Baker-Akhiezer functions and configurations of hyperplanes

Transcription

1 Baker-Akhiezer functions and configurations of hyperplanes Alexander Veselov, Loughborough University ENIGMA conference on Geometry and Integrability, Obergurgl, December 2008

2 Plan BA function related to configuration of hyperplanes Trivial monodromy and locus configurations Coxeter configurations and deformed root systems BA function as iterated residue and Selberg-type integral Applications to Hadamard s problem Some open problems References O. Chalykh, M. Feigin, A.V. Comm. Math. Phys. 206 (1999), O. Chalykh, A.V. Phys. Letters A 285 (2001), M. Feigin, A.V. IMRN 10 (2002), G. Felder, A.V. Moscow Math J. 3 (2003), A.N. Sergeev, A.V. Comm. Math Phys 245 (2004), G. Felder, A.V. arxiv (2008)

3 Brief history Clebsch, Gordan (1860s): generalisations of the exponential function on a Riemann surface of arbitrary genus Burchnall-Chaundy, Baker (1920s): relation with commuting differential operators Akhiezer (1961): relations with spectral theory Novikov, Dubrovin, Its, Matveev ( ): relations with the theory of KdV equation and finite-gap theory Krichever (1976): general notion of BA function Chalykh, M. Feigin, Veselov (1998): BA function related to configuration of hyperplanes Motivation: links with Hadamard s problem (Berest, Veselov (1993))

4 Simplest example in dimension 1 The simplest BA function is ψ(k, x) = (1 1 kx )ekx = 1 k (k 1 x )ekx It can be defined uniquely as the function of the form with the property that when k = 0 for all x. ψ(k, x) = k a(x) e kx k (kψ(k, x)) = 0 k

5 Simplest example in dimension 1 The simplest BA function is ψ(k, x) = (1 1 kx )ekx = 1 k (k 1 x )ekx It can be defined uniquely as the function of the form ψ(k, x) = k a(x) e kx k with the property that (kψ(k, x)) = 0 k when k = 0 for all x. It satisfies the Schrödinger equation Lψ = k 2 ψ, where L = D x 2.

6 Multi-dimensional case: configurations of hyperplanes Theorem [Berest-V.] Suppose that the Schrödinger operator L = + u(x), x C n with meromorphic potential u(x) has an eigenfunction of the form ϕ(x, k) = P(k, x)e (k,x), where P(k, x) is a polynomial in k with coefficients meromorphic in x, then the singularities of the potential lie on a configuration of hyperplanes (possibly, infinite).

7 Multi-dimensional case: configurations of hyperplanes Theorem [Berest-V.] Suppose that the Schrödinger operator L = + u(x), x C n with meromorphic potential u(x) has an eigenfunction of the form ϕ(x, k) = P(k, x)e (k,x), where P(k, x) is a polynomial in k with coefficients meromorphic in x, then the singularities of the potential lie on a configuration of hyperplanes (possibly, infinite). In the rational case the potential has a form where m i Z +. u(x) = NX i=1 m i (m i + 1)(α i, α i ) ((α i, x) + c i ) 2,

8 Reflections and quasi-invariance Let A be a finite set of non-isotropic vectors α in complex Euclidean space C n with multiplicities m α N, Σ be the corresponding linear configuration of hyperplanes Π α : (α, k) = 0.

9 Reflections and quasi-invariance Let A be a finite set of non-isotropic vectors α in complex Euclidean space C n with multiplicities m α N, Σ be the corresponding linear configuration of hyperplanes Π α : (α, k) = 0. Let s α be the reflection with respect to Π α. We say that a function f (k), k C n is quasi-invariant under s α if f (s α(k)) f (k) = O((α, k) 2mα ). Equivalently, all first m α odd normal derivatives vanish on the hyperplane Π α. αf (k) = 3 αf (k) =... = 2mα 1 α f (k) 0

10 Rational BA function related to configuration of hyperplanes Definition. A function ψ(k, x), k, x C n is called rational Baker-Akhiezer function related to configuration of hyperplanes Σ if 1) ψ(k, x) has a form P(k, x) ψ(k, x) = A(k) e(k,x), where P(k, x) is a polynomial in k with highest term A(k) = Q α A (k, α)mα 2) for all α A the function ψ(k, x)(k, α) mα is quasi-invariant under s α.

11 Rational BA function related to configuration of hyperplanes Definition. A function ψ(k, x), k, x C n is called rational Baker-Akhiezer function related to configuration of hyperplanes Σ if 1) ψ(k, x) has a form P(k, x) ψ(k, x) = A(k) e(k,x), where P(k, x) is a polynomial in k with highest term A(k) = Q α A (k, α)mα 2) for all α A the function ψ(k, x)(k, α) mα is quasi-invariant under s α. Theorem [CFV]. If BA function ψ exists then it is unique, symmetric with respect to x and k and satisfies the Schrödinger equation Lψ = k 2 ψ, where L = + X α A m α(m α + 1)(α, α) (α, x) 2 is a generalised Calogero-Moser operator. Conversely, if the Schrödinger equation Lψ = k 2 ψ has a solution ψ(k, x) of the form above, then ψ(k, x) has to be BA function.

12 Quasi-invariants and Harish-Chandra homomorphism Let Q m be the algebra of quasi-invariants, consisting of polynomials f (k) satisfying αf (k) = αf 3 (k) =... = α 2mα 1 f (k) 0 on the hyperplane (α, k) = 0 for any α A.

13 Quasi-invariants and Harish-Chandra homomorphism Let Q m be the algebra of quasi-invariants, consisting of polynomials f (k) satisfying αf (k) = αf 3 (k) =... = α 2mα 1 f (k) 0 on the hyperplane (α, k) = 0 for any α A. Theorem [CFV, Berest] If the BA function ψ(k, x) exists then for any quasi-invariant f (k) Q m there exists some differential operator L f (x, x ) such that L f ψ(k, x) = f (k)ψ(k, x). The corresponding commuting operators L f L f = c N (ad L ) N [ˆf (x)], for f Q m can be given by where c N = ( 1) N /2 N N!, N = degf, ˆf is the operator of multiplication by f (x) and ad A B = AB BA.

14 Quasi-invariants and m-harmonic polynomials We have Q = S G... Q 2 Q 1 Q 0 = S(V ). Let I m Q m be the ideal generated by Casimirs σ 1,..., σ n. The joint kernel of Calogero-Moser integrals L i = L σi is G -dimensional space H m of polynomials called m-harmonics. When m = 0 they satisfy σ 1( )f = σ 2( )f = = σ n( )f = 0.

15 Quasi-invariants and m-harmonic polynomials We have Q = S G... Q 2 Q 1 Q 0 = S(V ). Let I m Q m be the ideal generated by Casimirs σ 1,..., σ n. The joint kernel of Calogero-Moser integrals L i = L σi is G -dimensional space H m of polynomials called m-harmonics. When m = 0 they satisfy σ 1( )f = σ 2( )f = = σ n( )f = 0. Feigin-V, Etingof-Ginzburg: Q m is free module over S G of rank G. The action of G on H m = Q m/i m is regular.

16 Quasi-invariants and m-harmonic polynomials We have Q = S G... Q 2 Q 1 Q 0 = S(V ). Let I m Q m be the ideal generated by Casimirs σ 1,..., σ n. The joint kernel of Calogero-Moser integrals L i = L σi is G -dimensional space H m of polynomials called m-harmonics. When m = 0 they satisfy σ 1( )f = σ 2( )f = = σ n( )f = 0. Feigin-V, Etingof-Ginzburg: Q m is free module over S G of rank G. The action of G on H m = Q m/i m is regular. When m = 0 and Weyl group G the quotient H = Q 0/I 0 = S(V )/I 0 can be interpreted as the cohomology ring of the corresponding flag manifold. Question. Is there natural topological interpretation of H m = Q m/i m? Partial results: M. Feigin-Feldman (2004)

17 Quasi-invariants and m-harmonic polynomials We have Q = S G... Q 2 Q 1 Q 0 = S(V ). Let I m Q m be the ideal generated by Casimirs σ 1,..., σ n. The joint kernel of Calogero-Moser integrals L i = L σi is G -dimensional space H m of polynomials called m-harmonics. When m = 0 they satisfy σ 1( )f = σ 2( )f = = σ n( )f = 0. Feigin-V, Etingof-Ginzburg: Q m is free module over S G of rank G. The action of G on H m = Q m/i m is regular. When m = 0 and Weyl group G the quotient H = Q 0/I 0 = S(V )/I 0 can be interpreted as the cohomology ring of the corresponding flag manifold. Question. Is there natural topological interpretation of H m = Q m/i m? Partial results: M. Feigin-Feldman (2004) Hilbert-Poincare series (Felder-V, 2003): for G = S n P(Q m, t) = n! t mn(n 1) 2 Idea: relation with KZ equation. X ny λ Y n k=1 t m(l k a k )+l k 1 h k (1 t h k ).

18 Monodromy-free Schrödinger operators Consider first 1D Schrödinger operator L = D 2 + u(z), D = d dz, where u = u(z) is a meromorphic function of z C.

19 Monodromy-free Schrödinger operators Consider first 1D Schrödinger operator L = D 2 + u(z), D = d dz, where u = u(z) is a meromorphic function of z C. Definition. Operator L is called monodromy free if all solutions of the corresponding Schrödinger equation in the complex domain ψ + u(z)ψ = kψ are meromorphic (and hence single-valued) for all k C.

20 Monodromy-free Schrödinger operators Consider first 1D Schrödinger operator L = D 2 + u(z), D = d dz, where u = u(z) is a meromorphic function of z C. Definition. Operator L is called monodromy free if all solutions of the corresponding Schrödinger equation in the complex domain ψ + u(z)ψ = kψ are meromorphic (and hence single-valued) for all k C. Theorem [Duistermaat Grünbaum]. A Schrödinger operator L is monodromy-free iff all poles of u are of second order and the coefficients of its Laurent expansion u(z) = c 2 (z z + X c n(z z 0) n 0) 2 around any pole z 0 satisfy the conditions n 1 i) c 2 = m(m + 1) for a positive integer m and ii) c 2k 1 = 0 for k = 0,..., m.

21 Locus configurations and BA function A configuration of hyperplanes Σ is called a locus configuration if the corresponding potential u Σ (x) = X α A satisfies the quasi-invariance conditions m α(m α + 1)(α, α) (α, x) 2 u Σ (x) u Σ (s i (x)) = O((α, x) 2mα ) for all α A. This is a condition that the Schrödinger operator L = + u Σ has trivial monodromy.

22 Locus configurations and BA function A configuration of hyperplanes Σ is called a locus configuration if the corresponding potential u Σ (x) = X α A satisfies the quasi-invariance conditions m α(m α + 1)(α, α) (α, x) 2 u Σ (x) u Σ (s i (x)) = O((α, x) 2mα ) for all α A. This is a condition that the Schrödinger operator L = + u Σ has trivial monodromy. Theorem [CFV]. The BA function exists iff the corresponding Σ is a locus configuration. In that case it can be given by the Berest formula ψ(k, x) = [( 2) M M!A(k)] 1 (L + k 2 ) M [ Y α A(α, x) mα exp(k, x)], where M = P α A mα, A(k) = Q α A (α, k)mα

23 Examples of locus configurations Coxeter configurations: reflection hyperplanes of a Coxeter group G taken with integer G-invariant multiplicities

24 Examples of locus configurations Coxeter configurations: reflection hyperplanes of a Coxeter group G taken with integer G-invariant multiplicities Deformed root systems [CFV] j ei e j with multiplicity m A n,1(m) = e i me n+1 with multiplicity 1 nx L (n,1) 2 nx m = m 2 xi 2 y + 2m(m + 1) nx 2(m + 1) + 2 (x i x j ) 2 (x i y) 2 i=1 i<j i=1

25 Examples of locus configurations Coxeter configurations: reflection hyperplanes of a Coxeter group G taken with integer G-invariant multiplicities Deformed root systems [CFV] j ei e j with multiplicity m A n,1(m) = e i me n+1 with multiplicity 1 nx L (n,1) 2 nx m = m 2 xi 2 y + 2m(m + 1) nx 2(m + 1) + 2 (x i x j ) 2 (x i y) 2 i=1 8 e i ± e j with multiplicity k >< 2e i with multiplicity m C n,1(m, l) = 2 ke n+1 with multiplicity l >: e i ± ke n+1 with multiplicity 1 where l and m are integer parameters such that k = 2m+1 2l+1 i<j i=1 Z, 1 i < j n.

26 2D examples A 2 (m) 1 1 C 2 (m,l) 1 l 1 θ cosθ = m m m +1 ϕ m m l cos2ϕ = m +l+1

27 Other locus configurations Berest-Lutsenko configurations are 2D configurations with the potential where u(r, ϕ) = 1 r 2 2 ϕ 2 log W (χ1,..., χ N), χ i (ϕ) = cos(k i ϕ + θ i ), i = 1,..., n, where 0 < k 1 < < k N are positive integers, θ i C are arbitrary complex parameters and W denotes the Wronskian. They give all linear locus configurations in dimension 2.

28 Other locus configurations Berest-Lutsenko configurations are 2D configurations with the potential where u(r, ϕ) = 1 r 2 2 ϕ 2 log W (χ1,..., χ N), χ i (ϕ) = cos(k i ϕ + θ i ), i = 1,..., n, where 0 < k 1 < < k N are positive integers, θ i C are arbitrary complex parameters and W denotes the Wronskian. They give all linear locus configurations in dimension 2. A special complex series [Chalykh-V]: 8 e i e j, 1 i < j n, with multiplicity m, >< e i me n+1, i = 1,..., n with multiplicity 1, A n 1,2(m) = e i 1 me n+2, i = 1,..., n with multiplicity 1, >: men+1 1 me n+2 with multiplicity 1.

29 BA function as iterated residue Felder-V: The rational Baker-Akhiezer function for the configuration A n(m) can be given by the following iterated residue formula where ω m = «n(n 1) I ψ m (n) m! 2 (x, k) = e kn(x 1+ +x n) A(x) 1+m A(k) m ω m, 2πi Σ Y i j,l j+1 n (t i,j t l,j+1 ) m 1 Y 1 i<l j<n (t i,j t l,j ) 2+2m Y l j<n e (k j k j+1 )t l,j dt l,j, Σ as the product of circles t k,j x k = ɛ(n j) with ɛ small enough and t k,n = x k. Based on Stanley identity: cf. Awata et al, Kazarnovski-Krol, Okounkov-Olshanski, Kuznetsov-Mangazeev-Sklyanin, Langmann

30 BA function as Selberg-type integral The rational BA function can be given by the following Selberg-type integral Z ψ m (n) (x, k) = (( 1) m+1 m!) n(n 1) 2 e kn(x 1+ +x n) A(x) m A(k) m+1 α m with α m = Y i j,l j+1 n (t i,j t l,j+1 ) m Y 1 i<l j<n (t i,j t l,j ) 2m Y l j<n Γ e (k j k j+1 )t l,j dt l,j and the integration contour Γ such that t i,j = t i,j+1 + τ i,j with real variables τ i,j, 1 i j = 1,..., n 1 changing from zero to infinity.

31 BA function as Selberg-type integral The rational BA function can be given by the following Selberg-type integral Z ψ m (n) (x, k) = (( 1) m+1 m!) n(n 1) 2 e kn(x 1+ +x n) A(x) m A(k) m+1 α m with α m = Y i j,l j+1 n (t i,j t l,j+1 ) m Y 1 i<l j<n (t i,j t l,j ) 2m Y l j<n Γ e (k j k j+1 )t l,j dt l,j and the integration contour Γ such that t i,j = t i,j+1 + τ i,j with real variables τ i,j, 1 i j = 1,..., n 1 changing from zero to infinity. Remark 1. This may be considered as a new case of explicit calculation of Selberg-type integrals.

32 BA function as Selberg-type integral The rational BA function can be given by the following Selberg-type integral Z ψ m (n) (x, k) = (( 1) m+1 m!) n(n 1) 2 e kn(x 1+ +x n) A(x) m A(k) m+1 α m with α m = Y i j,l j+1 n (t i,j t l,j+1 ) m Y 1 i<l j<n (t i,j t l,j ) 2m Y l j<n Γ e (k j k j+1 )t l,j dt l,j and the integration contour Γ such that t i,j = t i,j+1 + τ i,j with real variables τ i,j, 1 i j = 1,..., n 1 changing from zero to infinity. Remark 1. This may be considered as a new case of explicit calculation of Selberg-type integrals. Remark 2. These two representations can be related by an analytic continuation from m to m 1, which is very similar to the Riemann s proof of the reflection property of the Riemann zeta-function.

33 Example: two-particle case In that case the rational BA function is known to be Ψ (2) m = (k 1 k 2) m (D 2m 2(m 1) 12 )(D 12 )... (D 2 12 ) exp(k 1x 1+k 2x 2), x 1 x 2 x 1 x 2 x 1 x 2 where D 12 =. x 1 x 2

34 Example: two-particle case In that case the rational BA function is known to be Ψ (2) m = (k 1 k 2) m (D 2m 2(m 1) 12 )(D 12 )... (D 2 12 ) exp(k 1x 1+k 2x 2), x 1 x 2 x 1 x 2 x 1 x 2 where D 12 =. x 1 x 2 We have two different representations for it. The first one is as a residue Ψ (2) m = the second one is the integral Ψ (2) m = m!(x1 x2)m+1 (k 1 k 2) m e k 2(x 1 +x 2 ) Res z=x1 e (k1 k2)z (z x 1) m+1 (z x 2) m+1, (k2 k1)m+1 m!(x 1 x 2) m ek 2(x 1 +x 2 ) Z + x 1 (z x 1) m (z x 2) m e (k 1 k 2)z dz, which in this case can be effectively computed using the Γ-integral Γ(a) = Z + 0 z a 1 e z dz = (a 1)!

35 Application: Huygens principle and Hadamard s problem Huygens Principle in the narrow sense: an instantaneous signal remains instantaneous for every observer at each later time. Mathematically: the fundamental solution of the corresponding hyperbolic equation is located on the characteristic conoid.

36 Application: Huygens principle and Hadamard s problem Huygens Principle in the narrow sense: an instantaneous signal remains instantaneous for every observer at each later time. Mathematically: the fundamental solution of the corresponding hyperbolic equation is located on the characteristic conoid. Example: pure wave equation in R n : n(φ) = 0, n = n, i = x i, x 0 = t. Huygens Principle holds only in odd dimensions starting from 3. Fundamental solution in that case is Φ = C(n)δ (k) (t 2 x 2 ), k = n 3. 2

37 Hadamard s problem Describe all second-order hyperbolic equations for which Huygens Principle holds. Special case: hyperbolic equations of the form ( n + u(x))φ = 0.

38 Hadamard s problem Describe all second-order hyperbolic equations for which Huygens Principle holds. Special case: hyperbolic equations of the form ( n + u(x))φ = Hadamard: Dimension n must be odd and larger than 1. Hadamard s Conjecture : HP holds only for pure wave equations

39 Hadamard s problem Describe all second-order hyperbolic equations for which Huygens Principle holds. Special case: hyperbolic equations of the form ( n + u(x))φ = Hadamard: Dimension n must be odd and larger than 1. Hadamard s Conjecture : HP holds only for pure wave equations Mathisson, Asgeirsson, Hadamard: If n = 3 then u must be zero.

40 Hadamard s problem Describe all second-order hyperbolic equations for which Huygens Principle holds. Special case: hyperbolic equations of the form ( n + u(x))φ = Hadamard: Dimension n must be odd and larger than 1. Hadamard s Conjecture : HP holds only for pure wave equations Mathisson, Asgeirsson, Hadamard: If n = 3 then u must be zero Stellmacher: If u = m(m+1) with integer m then HP holds in x1 2 any odd dimension starting from 2m + 3.

41 Hadamard s problem Describe all second-order hyperbolic equations for which Huygens Principle holds. Special case: hyperbolic equations of the form ( n + u(x))φ = Hadamard: Dimension n must be odd and larger than 1. Hadamard s Conjecture : HP holds only for pure wave equations Mathisson, Asgeirsson, Hadamard: If n = 3 then u must be zero Stellmacher: If u = m(m+1) with integer m then HP holds in x1 2 any odd dimension starting from 2m Stellmacher and Lagnese: solution of the Hadamard problem in the class ( + u(x 1))φ = 0

42 Hadamard s problem Describe all second-order hyperbolic equations for which Huygens Principle holds. Special case: hyperbolic equations of the form ( n + u(x))φ = Hadamard: Dimension n must be odd and larger than 1. Hadamard s Conjecture : HP holds only for pure wave equations Mathisson, Asgeirsson, Hadamard: If n = 3 then u must be zero Stellmacher: If u = m(m+1) with integer m then HP holds in x1 2 any odd dimension starting from 2m Stellmacher and Lagnese: solution of the Hadamard problem in the class ( + u(x 1))φ = Berest-V: examples related to Coxeter groups

43 Main result Theorem [CFV] Hyperbolic equation ( + u(x))φ = 0 with the potential u(x) = KX j=1 m j (m j + 1)(α j, α j ) ((α j, x) + c j ) 2 related to any locus configuration satisfies HP if n is odd and large enough: n 2M + 3, M = P K j=1 m j.

44 Main result Theorem [CFV] Hyperbolic equation ( + u(x))φ = 0 with the potential u(x) = KX j=1 m j (m j + 1)(α j, α j ) ((α j, x) + c j ) 2 related to any locus configuration satisfies HP if n is odd and large enough: n 2M + 3, M = P K j=1 m j. Conversely, if the equation ( + u(x))φ = 0 satisfies HP and all the Hadamard s coefficients are rational functions, then the potential u(x) must be related to locus configuration.

45 Some open problems

46 Some open problems Classification of locus configurations Partial results: CFV, Sergeev-V

47 Some open problems Classification of locus configurations Partial results: CFV, Sergeev-V Effective description of quasi-invariants and m-harmonic polynomials Partial results: Feigin-V, Felder-V, Etingof-Ginzburg

48 Some open problems Classification of locus configurations Partial results: CFV, Sergeev-V Effective description of quasi-invariants and m-harmonic polynomials Partial results: Feigin-V, Felder-V, Etingof-Ginzburg Elliptic case: generalised Lamè operators Partial results: Chalykh-Etingof-Oblomkov

49 Some open problems Classification of locus configurations Partial results: CFV, Sergeev-V Effective description of quasi-invariants and m-harmonic polynomials Partial results: Feigin-V, Felder-V, Etingof-Ginzburg Elliptic case: generalised Lamè operators Partial results: Chalykh-Etingof-Oblomkov Spectral theory of the deformed Calogero-Moser systems