cid:19) (cid:0) ad i ( D ) f (cid:1) D ( n − i ) = X ≤ i ≤ n (cid:18) ni (cid:19) f ( i ) D ( n − i ) . q.e.d. If P and Q are two elements of A ( I ), then due to this formula theproduct P Q is again an element of A ( I ). Since this product was derivedfrom the action of the ordinary differential operators on the smoothfunctions, it endows A ( I ) with the structure of an associative algebra(a subalgebra of the operators on the smooth functions on I ). The or-der of an element P of A ( I ) is the highest number m whose coefficient α m does not vanish identically on I . Let us first use two transforma-tions in order to bring a commutative subalgebra into standard form.The first transformation changes the domain I of the correspondingcoefficient functions of A ( I ). If ξ is a smooth resp. holomorphic, in-vertible function on I , the second transformation P ξ − P ξ is aninner automorphism of A ( I ). Now these two transformations may beused in order to bring a commuting pair of differential operators intostandard form: URCHNALL-CHAUNDY THEORY 5
Proposition 2.2.
Let P ∈ A ( I ) be a differential operator of order m .Then P can be transformed into an element e P ∈ A ( ˜ I ) with highestcoefficient equal to and vanishing second highest coefficient. Therebyall operators Q ∈ A ( I ) which commute with P are transformed into e Q ∈ A ( ˜ I ) which commute with e P and have constant highest and secondhighest coefficient.Proof. We first consider case 1. If ξ ( t ) is a diffeomorphism of an openinterval I onto another open interval ˜ I , the vector field ddt is trans-formed under this diffeomorphism onto the vector field 1 /ξ ′ ( t ) ddξ . Thistransformation therefore induces an isomorphism of the algebras A ( I )and A ( ˜ I ). We now construct such a deformation so that the highestcoefficients of e P and ultimately e Q are equal to 1.The highest coefficient α m of P cannot vanish identically. So thereexists a subinterval of I on which α m has no roots. Then there ex-ists an invertible function χ such that α m = χ m . Hence, dξdt = χ − and ξ = R χ − dt is strictly monotonous and therefore a diffeomor-phism of some subinterval of I with some interval ˜ I with the desiredproperties. Then P ∈ A ( ˜ I ) has highest coefficient one. We define η := α m − m . Then the inner automorphism corresponding to the func-tion ζ = exp (cid:0) − R η dt (cid:1) transforms P into a differential operator of thedesired form. If an operator Q ∈ A ( I ) of order n commutes with P ,then the above transformation maps Q to an operator e Q ∈ A ( ˜ I ). Sincethis transformation is an algebra homomorphism, e Q commutes with e P .Lemma 2.1 yields that the coefficient of D m + n − in e Q e P − e P e Q is equalto n e β n e α ′ m − m e α m e β ′ n = 0 , (1)where e α m and e β n are the highest coefficients of e P and e Q , respectively.Therefore, e β n is constant if e α m is constant. The coefficient of D m + n − in e Q e P − e P e Q is due to Lemma 2.1 proportional to n e α ′ m − − m e β ′ n − . Therefore, e β n − is constant if e α m − is constant.In case 2, the proof is essentially the same as in case 1 with all(sub)intervals replaced by (sub)manifolds of C . We now choose ξ = R η − dz which is complex-valued. Due to the inverse function theorem, ξ defines a diffeomorphism on a possibly smaller submanifold of I ontoanother submanifold ˜ I .In case 3, there are no big changes from the proof of case 1 either.The (sub)intervals are replaced by open subsets of C , the smooth maps S. KLEIN, E. L ¨UBCKE, M. SCHMIDT, AND T. SIMON are replaced by holomorphic maps and the diffeomorphisms by biholo-morphic maps. Again the inverse function theorem gives that ξ is abiholomorphic function on a possibly smaller open subset of I . q.e.d. We remark that if m is equal to one, an inductive application ofProposition 2.2 shows that the coefficients of Q are constant and Q isa polynomial with respect to P . In the sequel, we shall only considerelements of a commutative subalgebra of A ( I ). Definition 2.3.
A commutative subalgebra of A ( I ) is called of rank if it contains two differential operators of coprime orders. We denotethe set of such subalgebras by R in the following. In the sequel, we consider only commutative subalgebras of rank 1.
Lemma 2.4.
A commutative subalgebra A of A ( I ) is of rank 1 if andonly if there exists d ∈ N so that A contains elements of every degree d ≥ d .Proof. For A ∈ R , A contains a pair ( P, Q ) of differential operators ofcoprime orders n, m . By B´ezout’s identity, there exist integers a, b with1 = an + bm . If n = 1 or m = 1, then A contains operators of everypositive order, so we now suppose n, m ≥
2. Then ab <
0, and wesuppose without loss of generality that a > b <
0. For d ≥ nm there exists an integer l so that dan − d ≤ lmn ≤ dan , and then d =( da − lm ) n + ( db + ln ) m with da − lm ≥ db + ln = d − dan + lnmm ≥ P da − lm · Q db + ln ∈ A is of order d .The converse follows because for every d ∈ N there exist two coprimenumbers n, m ≥ d . q.e.d. The observation that the highest coefficients of all elements of A ∈ R are constant allows to define the degrees of the elements intrinsically. Lemma 2.5.
For A ∈ R and P ∈ A \ { } the degree deg( P ) is equalto dim( A/P A ) with P A = { P Q | Q ∈ A } Proof.
By Lemma 2.4 for A ∈ R the complement of the set N = { deg( P ) | P ∈ A } in N is finite. Since the highest coefficients of the elements of A areconstant, for P ∈ A \ { } the map A → A with Q P Q is injective.Therefore { deg( P Q ) | Q ∈ A } is equal to N + deg( P ) = { d + deg( P ) | d ∈ N } . Furthermore, dim( A/P A ) = N \ ( N + deg( P )). The set N + deg( P ) is a subset of N , since P A is a subspace of A . Therefore N \ ( N + deg( P )) is equal to ( N \ ( N + deg( P ))) \ ( N \ N ) and thecardinality of both sets are equal to N \ ( N + deg( P ))) − N \ N ). URCHNALL-CHAUNDY THEORY 7
This difference equals deg( P ), since N \ ( N + deg( P )) is the disjointunion { , . . . , deg( P ) − } ˙ ∪ (( N \ N ) + deg( P )). q.e.d. We will always assume that differential operators are in standardform as in Proposition 2.2, which means that highest and second high-est coefficients are constant. In constructing this standard form, wepossibly made the domain I smaller. Our solution of the inverse prob-lem (Theorem 4.4(1)) will show that such commutative algebras arealready uniquely determined by their restriction to arbitrarily small,open subsets of I . 3. The direct problem
The following Theorem, together with some kind of converse, hasalready been shown in [B-C-I]. We give a proof here with methodswhich better match the point of view we want to take in this work.
Theorem 3.1.
For two commuting ordinary differential operators P and Q of orders m respectively n , there exists a polynomial f ( λ, µ ) withconstant coefficients with the following properties:(i) If we associate to λ the degree m and to µ the degree n , then thecommon degree of f ( λ, µ ) is equal to mn . Moreover, the highestcoefficient is equal to µ m + cλ n with some non-zero constant c .(ii) The differential operator f ( P, Q ) is identically equal to zero.Proof. In this proof we index the coefficients of P and Q differentlyfrom Section 2 to simplify the characterisation of the degree of thecoefficients of f . Specifically we write P = D m + α D m − + . . . + α m and Q = β D n + β D n − + . . . + β n .For λ ∈ C , we collect the derivatives ψ, ψ ′ , . . . , ψ ( m − of the solutionsof the differential equation ( λ − P ) ψ = 0 to a column-vector-valuedfunction b ψ = ( b ψ , . . . , b ψ m − ) T . Consequently, the differential equa-tion ( λ − P ) ψ = 0 is equivalent to the first order differential equation( D − U ( · , λ )) b ψ = 0 with the m × m -matrix-valued function U ( t, λ ) = . . .
00 0 1 . . . . . . ... λ − α m ( t ) − α m − ( t ) − α m − ( t ) . . . − α ( t ) . (2)With the help of the equation P ψ = λψ we may express all derivativesof ψ of order higher than m − b ψ : That S. KLEIN, E. L ¨UBCKE, M. SCHMIDT, AND T. SIMON equation is equivalent to ψ ( m ) = λψ − m − X l =0 α m − l ψ ( l ) = λ ˆ ψ − m − X l =0 α m − l ˆ ψ l Differentiating this formula yields ψ ( m +1) = λψ ′ − m − X l =0 α m − l ψ ( l +1) − m − X l =0 α ′ m − l ψ ( l ) = λ ˆ ψ − ( α ′ m + α λ − α α m ) b ψ + m − X l =1 ( α α m − l − α m − l +1 − α ′ m − l ) ˆ ψ l . The higher derivatives of ψ can be obtained by inductively repeatingthis procedure. Note that the sum of the indices on the right hand sidein each term plus the order of the derivative in this term always equalsthe order of the derivative on the left hand side.In particular, there exists a unique m × m -matrix V ( t, λ ) whosecoefficients are polynomials in λ and differential polynomials of thecoefficients α i of P and β j of Q such that V ( · , λ ) b ψ = Q b ψ for all ψ inthe kernel of λ − P .For any complex number λ ∈ C we consider the solutions of thedifferential equation ( P − λ ) ψ = 0. Due to the theory of ordinary dif-ferential equations, there exist exactly m linear independent solutions ψ , . . . , ψ m of this ordinary differential equation on I . Two solutionscoincide if and only if the corresponding values of ψ, ψ ′ , . . . , ψ ( m − atany element of the domain coincide. In particular, a base ψ , . . . , ψ m ofall solutions is uniquely determined by the condition that the deriva-tives up to order m − ψ i vanish at the marked point t with theexception of the ( i − t . Since Q commutes with P , the span of ψ , . . . , ψ m is invariant with respect to Q . Hence Q acts as right multiplication with an m × m -matrix M ( λ )on the row vector ( ψ , . . . , ψ m ).The vectors b ψ corresponding to the basis ψ , . . . , ψ m build the funda-mental solution g ( t, λ ), i.e. an m × m -matrix-valued function dependingon ( t, λ ) ∈ I × C :( D − U ( · , λ )) g ( · , λ ) = 0 and g ( t , λ ) = 1l , where g ( t, λ ) is invertible for all ( t, λ ) ∈ I × C . We conclude V ( t, λ ) g ( t, λ ) = g ( t, λ ) M ( λ ) ⇔ M ( λ ) = g − ( t, λ ) V ( t, λ ) g ( t, λ ) (3)where M ( λ ) does not depend on t , but it does depend on the choiceof the marked point t . Since g ( t, λ ) is a solution of the differential URCHNALL-CHAUNDY THEORY 9 equation ( D − U ( · , λ )) g ( · , λ ) = 0 one has0 = Dg − ( · , λ ) V ( · , λ ) g ( · , λ ) = − g − ( · , λ ) U ( · , λ ) g ( · , λ )++ g − ( · , λ ) ∂V ( · , λ ) ∂t g ( · , λ ) + g − ( · , λ ) V ( · , λ ) U ( · , λ ) g ( · , λ ) = g − ( · , λ ) [ D − U ( · , λ ) , V ( · , λ )] g ( · , λ ) . Therefore, the commutativity of the operators P and Q implies[ D − U ( · , λ ) , V ( · , λ )] = 0 . Since the characteristic polynomial of V ( t, λ ) is invariant under conju-gation of V ( t, λ ) with g ( t, λ ), this yields f ( λ, µ ) := det ( µ − V ( t, λ )) = det ( µ − M ( λ )) . So f ( λ, µ ) does not depend on t . Due to our construction, U ( t, λ ), V ( t, λ ) and M ( λ ) = V ( t , λ ) are polynomials with respect to λ .Let us determine the highest coefficients of these polynomials. Inorder to regard P as homogeneous of degree m and Q as homogeneousof degree n , we assign the weight i to α i for i = 0 , . . . , m , the weight j to β j for j = 0 , . . . , n and to every derivative the weight 1. Thisassignment is in accordance with the weight m for λ and the weight n for µ as stated in the theorem. The k -th row of V ( · , λ ) describesthe action of Q on ψ ( k ) . So the entry V kl of the matrix V ( · , λ ) has theweight n + k − l since V kl b ψ l contains at most n + k derivatives. Theentries V kl are therefore homogeneous polynomials of degree n + k − l with respect to λ and derivatives of α i and β j , where deg( α ( r ) i ) = i + r ,deg( β ( s ) j ) = j + s and deg( λ ) = m . So det( V ( t, λ )) has the degree mn .Moreover the ( k, l )-th entry of µ − V ( t, λ ) has the weight n + k − l ,so the characteristic polynomial of V ( t, λ ) is homogeneous of weight mn . In particular the highest coefficients of f ( λ, µ ) depend only onthe coefficients of P and Q of weight zero, which are the highestcoefficients.Therefore the highest coefficients of f ( λ, µ ) for general P and Q arealready obtained by considering the “free case” P = D m and Q = β D n with a constant β = 0 . In this case, the matrices are U free ( t, λ ) = . . .
00 0 1 . . . . . . ... λ . . . , V free ( t, λ ) = M free ( λ ) = β · U n free ( t, λ ) . (4) Hence det( U free ( t, λ )) = ( − m − λ and the highest coefficient of f ( λ, µ )is equal to µ m − β m λ n .Now we claim that the differential operator f ( P, Q ) vanishes identi-cally. Due to the commutativity of P and Q , this differential operatordoes not depend on the order in which the operators are inserted intothe polynomial f ( P, Q ). If ψ is any common solution of the equations( P − λ ) ψ = 0 and ( Q − µ ) ψ = 0 with f ( λ, µ ) = 0, then P acts on ψ asthe multiplication with λ and Q acts on ψ as the multiplication with µ . Consequently, the action of f ( P, Q ) on ψ is the same as the actionof f ( λ, µ ) on ψ and hence vanishes. For any roots ( λ , µ ) , . . . , ( λ r , µ r )in C of f with pairwise different λ , . . . , λ r , any choice of non-trivialcommon solutions ψ , . . . , ψ r of ( P − λ k ) ψ k = 0 and ( Q − µ k ) ψ k = 0for k = 1 , . . . , r are linear independent. In fact, suppose that thesesolutions obey a linear relation a ψ + . . . + a r ψ r = 0 . Then the action of
P, P , . . . , P r − on the relation adds r − a λ s ψ + . . . + a r λ sr ψ r = 0 for s = 1 , . . . , r − . Altogether, we have r linear relations on the functions a ψ , . . . , a r ψ r .Since λ , . . . , λ r are pairwise different, the determinant of coefficientsof these relations is a non-vanishing Vandermonde determinant. Thisimplies that all these functions a k ψ k vanish identically. A solution ψ of ( P − λ ) ψ = 0 vanishes identically on ˜ I if the first m derivatives of ψ vanish at t ∈ ˜ I . Hence, the complements of the sets of roots of ψ k are open and dense and there exists t ∈ ˜ I with non-vanishing values ψ k ( t ). Therefore a , . . . , a r vanish, and thus the ψ k are indeed linearindependent. We conclude that the differential operator f ( P, Q ) hasinfinitely many solutions f ( P, Q ) ψ = 0. Since the order of f ( P, Q ) isbounded by nm , this implies that this differential operator vanishesidentically. q.e.d. A partial converse has been shown by Burchnall and Chaundy in[B-C-I] which includes pairs of differential operators
P, Q of co-primeorders, but not the general case. In the sequel, we shall associate asingular curve to a commutative algebra of differential operators.From now on, we want to investigate pairs of commuting operators P and Q whose orders m and n are co-prime. We use the results andnotation of [K-L-S-S]. In the proof of Theorem 3.1, we constructeda holomorphic matrix-valued function M : C → C m × m . In [K-L-S-S,Section 4] we have described how this matrix can be associated with apair ( X ′ , S ′ ) where X ′ is a complex curve and S ′ a generalized divisor. URCHNALL-CHAUNDY THEORY 11
In Theorem 3.1 a pair of commuting differential operators is given.For such a pair, we define the singular curve X ′ as the one-point com-pactification of (cid:8) ( λ, µ ) ∈ C × C | det (cid:0) µ · − M ( λ ) (cid:1) = 0 (cid:9) . (5) X ′ does not depend on the choice of the marked point t because ofEquation (3).We claim that X ′ is a singular curve and the point at infinity ∞ a smooth point. Since P and Q are in standard form with highestcoefficient equal to 1, M ( λ ) has the following form: M ( λ ) = . . .
00 0 1 . . . . . . ... λ . . . n + O ( λ n − ) , as λ → ∞ . (6)Therefore, there exists a local parameter z defined for large | λ | such that λ = ( z/ πi ) − m , µ = ( z/ πi ) − n + O ( z − n ) as z →
0. If m and n arerelatively prime, then z is uniquely characterised by these conditions.The single root of z , which is added at infinity to (5), is a smooth point ∞ of X ′ . The eigenvalue λ corresponds to a meromorphic function on X ′ with a single pole of order m at ∞ and µ to a meromorphic functionwith a single pole of order n at ∞ . In order to define the generalizeddivisor S ′ we normalize the eigenfunction of M ( λ ) by ℓ ( ψ ) = 1 with ℓ : C m → C , ( ψ , . . . , ψ m ) ψ . Lemma 3.2.
There exist only finitely many ( λ, µ ) ∈ X ′ \ {∞} forwhich the kernel of ℓ contains non-trivial eigenvectors of M ( λ ) witheigenvalue µ .Proof. We consider V ( t, λ ) as defined in the proof of Theorem 3.1 and V free ( t, λ ) as defined in equation (4) which corresponds to the free case P = D m and Q = D n . The normalized free eigenvector ϕ obeys V free ( t, λ ) ϕ = λ n/m ϕ with ϕ = (cid:0) , λ /m , . . . , λ ( m − /m (cid:1) T . (7)Due to the weights of V kl , as introduced in the proof of Theorem 3.1, V kl is a polynomial of degree n + k − l . Therefore, all contributionsto V kl which do not contribute to V free ,kl include at least one of thecoefficients α i for i ≤ m − β j for j ≤ n −
1, or a derivative of sucha coefficient. This implies | ( V kl − V free ,kl ) | = O (cid:0) λ − /m (cid:1) λ ( n + k − l ) /m . Let T be the diagonal matrix diag (cid:0) , λ − /m , . . . , λ − ( m − /m (cid:1) . There-fore, | ( T ( V − V free ) T − ) kl | = | ( V − V free ) kl λ ( l − k ) /m | = | λ | n/m O ( λ − /m ) , Due to equation (4) λ − n/m · T V free T − = . . .
00 0 1 . . . . . . ...1 0 0 . . . n and hence | λ | − n/m · k T V T − − T V free T − k ≤ O ( λ − /m ) . Since
T V free T − has m pairwise different eigenvalues, it is diagonalis-able. Now we show that the distance k T ψ − T φ k of the eigenfunctionsis also of order O ( λ − /m ). Due to the implicit function theorem ap-plied to ( T V T − − µ T ϕ , the normalized eigenfunction
T ϕ and theeigenvalue µ depend nearby T V free T − continuously differentiably onthe entries of T V T − . Since ℓ ◦ T = ℓ and ϕ ker( ℓ ), for sufficientlylarge λ also ψ ker( ℓ ). This shows that the set of ( λ, µ ) ∈ X ′ \{∞} sothat ker( ℓ ) contains a non-trivial eigenvector of M ( λ ) for the eigen-value µ is a subvariety of X ′ \ {∞} of codimension at least 1 , andhence finite. q.e.d. By evaluating ψ at the marked point t = t , we obtain the globalmeromorphic function χ := ψ ( · , t ) = ( χ , . . . , χ m ) T : X ′ \{∞} → C m . χ is characterised uniquely by M χ = µχ and ℓ ( χ ) = 1 , where we regard also M and µ as functions on X ′ \ {∞} . Locally, χ can be obtained from any holomorphic eigenfunction ˜ χ by taking χ = ˜ χ/ ˜ χ .In the sequel we use generalised divisors on the spectral curve X ′ .For this purpose we again apply the notations introduced in [K-L-S-S].We define the generalised divisor S ′ corresponding to χ on X ′ \{∞} asthe subsheaf of the sheaf of meromorphic functions on X ′ \ {∞} whichis generated over O X ′ \{∞} by χ , . . . , χ m . Because of Lemma 3.2, S ′ isequal to O X ′ on a punctured neighborhood of ∞ . We therefore extend S ′ to ∞ by defining S ′∞ = O X ′ , ∞ .In Section 4 we will describe the dependence of S ′ on the markedpoint t by means of the Krichever construction.Note that the eigenspace of M ( λ ) with eigenvalue µ is one-dimensionalat all points of X ′ where the map X ′ → P , ( λ, µ ) λ is not ramified. URCHNALL-CHAUNDY THEORY 13
If the eigenspaces of M define a line bundle on X ′ , then S ′ describesthe dual eigenline bundle in the sense of the correspondence betweendivisors and line bundles, see [Fo, § X ′ , S ′ ) to the pair ( P, Q )of commuting differential operators of co-prime orders.In [K-L-S-S, Definition 4.2] we have introduced the S ′ -halfway nor-malisation X ( S ′ ) for the pair ( X ′ , S ′ ): For a generalised divisor S ′ ona singular curve X ′ , the S ′ -halfway normalisation of X ′ is the uniqueone-sheeted covering π X ( S ′ ) : X ( S ′ ) → X ′ such that( π X ( S ′ ) ) ∗ O X ( S ′ ) = { f ∈ ¯ O X ′ | f · g ∈ S ′ for all g ∈ S ′ } . On X ( S ′ ), there exists a unique generalized divisor S ( S ′ ) whose directimage with respect to π X ( S ′ ) equals S ′ . For any q ∈ X ′ , we choose localgenerators φ , . . . , φ m of S ′ q . Then for any q ′ ∈ π − X ( S ′ ) [ { q } ], S ( S ′ ) q ′ isthe X ( S ′ )-submodule of M q ′ generated by φ ◦ π X ( S ) , . . . , φ m ◦ π X ( S ) ,see [K-L-S-S].Until now, we have assigned the quadruple ( X ′ , S ′ , ∞ , z ) to commut-ing differential operators P and Q of coprime orders m and n . Here, P and Q correspond to meromorphic functions λ and µ on X ′ withpoles of orders m and n only at ∞ . The meromorphic functions on X ′ having only poles at ∞ are equal to the algebra C [ λ, µ ] / ( f ) with f defined in Theorem 3.1. In the sequel, we will assign such quadru-ples ( X ′ , S ′ , ∞ , z ) to commutative algebras. This assignment has theproperty that the commutative algebra is isomorphic to the algebraof meromorphic functions on X ′ which have poles only at ∞ . Thequadruple ( X ′ , S ′ , ∞ , z ) constructed above is assigned to the commu-tative algebra generated by the two differential operators P and Q . Ourmain result gives an essentially 1 − X ′ , S ′ , ∞ , z )of a compact singular curve X ′ with smooth marked point ∞ and co-ordinate z near ∞ and a generalized divisor S ′ on X ′ whose degreeis equal to the arithmetic genus of X ′ . In this section, we investigatethe map from the algebra to the triple and in the following section theinverse of this map.We will see that the following definition describes the maximal com-mutative subalgebras of A ( I ) which are of rank 1. Definition 3.3 (centraliser) . For each subalgebra A of A ( I ) the algebra C ( A ) := { P ∈ A ( I ) | ∀ Q ∈ A : [ P, Q ] = 0 } is called the centraliser of A . Lemma 3.4.
For each A ∈ R , we have C ( A ) ∈ R , and A has finitecodimension in C ( A ) .Proof. Because we consider only differential algebras where the highestorder coefficients of the member operators are constant, Lemma 2.4implies dim(
B/A ) ≤ d < ∞ for any B ∈ R with B ⊃ A . Thereforeit suffices to prove that C ( A ) is commutative because C ( A ) contains A .Let A ∈ R and P, Q ∈ A be two differential operators of coprimeorders m and n . For λ ∈ C , let V λ be the m -dimensional space ofsolutions of P ψ = λψ . As in the proof of Theorem 3.1, let M ( λ ) denotethe endomorphism V λ → V λ induced by Q . Due to Theorem 3.1, f ( λ, µ )has weighted degree mn with highest term µ m + cλ n with c = 0. Forcoprime m and n , the m -th roots of λ n are all pairwise different. Hence,for large λ the m solutions of f ( λ, µ ) = 0 are also pairwise different and M ( λ ) has m pairwise different eigenvalues. Because the discriminantis holomorphic, the same holds for λ in an open and dense subset of C .Now, let R, S ∈ C ( A ). Since they commute with P and Q , they de-fine endomorphisms B ( λ ) and C ( λ ) of V λ commuting with M ( λ ). Thecommutator [ R, S ] ∈ C ( A ) induces the endomorphism [ B ( λ ) , C ( λ )] of V λ . If M ( λ ) has pairwise different eigenvalues, it is diagonal with re-spect to an appropriate basis. Since B ( λ ) and C ( λ ) commute with M ( λ ), this basis also diagonalises B ( λ ) and C ( λ ). Therefore, B ( λ )and C ( λ ) commute. This implies that the vector spaces V λ belong tothe kernel of [ R, S ] if M ( λ ) has pairwise different eigenvalues. By def-inition, V λ ∩ V λ ′ = { } for λ = λ ′ . As in the proof of Theorem 3.1,[ R, S ] has an infinite dimensional kernel. Since [
R, S ] has finite order,its kernel can only be infinite dimensional if [
R, S ] = 0. q.e.d.Definition 3.5.
Spectral data are a quadruple ( X ′ , S ′ , ∞ , z ) , where X ′ is a compact singular curve, S ′ is a generalised divisor on X ′ whose degree is equal to the arithmetic genus of X ′ , ∞ is a smoothpoint of X ′ and z is a local coordinate of X ′ near ∞ . We show in the following theorem that C ( A ) has spectral data( X ( S ′ ) , S ( S ′ ) , ∞ , z ). This will lay the foundation to construct thespectral data assigned to general algebras A ∈ R . In the sequel, M denotes the sheaf of meromorphic functions on X ( S ′ ) . We omit thesubscript X ( S ′ ) because ( π X ( S ′ ) ) ∗ is an isomorphism of sheaves from M X ( S ′ ) onto M X ′ . Theorem 3.6.
Let A ∈ R and P, Q ∈ C ( A ) two differential operatorsof coprime orders and ( X ′ , S ′ , ∞ , z ) be the spectral data correspond-ing to the subalgebra of C ( A ) generated by P and Q . Then the triple URCHNALL-CHAUNDY THEORY 15 ( X ( S ′ ) , S ( S ′ ) , ∞ ) and the value d z ( ∞ ) are independent of the choiceof P and Q , C ( A ) is isomorphic to the algebra B := { f ∈ H ( X ( S ′ ) , M ) | ∀ p ∈ X ( S ′ ) \ {∞} : f p ∈ O X ( S ′ ) ,p } (8) and ( π X ( S ′ ) ) ∗ S ( S ′ ) = S ′ . We will see in Section 4 that the solution of the inverse problem forgiven spectral data ( X ′ , S ′ , ∞ , z ) depends on z only in terms of thevalue of d z ( ∞ ) . Proof.
Let
P, Q ∈ A be the differential operators of coprime orders m and n with the corresponding matrix M ( λ ). The subalgebra h P, Q i of A ( I ) generated by P and Q is commutative, hence R ∋ h
P, Q i ⊂ A and therefore A ⊂ C ( A ) ⊂ C ( h P, Q i ) . By Lemma 3.4, C ( h P, Q i )is commutative and therefore contained in C ( A ) . This implies that C ( h P, Q i ) = C ( A ) .We now show that C ( A ) is isomorphic to the algebra B in (8). Let R ∈ C ( A ). Since [ R, P ] = 0, there exists for each λ ∈ C a matrix N ( λ ) ∈ C m × m which describes the action of R on the kernel of P − λ · R, Q ] = 0, we have [ M ( λ ) , N ( λ )] = 0 for all λ ∈ C . Therefore, foreach ( λ, µ ) ∈ X ′ , N ( λ ) acts on the kernel of M ( λ ) − µ · λ, µ )with one-dimensional ker( M ( λ ) − µ · , N ( λ ) acts as multiplication witha complex number ν . Such ( λ, µ ) build an open and dense subset of X ′ .Since the entries of N ( λ ) are meromorphic, ν extends to a meromorphicfunction on X ′ . For all λ ∈ C , all entries of N ( λ ) are bounded andtherefore also the eigenvalue ν of the N ( λ ). This implies ν ∈ B .Conversely, in [K-L-S-S], it has been proven that (ˆ λ ◦ π X ( S ′ ) ) ∗ O X ( S ′ ) is isomorphic to the sheaf of holomorphic n × n matrices on P whichcommute with M ( λ ). Here, ˆ λ is the map ˆ λ : X ′ → C such that( λ, µ ) λ .This algebra isomorphism extends to an isomorphism of B with ma-trices N ( λ ) whose entries are polynomials with respect to λ and com-mute with M ( λ ). These are the matrices which describe the action ofthe elements of C ( A ) on the kernel of P − λ · P ′ , Q ′ ∈ C ( A ) be another pair of differential operators of coprimeorders. Since h P, Q i ∩ h P ′ , Q ′ i contains all R ∈ C ( A ) of sufficientlylarge order by Lemma 2.4, there exists a third pair P ′′ , Q ′′ ∈ A suchthat h P ′′ , Q ′′ i ⊂ h P, Q i ∩ h P ′ , Q ′ i . Without loss of generality, we maytherefore suppose P ′ , Q ′ ∈ h P, Q i , meaning that P ′ and Q ′ can beregarded as polynomials in P and Q .The spectral data of P ′ , Q ′ is a quadruple ( X ′′ , S ′′ , ∞ ′ , z ′ ) togetherwith two meromorphic eigenfunctions λ ′ , µ ′ on X ′′ . λ ′ and µ ′ can be regarded as polynomials in λ and µ , and in this way we obtaina holomorphic map X ′ \ {∞} → X ′′ \ {∞ ′ } . Because ∞ and ∞ ′ are smooth points of X ′ and X ′′ , respectively, this map extends to aholomorphic map X ′ → X ′′ , which is biholomorphic on an open anddense subset of X ′ by Lemma 3.2. Therefore this map is a one-foldcovering. The pullback of a common eigenfunction of P ′ and Q ′ is acommon eigenfunction of P and Q , or equivalently, the direct imageof S ′ is S ′′ . By definition of X ( S ′ ) , it follows that ( X ( S ′′ ) , S ( S ′′ ) , ∞ )is isomorphic to ( X ( S ′ ) , S ( S ′ ) , ∞ ′ ) . Because P and P ′ have highestcoefficient 1 , it follows from the definition of z that the correspondingbiholomorphic map maps d z ( ∞ ) onto d z ′ ( ∞ ′ ) . q.e.d.Theorem 3.7. For A ∈ R and P, Q ∈ A of coprime orders, let ( X ′ , S ′ , ∞ , z ) be the corresponding spectral data. Then up to isomor-phy, there exists a unique one sheeted covering π ′′ : X ′′ → X ′ and ageneralized divisor S ′′ on X ′′ with the following properties:(i) π ′′∗ S ′′ = S ′ .(ii) The following diagram commutes h P, Q i ֒ → A ֒ → C ( A ) g ↓∼ = g ↓∼ = g ↓∼ = C [ λ,µ ]( f ) ֒ → C ֒ → B .
Here, the polynomial f and the isomorphism g are defined inTheorem 3.1, and B and g are defined in Theorem 3.6. We alsoset C := { f ∈ H ( X ′′ , M ) | ∀ p ∈ X ′′ \ ( π ′′ ) − [ {∞} ] : f p ∈ O X ′′ ,p } , (9) and g is defined by the above diagram. ( X ′′ , S ′′ , ∞ , z ) does not depend on the choice of P, Q ∈ A .Proof. In Theorems 3.1 and 3.6, we have shown that g and g areisomorphisms, respectively. The sheaf O ′ X ′ is contained in π ( S ′ ) ∗ O X ( S ′ ) .Note that A is a subalgebra of C ( A ). Hence the image of A under g iscontained in B . We identify the meromorphic functions on X ( S ′ ) withthe meromorphic functions on X ′ , in this way B becomes a subalgebraof H ( X ′ \ {∞} , π ( S ′ ) ∗ O X ( S ′ ) ). The image of A in B generates on X ′ \ {∞} a subsheaf A of subrings of π ( S ′ ) ∗ O X ( S ′ ) . It contains O X ′ since A contains P and Q . The stalks of the latter subsheaf havefinite codimension in the stalks of π ( S ′ ) ∗ O X ( S ′ ) , and the codimensionis 0 away from the singularities of X ′ . Therefore we may extend A to X ′ by A = O X ′ near the smooth point ∞ . By definition of X ( S ′ ), π ( S ′ ) ∗ O X ( S ′ ) acts on S ′ . Due to [K-L-S-S, Lemma 4.1], there exists aunique one-sheeted covering π ′′ : X ′′ → X ′ such that π ′′∗ ( O X ′′ ) = A . URCHNALL-CHAUNDY THEORY 17
The sequence of one-sheeted coverings X ( S ′ ) → X ′′ → X ′ induces theembeddings in the lower row of the diagram. The embeddings of theupper row are obvious.It remains to show that there exists an isomorphism g as in thediagram. Because g is an isomorphism, there exists a subalgebra of C ( A ) which is mapped isomorphically onto C by g . It suffices to showthat this subalgebra equals A .On the one hand, since the sheaf of subrings π ′′∗ ( O X ′′ ) of π ( S ′ ) ∗ O X ( S ′ ) is generated by the image of A under g in B , this algebra is containedin A . On the other hand, the image of every element of A with respectto g in B belongs to the subalgebra which generates π ′′∗ ( O X ′′ ) andtherefore to the image of C in B . So this subalgebra contains A .The only choice that was made in this construction was that of P and Q . The independence of ( X ′′ , S ′′ ) from the choice of P and Q follows by the same argument as in the proof of Theorem 3.6. q.e.d.Lemma 3.8. The generalized divisors S ′ , S ( S ′ ) and S ′′ have degreeequal to the arithmetic genus of X ′ , X ( S ′ ) , X ′′ respectively, and theyare non-special.Proof. We first show the claim for X ′ and S ′ . At ∞ , the function ψ k has a pole of order k − ψ k that is holomorphic at ∞ is a multiple of ψ . This shows thatdim H ( X ′ , S ′ ) = 1. Let U ′ := { ( λ, µ ) ∈ X ′ \ {∞} (cid:12)(cid:12) | λ | > R } , wherewe choose R > S ′ is equal to O X ′ on U ′ . Then U := ( X ′ \ {∞} , U ′ ∪ {∞} ) is a Leray covering of ( X ′ , S ′ ) by [K-L-S-S,Proposition 4.5]. We use this covering to show that H ( X ′ , S ′ ) =0. Let f ∈ H ( U , S ′ ). By [K-L-S-S, Proposition 4.5], we have f = f ψ + . . . + f m ψ m with holomorphic functions f , . . . , f m on U := { λ ∈ C (cid:12)(cid:12) | λ | > R } . We can write f k = g k − λ − h k , where g k is anentire function, and h k is a holomorphic function on U ∪ {∞} . h := h λ − ψ + . . . + h m λ − ψ m is holomorphic on U ′ ∪{∞} , because the poleorder of ψ k at ∞ is at most m −
1, so λ − ψ k is holomorphic at ∞ . With g := g ψ + . . . + g m ψ m , we have f = g − h , and therefore f is a boundarywith respect to ( U , S ′ ). This shows that H ( X ′ , S ′ ) = H ( U , S ′ ) = 0.By Riemann-Roch’s Theorem [K-L-S-S, Theorem 5.2], it follows thatdeg( S ′ ) equals the arithmetic genus of X ′ , and that S ′ is non-special.We now consider S ′′ on X ′′ . On one hand, we have H ( X ′′ , S ′′ ) = H ( X ′ , S ′ ) because of π ′′∗ S ′′ = S ′ . On the other hand, because of S ′ ⊃ π ′′∗ O X ′′ , we havedeg( S ′ ) = dim H ( X ′ , S ′ / O X ′ )= dim H ( X ′ , S ′ /π ′′∗ O X ′′ ) + dim H ( X ′ , π ′′∗ O X ′′ / O X ′ )= deg( S ′′ ) + ( g ( X ′ ) − g ( X ′′ )) . Because deg( S ′ ) equals the arithmetic genus g ( X ′ ) by the previous partof the proof, deg( S ′′ ) = g ( X ′′ ) follows. Therefore also S ′′ is non-special.This argument likewise applies to ( X ( S ′ ) , S ( S ′ )). q.e.d.Proposition 3.9. Suppose that we are in case 1, i.e. A = C ∞ ( I, R )[ D ] with an open interval I ⊂ R . Then the spectral data ( X ′′ , S ′′ , ∞ , z ) of A ∈ R satisfy the following reality conditions:(1) There exists an anti-holomorphic involution ρ on X ′′ so that ∞ is a smooth point of the real singular curve given by the fixedpoint set of ρ , and ρ ∗ z = − z . For any P, Q ∈ A of co-primeorder, ρ acts on the eigenvalues ( λ, µ ) as ( λ, µ ) (¯ λ, ¯ µ ) .(2) We have ρ ∗ S ′′ = S ′′ , where the generalised divisor ρ ∗ S ′′ ischaracterised by H ( U, ρ ∗ S ′′ ) = { f ◦ ρ | f ∈ H ( ρ ( U ) , S ′′ ) } for any open subset U ⊂ X ′′ .Proof. We consider differential operators
P, Q ∈ A of co-prime order m and n , respectively. Because we are in case 1, P and Q havereal coefficients. Therefore the matrices U and V from the proofof Theorem 3.1 are real for λ ∈ R and thus also M ( λ ) is real for λ ∈ R . This shows that M (¯ λ ) = M ( λ ) for all λ ∈ C . Therefore thepolynomial f has real coefficients, and hence ρ : ( λ, µ ) (¯ λ, ¯ µ ) is ananti-holomorphic involution on the singular curve X ′ \ {∞} .We extend ρ to X ′ by setting ρ ( ∞ ) = ∞ . It was shown in Theo-rem 3.1 that the highest coefficient of the polynomial f is of the form µ m + cλ n with a non-zero constant c , which is real in the present set-ting. Therefore ∞ is a smooth point of the fixed point set of ρ , and ρ ∗ z = − z .Because X ′′ → X ′ is a one-sheeted covering, we obtain an anti-holomorphic map ρ on X ′′ with the desired properties.As the linear form ℓ is real, the normalised section ψ also satisfies ψ ◦ ρ = ψ , and therefore ρ ∗ S ′′ = S ′′ holds. q.e.d. The inverse problem
We now solve the corresponding inverse problem. We let spectraldata ( X ′ , S ′ , ∞ , z ) be given. This means that X ′ is a singular curve URCHNALL-CHAUNDY THEORY 19 with a marked smooth point ∞ and a local parameter z defined onan open neighbourhood U of ∞ , and S ′ is a generalised divisor on X ′ of degree equal to the arithmetic genus of X ′ .We will use the Krichever construction as in [K-L-S-S, Section 7].In particular we define the one-parameter group of invertible sheaves L /z ( t ) with t ∈ C : Let U := X ′ \ {∞} , then ( U , U ) is a coveringof X ′ and the cocycle z ∗ exp( − πit/z ) defines L /z ( t ) with respectto this covering. On an open subset O ⊂ C , the same cocycles withvariable t ∈ O also define a sheaf L /z on X ′ × O .The Krichever construction depends on the choice of the local pa-rameter z only via the Mittag-Leffler distribution induced by z . Forany two different local parameters z , z on X ′ around ∞ there ex-ists a constant c = d z d z ( ∞ ) = 0 so that z − cz is holomorphic. Thisshows that our construction in fact depends on the choice of the localcoordinate z only in terms of the Taylor coefficient d z ( ∞ ) .As in [K-L-S-S, Equation (31)] we define T := { t ∈ C | H ( X ′ , S ′−∞ ⊗ L /z ( t )) = 0 } , (10)where S ′−∞ is the generalised divisor obtained by multiplying S ′ withthe invertible sheaf defined by the classical divisor −∞ . In [K-L-S-S,Theorem 8.6] it was shown that T is a subvariety of C . For our specificsituation we improve that result by the following lemma. Lemma 4.1. T is discrete.Proof. We assume on the contrary that T is not discrete. Because T is a subvariety of C by [K-L-S-S, Theorem 8.6], this means that itcontains an open subset O ⊂ C .Let k > H ( X ′ , S ′−∞ ⊗ L /z ( t )) for t ∈ O . The sheaf S ′−∞ on X ′ induces a sheaf on X ′ × O , whichwe also denote by S ′−∞ . Then the sheaf S ′−∞ ⊗ L /z on X ′ × O is flat with respect to the projection X ′ × O → O by [K-L-S-S,Lemma 8.5]. Because of [G-P-R, Chapter III Theorem 4.7 (a)], themap t dim( H ( X ′ , S ′−∞ ⊗ L /z ( t ))) is upper semi-continuous, andtherefore the subset O ⊂ O on which this dimension is equal to k is open. Due to [G-P-R, Chapter III Theorem 4.7 (d)] the spaces H ( X ′ , S ′−∞ ⊗ L /z ( t )) are the fibres of a vector bundle over t ∈ O . Inparticular there exists a non-trivial section of S ′−∞ ⊗ L /z on X ′ × O .By definition of L /z , this section corresponds to a section ψ of S ′−∞ on U × O such that the function φ ( x, t ) = ψ ( x, t ) z exp( − πi t/z ) (11)is holomorphic on U × O and vanishes on {∞} × O . Let φ ( x, t ) = P n ≥ z n φ n ( t ) be the Taylor expansion of φ with re-spect to the local coordinate z at ∞ . We let N ≥ φ N does not vanish identically on {∞} × O . Then O := { t ∈ O | φ N ( ∞ , t ) = 0 } is an open subset of O . Due to Equa-tion (11), the m -th derivative x ∂ m ∂t m ψ ( x, t ) has for every m ∈ N and t ∈ O an ( m − N )-th order pole at x = ∞ . Furthermore, on U these m -th derivatives are section of S ′ . This implies that for all m ∈ N and t ∈ O the sheaf S ′ ( m − N ) ∞ ⊗ L /z ( t ) has a non-trivial sec-tion which does not belong to S ′ ( m − N − ∞ ⊗ L /z ( t ). For sufficientlylarge m the degree of S ′ ( m − N ) ∞ is greater than 2 g −
2, and due to S´erreDuality [K-L-S-S, Corollary 6.6(c)] H ( X ′ , S ′ ( m − N ) ∞ ) is trivial. NowRiemann-Roch implies dim H ( X ′ , S ′ ( m − N ) ∞ ) = m − N + 1. Becausethe derivatives ∂ l ∂t l ψ ( ∞ , t ) belong to this space for l = 0 , . . . , m , we have m − N + 1 ≥ m + 1. This implies N ≤
0, which contradicts N ≥ q.e.d. For every t ∈ C \ T , S ′ ⊗ L /z ( t ) is equivalent to a generaliseddivisor S ′′ with O X ′ ⊂ S ′′ and support contained in X ′ \ {∞} by[K-L-S-S, Lemma 8.4]. Because S ′′ ⊗ L /z ( t ) is equivalent to S ′ ⊗L /z ( t + t ) , we then have { t ∈ C | H ( X ′ , S ′′−∞ ⊗ L /z ( t )) = 0 } = t + T .
By [K-L-S-S, Theorem 8.8], S ′ therefore induces a Baker-Akhiezerfunction ψ : ( X ′ \ {∞} ) × ( C \ T ) → C such that the holomorphicextension of the function ψ ( x, t ) · exp( − πit/z ) takes the value 1 at x = ∞ . Theorem 4.2.
For given spectral data ( X ′ , S ′ , ∞ , z ) , there exists anmonomorphism of algebras Φ : { f ∈ H ( X ′ , M ) | ∀ p ∈ X ′ \ {∞} : f p ∈ O X ′ ,p } −→ A ( C \ T ) f Φ( f ) so that Φ( f ) ψ = f · ψ , where ψ is the Baker-Akhiezer function corresponding to S ′ . Thetwo highest coefficients of Φ( f ) are constant. The image A of Φ in A ( C \ T ) belongs to R .Proof. We conclude that for all t ∈ C \ T the sheaf S ′ ⊗ L /z ( t ) hasa one-dimensional space of global sections on X ′ , and all non-trivialsections do not vanish at ∞ . Therefore, this sheaf is isomorphic togeneralised divisor S which contains the sheaf of holomorphic functions O X ′ . The support of the sheaf S / O X ′ is contained in X \ {∞} . Due to[K-L-S-S, Theorem 8.8] there exists a unique Baker-Akhiezer function
URCHNALL-CHAUNDY THEORY 21 on X × { t ∈ C | t + t T } corresponding to the one-dimensional familyof sheaves S ⊗ L /z ( t ) ≃ S ′ ⊗ L /z ( t + t ) with t ∈ C . The differential operator D l acts on exp (2 πit/z ) as the multiplica-tion with (2 πi/z ) l . Therefore the uniqueness of the Baker-Akhiezerfunction implies that for all meromorphic functions f on X ′ which areholomorphic on X ′ \{∞} , there exists a unique holomorphic differentialoperator P = Φ( f ) on C \ T , such that for all y ∈ X ′ \ {∞} , the value ψ ( y, · ) of the Baker-Akhiezer function solves the holomorphic differen-tial equation f ( y ) ψ ( y, · ) = P ψ ( y, · ) . More precisely, the order of P isequal to the degree of f . If f = P i ≥− m a i z i denotes the Laurent seriesof the function f in some neighbourhood of ∞ with respect to the localparameter z , then the highest coefficient of P is equal to a − m (2 πi ) m .Moreover, since the values of exp( − πit/z ) ψ ( x, t ) at ∞ are equal to1, we have in a neighbourhood of ∞ the following equation of Laurentseries with respect to z :exp( − πit/z ) ( f ( x ) ψ ( x, · ) − (2 πi ) m D m ψ ( x, · )) = a m − z − m + O ( z − m ) . Therefore the coefficient of D m − in P is equal to a − m (2 πi ) m − .If λ and µ are two meromorphic functions on X , which are holomor-phic on X ′ \{∞} , then the values of the Baker-Akhiezer function at anyelement x ∈ X ′ \{∞} yields a common solution of ( P − λ ( x )) ψ ( x, · ) = 0and ( Q − µ ( x )) ψ ( x, · ) = 0, where P and Q denotes the differential op-erators corresponding to λ and µ . Since the commutator of P and Q on I is a differential operator of finite order, the same arguments as inthe proof of Theorem 3.1 show that the commutator is equal to zero.By construction of Φ , the degree of Φ( f ) is equal to the degreeof f , i.e. the pole order of f at ∞ . For d > g ′ − H ( X ′ , O d ·∞ ) = 0 by [K-L-S-S, Corollary 6.6(c)], and therefore byRiemann-Roch H ( X ′ , O d · x ) = d − g ′ + 1 . It follows that for every d > g ′ − f on X ′ with poleorder d at ∞ , and therefore A contains the element Φ( f ) of degree d . Lemma 2.4 thus shows A ∈ R . q.e.d.Proposition 4.3. Let spectral data ( X ′ , S ′ , ∞ , z ) be given, such that X ′ is endowed with an anti-holomorphic involution ρ so that ∞ is asmooth point of the fixed point set of ρ and ρ ∗ z = − z and ρ ∗ S ′ = S ′ .Then the restriction of the elements Φ( f ) with ρ ∗ f = f to anyconnected component I of i R \ T defines a subalgebra of A ( I ) (case 1)which belongs to R . Proof. ρ ∗ ¯ ψ is another function which satisfies the properties of theBaker-Akhiezer function ψ including the normalisation condition. Dueto the uniqueness of the Baker-Akhiezer function we therefore have ρ ∗ ¯ ψ = ψ . Therefore the differential operators Φ( f ) where ρ ∗ ¯ f = f have real coefficients on any connected component I of i R \ T . Forevery f ∈ H ( X ′ , M ) which is holomorphic on X ′ \{∞} , the degree ofΦ( f + ρ ∗ ¯ f ) is the same as the degree of Φ( f ) . Therefore the subalgebraof A ( I ) (case 1) belongs to R . q.e.d. The following theorem shows that the constructions of the directproblem in Section 3 and the inverse problem in Section 4 are essentiallyinverse to each other.
Theorem 4.4. (1) Let I be a domain as in one of the three cases, t ∈ I and A ∈ R , ( X ′′ , S ′′ , ∞ , z ) be the corresponding spectraldata constructed in Theorem 3.7, and A ( X ′′ , S ′′ , ∞ , z ) ∈ R bethe algebra corresponding to these spectral data by Theorem 4.2.Then T , I ⊂ C \ ( t + T ) , and the differential algebra A ( X ′′ , S ′′ , ∞ , z ) is isomorphic to A via the translation t t + t and the restriction to I .(2) Let ( X ′ , S ′ , ∞ , z ) be spectral data as considered in Section 4and t ∈ I := C \ T . Let A be the algebra corresponding to ( X ′ , S ′ , ∞ , z ) as in Theorem 4.2. Then the spectral data corre-sponding to ( A, I, t ) by means of Theorem 3.7 are isomorphicto ( X ′ , S ′ ⊗ L /z ( t ) , ∞ , z ) .Proof. We first consider the case where A = h P, Q i is the algebragenerated by the two commuting differential operators P and Q ofco-prime order m and n , respectively. We again consider the lo-cal parameter z defined for large | λ | such that λ = ( z/ πi ) − m , µ =( z/ πi ) − n + O ( z − n ) as z →
0. Due to Equation (6), the eigenfunction b ψ of M ( λ ) has the asymptotic behaviour b ψ = ∆ · (1 , . . . , T · (1 + O ( z ))with ∆ := diag (cid:0) , πi/z, (2 πi/z ) , . . . , (2 πi/z ) m − (cid:1) . Then by Equation (2) it follows∆ · U ( · , λ ) · ∆ − = 2 πi/z · ...
00 0 1 ... ... ... ... ... ... ... ! − ...
00 0 ... ... ... ... ... ... α m ! + O ( z ) . URCHNALL-CHAUNDY THEORY 23
Because b ψ solves the differential equation ( D − U ) b ψ = 0 and α m isconstant, the asymptotic equation∆ − b ψ = (1 , . . . , , e − tα m ) T · e πit/z · (cid:0) O ( z ) (cid:1) follows. In particular e − πit/z ψ = e − πit/z b ψ is holomorphic near ∞ and equal to 1 there. By the definition of S ′ , for all t ∈ Iψ ( · , t ) = g ( t, λ ) · χ is a section of S ′ on X ′ \ {∞} . By uniqueness of the Baker-Akhiezerfunction it follows that ψ is equal to the Baker-Akhiezer function of( X ′ , S ′ , ∞ , z ) . This proves (1) for A = h P, Q i .For general A ∈ R , we apply Theorem 3.7 and choose differen-tial operators P, Q ∈ A of co-prime order. In this situation, for all t ∈ C we have H ( X ′ , S ′−∞ ⊗ L /z ( t )) ≃ H ( X ′′ , S ′′−∞ ⊗ L /z ( t )) , be-cause π ′′∗ ( S ′′−∞ ⊗ L /z ( t )) = S ′−∞ ⊗ L /z ( t ) . Moreover, that theoremshows that the Baker-Akhiezer function of ( X ′′ , S ′′ , ∞ , z ) is equal tothe composition of π ′′ × id C \ T with the Baker-Akhiezer function of( X ′ , S ′ , ∞ , z ) . This implies (1) for general A .In the situation of (2), choose two differential operators P, Q ∈ A of co-prime order. Let ( X ′′ , S ′′ , ∞ , z ) be the spectral data of h P, Q i defined after Lemma 3.2. Then there exists a one-sheeted covering π ′ : X ′ → X ′′ . The arguments from the proof of (1) show that theBaker-Akhiezer function of ( X ′ , S ′ , ∞ , z ) is equal to the composition of π ′ × id C \ T with the Baker-Akhiezer function of ( X ′′ , S ′′ , ∞ , z ) . Becauseof Theorem 3.7, this proves (2). q.e.d. Theorem 3.7 shows that any A ∈ R is isomorphic to the algebra M ( X ′ , ∞ ) of meromorphic functions on the spectral curve X ′ of A with pole at most at ∞ . In particular, if two spectral curves ( X ′ , ∞ )and ( X ′′ , ∞ ) with marked points are biholomorphic, then the corre-sponding algebras are also isomorphic. Let us now prove the converse. Proposition 4.5.
Let ( X ′ , ∞ ) and ( X ′′ , ∞ ) be two singular curveswith a smooth point and isomorphic algebras M ( X ′ , ∞ ) and M ( X ′′ , ∞ ) .Then ( X ′ , ∞ ) is biholomorphic to ( X ′′ , ∞ ) .Proof. Due to Lemma 2.5 the degrees of two elements of two algebras M ( X ′ , ∞ ) and M ( X ′′ , ∞ ), respectively, coincide if they are mappedonto each other by an isomorphism M ( X ′ , ∞ ) ≃ M ( X ′′ , ∞ ). Firstwe choose two elements λ and µ of M ( X ′ , ∞ ) ≃ M ( X ′′ , ∞ ) of co-prime order. Due to Theorem 3.1 there exists a polynomial f with f ( λ, µ ) = 0. This equation defines a singular curve X with smoothmarked point ∞ . Due to Theorem 3.7 both singular curves X ′ and X ′′ are one-sheeted coverings π ′ : X ′ → X and π ′′ : X ′′ → X of this curve. Let us now show that for all x ∈ X \ {∞} the algebras M ( X ′ , ∞ ) ≃M ( X ′′ , ∞ ) generate the subrings ( π ′∗ ( O X ′ ) x and ( π ′′∗ ( O X ′′ ) x , respec-tively. By symmetry it suffices to give the argument for M ( X ′ , ∞ ).As in [K-L-S-S] we denote the direct image of the sheaf of holomor-phic functions on the normalisation of X by ¯ O X . For x ∈ X \ {∞} let r x be the radical r x = { f ∈ ¯ O X,x | f ( x ) = 0 } . Due to [K-L-S-S,Proposition 2.1] ( π ′∗ ( O X ′ )) x ⊃ O X,x contains r nx for some n ∈ N . Sincethe multiplication is surjective from r nx × r nx onto r nx , any choice ofelements f , . . . , f m of ( π ′∗ ( O X ′ )) x ∩ r x which span ( π ′∗ ( O X ′ )) x ∩ r x /r nx define a surjective homomorphism C { f , . . . , f m } → ( π ′∗ ( O X ′ )) x . Let S ′ be the unique generalized divisor with support {∞} and degreedeg( S ′ ) = 2 g ′ − π ′∗ ( O X ′ )) x /r nx ). Let S be the unique subsheafof S ′ which coincides on X \{ x } with S ′ and with stalk S x = r nx . It hasthe degree deg( S ) = 2 g ′ −
1. By Serre duality, we have H ( X, S ′ ) = H ( X, S ) = 0 , and therefore the Riemann-Roch Theorem impliesdim( H ( X ′ , S ′ )) − dim( H ( X ′ , S )) = deg( S ′ ) − deg( S ) . Hence the natural projection of the subspace H ( X ′ , S ′ ) ⊂ M ( X ′ , ∞ )onto ( π ′∗ ( O X ′ )) x /r nx is surjective. Moreover, there exists f , . . . , f m ∈M ( X ′ , ∞ ), which vanish at x and induce a surjective homomorphism C { f , . . . , f m } → ( π ′∗ ( O X ′ )) x . this proves the claim. In particular, twopoints of the normalisation of X belong to the same point of X ′ if andonly if all functions of M ( X ′ , ∞ ) take at both points the same values.Consequently the sheaves π ′∗ ( O X ′ ) and π ′′∗ ( O X ′′ ) are isomorphic. Thisimplies first that X ′ and X ′′ are homeomorphic and second the sheaves O X ′ and O X ′′ are isomorphic. Now [K-L-S-S, Proposition 2.3] provesthat ( X ′ , ∞ ) and ( X ′′ , ∞ ) are biholomorphic. q.e.d. In the case where the spectral curve has geometric genus zero, thecommuting differential operators P and Q which generate the corre-sponding rank 1 algebra can be computed explicitly. We conclude thispaper with an example of this computation.Let us consider A ∈ R , generated by two commuting differentialoperators P and Q of co-prime degree m and n , respectively. We let( X ′ , S ′ , ∞ , z ) be the spectral data corresponding to A and supposethat the spectral curve X ′ has geometric genus zero. The simplestpossible case occurs when m = 2 , n = 3 , which we will investigate inthe sequel.Because X ′ has geometric genus zero, there exists a global coordi-nate of the normalisation X of X ′ , i.e. a global meromorphic function z on X which is zero at ∞ and nowhere else, so that the functions URCHNALL-CHAUNDY THEORY 25 λ and µ are given as λ = p ( z − ) and µ = q ( z − ) in terms of poly-nomials p of degree m = 2 and q of degree n = 3 . By choosing thegenerating operators P and Q and the coordinate z suitably, one canachieve λ = z − and µ = z − + b z − (12)with some b ∈ C . Indeed, z can be chosen such that p ( z − ) = z − + a with some a ∈ C . After subtracting the constant a from P andnormalising Q , we obtain p ( z − ) = z − and q ( z − ) = z − + b z − + b z − + b . By now subtracting b P + b from Q , we obtain (12).These λ , µ satisfy the relation f ( λ, µ ) = 0 with the polynomial f ( λ, µ ) given by f ( λ, µ ) = µ − λ ( λ + b ) = µ − λ − b λ − b λ . The complex curve defined by the equation f ( λ, µ ) = 0 , compactifiedby adding a smooth point at ∞ , is hyperelliptic, and has exactly onesingularity. This is a double point at λ = − b if b = 0 , and a cusp at λ = 0 if b = 0 . Thus X ′ is either equal to this curve (and then hasarithmetic genus 1 ), or to its normalisation (and then has arithmeticgenus 0 ).If X ′ has arithmetic genus zero, then X ′ is biholomorphic to theRiemann sphere and the corresponding Baker-Akhiezer function is holo-morphic outside ∞ . The corresponding ordinary differential operatorshave constant coefficients and are therefore equal to P = D Q = D + b D . If X ′ has arithmetic genus 1 , we at first consider the case b =0 . We will see that the case b = 0 can be treated via the samecalculations by taking the limit. Here X ′ is a one-fold cover belowthe Riemann sphere, obtained by identifying the two points z − = ± c with c := √− b as a double point z . In this case the correspondingBaker-Akhiezer function is holomorphic except for a single order poleat z = z and an essential singularity at ∞ , hence it is of the form ψ ( z, t ) = exp(2 πitz − ) z − + d ( t ) z − − z − = exp(2 πitz − ) z + d ( t ) z zz − z , with a suitable function d depending on t . Because ψ ( z, t ) has to takethe same values at z − = ± c , we haveexp(2 πict ) c + d ( t ) c − z − = exp( − πict ) − c + d ( t ) − c − z − , whence it follows that d ( t ) is given by d ( t ) = − c z − cos(2 πct ) + ci sin(2 πct ) c cos(2 πct ) + z − i sin(2 πct ) . Therefore the
Baker-Akhiezer function is equal to ψ ( z, t ) = exp(2 πitz − ) z − − z − (cid:18) z − − c z − cos(2 πct ) + ci sin(2 πct ) c cos(2 πct ) + z − i sin(2 πct ) (cid:19) . An explicit calculation shows that the Baker-Akhiezer function solvesthe differential equation ∂ ∂t ψ ( z, t ) = − π (cid:18) z − + 2 c ( z − − c )( c cos(2 πct ) + z − i sin(2 πct )) (cid:19) ψ ( z, t ) . This shows that the operator P corresponding to the function λ = z − is given by P = − / (4 π ) D − c ( z − − c )( c cos(2 πct ) + z − i sin(2 πct )) . We leave it to the reader to calculate the corresponding operator Q .Finally we consider the limit c →
0. In this case the Baker-Akhiezerfunction is equal to ψ ( z, t ) = exp(2 πitz − ) z − − z − (cid:18) z − − z − πiz − t (cid:19) , and P is equal to P = − / (4 π ) D − z − (1 + 2 πiz − t ) . References [B-C-I] J. L. Burchnall, T. W. Chaundy: Commutative ordinary differential op-erators. Proc. London Math. Soc. , 420-440 (1923).[B-C-II] J. L. Burchnall, T. W. Chaundy: Commutative ordinary differential op-erators. Proc. Roy. Soc. London A118 , 557-583 (1928).[B-C-III] J. L. Burchnall, T. W. Chaundy: Commutative ordinary differential op-erators II. The identity P n = Q n . Proc. Roy. Soc. London A134 , 471-485(1931).[Fo] O. Forster: Lectures on Riemann surfaces. Graduate Texts in Mathe-matics . Springer, New York (1981).[G-P-R] H. Grauert, Th. Peternell, R. Remmert (eds.): Several complex vari-ables VII. Encyclopedia of Mathematical Sciences . Springer, Berlin,Heidelberg (1994).[Ha-86] R. Hartshorne: Generalized divisors on Gorenstein curves and a theoremof Noether. J. Math. Kyoto Univ. No. 3, 375-386 (1986).
URCHNALL-CHAUNDY THEORY 27 [K-L-S-S] S. Klein, E. L¨ubcke, M. U. Schmidt, T. Simon: Singular curves andBaker-Akhiezer functions. Submitted for publication. arXiv:1609.07011
E-mail address : [email protected] Mathematics Chair III, Universit¨at Mannheim, D-68131 Mannheim,Germany
E-mail address : [email protected] Mathematics Chair III, Universit¨at Mannheim, D-68131 Mannheim,Germany
E-mail address : [email protected] Mathematics Chair III, Universit¨at Mannheim, D-68131 Mannheim,Germany
E-mail address : [email protected]@mail.uni-mannheim.de