Algebraic aspects of the Riemann-Hilbert problem for matrix orthogonal polynomials

Algebraic aspects of the Riemann-Hilbert problem for matrix orthogonal polynomials1 Manuel Dom´ınguez de la Iglesia Departamento de An´alisis Matem´atico, Universidad de Sevilla International Symposium on Orthogonal Polynomials and Special Functions: a Complex Analytic Perspective Copenhague, June 12, 2012 1joint work with F. A. Gr¨unbaum and A. Mart´ınez-Finkelshtein The RH problem for OP The RH problem for MOP Outline 1The Riemann-Hilbert problem for orthogonal polynomials The RHP for OPs The Lax pair Examples 2The Riemann-Hilbert problem for matrix orthogonal polynomials The RHP for MOPs The Lax pair Examples Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for OPs The Lax pair Examples Outline 1The Riemann-Hilbert problem for orthogonal polynomials The RHP for OPs The Lax pair Examples 2The Riemann-Hilbert problem for matrix orthogonal polynomials The RHP for MOPs The Lax pair Examples Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for OPs The Lax pair Examples Orthogonal polynomials Let dµbe a positive Borel measure supported on R. We will assume dµ(x) = ω(x)dx,ω≥0 and xiω, xjω0∈L1(R). We can then construct a family of orthonormal polynomials (pn)ns.t. (pn,pm)ω=ZR pn(x)pm(x)ω(x)dx =δn,m,n,m≥0 pn(x) = κn(xn+an,n−1xn−1+· · · ) = κnb pn(x) The monic polynomials b pn(x) satisfy a three-term recurrence relation xb pn(x) = b pn+1(x) + αnb pn(x) + βnb pn−1(x) Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for OPs The Lax pair Examples Orthogonal polynomials Let dµbe a positive Borel measure supported on R. We will assume dµ(x) = ω(x)dx,ω≥0 and xiω, xjω0∈L1(R). We can then construct a family of orthonormal polynomials (pn)ns.t. (pn,pm)ω=ZR pn(x)pm(x)ω(x)dx =δn,m,n,m≥0 pn(x) = κn(xn+an,n−1xn−1+· · · ) = κnb pn(x) The monic polynomials b pn(x) satisfy a three-term recurrence relation xb pn(x) = b pn+1(x) + αnb pn(x) + βnb pn−1(x) Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for OPs The Lax pair Examples Orthogonal polynomials Let dµbe a positive Borel measure supported on R. We will assume dµ(x) = ω(x)dx,ω≥0 and xiω, xjω0∈L1(R). We can then construct a family of orthonormal polynomials (pn)ns.t. (pn,pm)ω=ZR pn(x)pm(x)ω(x)dx =δn,m,n,m≥0 pn(x) = κn(xn+an,n−1xn−1+· · · ) = κnb pn(x) The monic polynomials b pn(x) satisfy a three-term recurrence relation xb pn(x) = b pn+1(x) + αnb pn(x) + βnb pn−1(x) Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for OPs The Lax pair Examples Solution of the RHP for orthogonal polynomials We try to find a 2 ×2 matrix-valued function Yn:C→C2×2such that 1Ynis analytic in C\R 2Yn +(x) = Yn −(x)1ω(x) 0 1 when x∈R 3Yn(z)=(I2+O(1/z)) zn0 0z−nas z→ ∞ For n≥1 the unique solution of the RHP above is given by Fokas-Its-Kitaev, 1990 Yn(z) = b pn(z)C(b pnω)(z) −2πiγn−1b pn−1(z)−2πiγn−1C(b pn−1ω)(z) where C(f)(z) = 1 2πiRR f(t) t−zdt is the Cauchy transform and γn=κ2 n. The existence and unicity is a consequence of the Morera’s theorem, Liouville’s theorem and detYn(z) = 1. Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for OPs The Lax pair Examples Solution of the RHP for orthogonal polynomials We try to find a 2 ×2 matrix-valued function Yn:C→C2×2such that 1Ynis analytic in C\R 2Yn +(x) = Yn −(x)1ω(x) 0 1 when x∈R 3Yn(z)=(I2+O(1/z)) zn0 0z−nas z→ ∞ For n≥1 the unique solution of the RHP above is given by Fokas-Its-Kitaev, 1990 Yn(z) = b pn(z)C(b pnω)(z) −2πiγn−1b pn−1(z)−2πiγn−1C(b pn−1ω)(z) where C(f)(z) = 1 2πiRR f(t) t−zdt is the Cauchy transform and γn=κ2 n. The existence and unicity is a consequence of the Morera’s theorem, Liouville’s theorem and detYn(z) = 1. Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for OPs The Lax pair Examples Solution of the RHP for orthogonal polynomials We try to find a 2 ×2 matrix-valued function Yn:C→C2×2such that 1Ynis analytic in C\R 2Yn +(x) = Yn −(x)1ω(x) 0 1 when x∈R 3Yn(z)=(I2+O(1/z)) zn0 0z−nas z→ ∞ For n≥1 the unique solution of the RHP above is given by Fokas-Its-Kitaev, 1990 Yn(z) = b pn(z)C(b pnω)(z) −2πiγn−1b pn−1(z)−2πiγn−1C(b pn−1ω)(z) where C(f)(z) = 1 2πiRR f(t) t−zdt is the Cauchy transform and γn=κ2 n. The existence and unicity is a consequence of the Morera’s theorem, Liouville’s theorem and detYn(z) = 1. Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for OPs The Lax pair Examples The Lax pair I We look for a pair of first-order difference/differential equations of the form Yn+1(z) = En(z)Yn(z),d dz Yn(z) = Fn(z)Yn(z) Problem. Typically, the coefficient Fn(z) is difficult to obtain. We can avoid that by transforming the RHP in another RHP with constant jump. Consider the transformation Xn(z) = Yn(z)ω1/20 0ω−1/2 We observe that Xnis invertible and that Xn +(x) = Xn −(x)1 1 0 1 That means that Xnhas a constant jump ⇒En(z) and Fn(z) are completely determined by their behavior at z→ ∞. Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for OPs The Lax pair Examples The Lax pair II If we additionally assume that (ω1/2)0 ω1/2is a polynomial of degree m, then Xn+1(z) = z−αn1 2πiγ−1 n −2πiγn0 | {z } En(z) Xn(z) d dz Xn(z) = −Bn(z)−1 2πiγ−1 nAn(z) 2πiAn−1(z)γn−1Bn(z) | {z } Fn(z) Xn(z) where An(z) and Bn(z) are polynomials of degree m−1 and mrespectively. Cross-differentiating the Lax pair yield Compatibility conditions E0 n(z) + En(z)Fn(z) = Fn+1(z)En(z) also known as string equations. Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for OPs The Lax pair Examples The Lax pair II If we additionally assume that (ω1/2)0 ω1/2is a polynomial of degree m, then Xn+1(z) = z−αn1 2πiγ−1 n −2πiγn0 | {z } En(z) Xn(z) d dz Xn(z) = −Bn(z)−1 2πiγ−1 nAn(z) 2πiAn−1(z)γn−1Bn(z) | {z } Fn(z) Xn(z) where An(z) and Bn(z) are polynomials of degree m−1 and mrespectively. Cross-differentiating the Lax pair yield Compatibility conditions E0 n(z) + En(z)Fn(z) = Fn+1(z)En(z) also known as string equations. Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for OPs The Lax pair Examples Example I: Hermite polynomials Consider ω(x) = e−x2⇒Hermite polynomials (Hn)n. The transformation Xn(z) = Yn(z) e−z2/20 0ez2/2!gives the following Lax pair Xn+1(z) = z1 2πiγ−1 n −2πiγn0Xn(z),d dz Xn(z) = −z−1 πiγ−1 n 4πiγn−1zXn(z) The difference equation gives (using βn=γn/γn+1) the TTRR xb Hn(x) = b Hn+1(x) + βnb Hn−1(x), while the differential equation gives the ladder operators b H0 n(x) = 2βnb Hn−1(x),b H0 n(x)−2xb Hn(x) = −2b Hn+1(x). The compatibility conditions are βn+1 −βn=1 2⇒βn=n 2 Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for OPs The Lax pair Examples Example I: Hermite polynomials Consider ω(x) = e−x2⇒Hermite polynomials (Hn)n. The transformation Xn(z) = Yn(z) e−z2/20 0ez2/2!gives the following Lax pair Xn+1(z) = z1 2πiγ−1 n −2πiγn0Xn(z),d dz Xn(z) = −z−1 πiγ−1 n 4πiγn−1zXn(z) The difference equation gives (using βn=γn/γn+1) the TTRR xb Hn(x) = b Hn+1(x) + βnb Hn−1(x), while the differential equation gives the ladder operators b H0 n(x) = 2βnb Hn−1(x),b H0 n(x)−2xb Hn(x) = −2b Hn+1(x). The compatibility conditions are βn+1 −βn=1 2⇒βn=n 2 Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for OPs The Lax pair Examples Example I: Hermite polynomials Consider ω(x) = e−x2⇒Hermite polynomials (Hn)n. The transformation Xn(z) = Yn(z) e−z2/20 0ez2/2!gives the following Lax pair Xn+1(z) = z1 2πiγ−1 n −2πiγn0Xn(z),d dz Xn(z) = −z−1 πiγ−1 n 4πiγn−1zXn(z) The difference equation gives (using βn=γn/γn+1) the TTRR xb Hn(x) = b Hn+1(x) + βnb Hn−1(x), while the differential equation gives the ladder operators b H0 n(x) = 2βnb Hn−1(x),b H0 n(x)−2xb Hn(x) = −2b Hn+1(x). The compatibility conditions are βn+1 −βn=1 2⇒βn=n 2 Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for OPs The Lax pair Examples Example I: Hermite polynomials Consider ω(x) = e−x2⇒Hermite polynomials (Hn)n. The transformation Xn(z) = Yn(z) e−z2/20 0ez2/2!gives the following Lax pair Xn+1(z) = z1 2πiγ−1 n −2πiγn0Xn(z),d dz Xn(z) = −z−1 πiγ−1 n 4πiγn−1zXn(z) The difference equation gives (using βn=γn/γn+1) the TTRR xb Hn(x) = b Hn+1(x) + βnb Hn−1(x), while the differential equation gives the ladder operators b H0 n(x) = 2βnb Hn−1(x),b H0 n(x)−2xb Hn(x) = −2b Hn+1(x). The compatibility conditions are βn+1 −βn=1 2⇒βn=n 2 Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for OPs The Lax pair Examples Example II: Freud orthogonal polynomials Consider ω(x) = e−x4⇒Freud polynomials (Pn)n. Xn(z) = Yn(z) e−z4/20 0ez4/2!satisfies the following Lax pair Xn+1(z) = z1 2πiγ−1 n −2πiγn0Xn(z) d dz Xn(z) = −2z3−4βnz−2 πiγ−1 n(z2+βn+βn+1) 8πiγn−1(z2+βn+βn+1) 2z3+ 4βnzXn(z) The ladder operators are b P0 n(x)+4βnxb Pn(x) = 4(x2+βn+βn+1)βnb Pn−1(x) b P0 n(x)+4x3b Pn(x) = −4(x2+βn+βn+1)b Pn+1(x) The compatibility conditions are n= 4βn(βn+1 +βn+βn−1) Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for OPs The Lax pair Examples Example II: Freud orthogonal polynomials Consider ω(x) = e−x4⇒Freud polynomials (Pn)n. Xn(z) = Yn(z) e−z4/20 0ez4/2!satisfies the following Lax pair Xn+1(z) = z1 2πiγ−1 n −2πiγn0Xn(z) d dz Xn(z) = −2z3−4βnz−2 πiγ−1 n(z2+βn+βn+1) 8πiγn−1(z2+βn+βn+1) 2z3+ 4βnzXn(z) The ladder operators are b P0 n(x)+4βnxb Pn(x) = 4(x2+βn+βn+1)βnb Pn−1(x) b P0 n(x)+4x3b Pn(x) = −4(x2+βn+βn+1)b Pn+1(x) The compatibility conditions are n= 4βn(βn+1 +βn+βn−1) Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for OPs The Lax pair Examples Example II: Freud orthogonal polynomials Consider ω(x) = e−x4⇒Freud polynomials (Pn)n. Xn(z) = Yn(z) e−z4/20 0ez4/2!satisfies the following Lax pair Xn+1(z) = z1 2πiγ−1 n −2πiγn0Xn(z) d dz Xn(z) = −2z3−4βnz−2 πiγ−1 n(z2+βn+βn+1) 8πiγn−1(z2+βn+βn+1) 2z3+ 4βnzXn(z) The ladder operators are b P0 n(x)+4βnxb Pn(x) = 4(x2+βn+βn+1)βnb Pn−1(x) b P0 n(x)+4x3b Pn(x) = −4(x2+βn+βn+1)b Pn+1(x) The compatibility conditions are n= 4βn(βn+1 +βn+βn−1) Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples Solution of the RHP for MOP Yn:C→C2N×2Nsuch that 1Ynis analytic in C\R 2Yn +(x) = Yn −(x)INW(x) 0 INwhen x∈R 3Yn(z)=(I2N+O(1/z)) znIN0 0z−nINas z→ ∞ For n≥1 the unique solution of the RH problem above is given by Yn(z) = b Pn(z)C(b PnW)(z) −2πiγn−1b Pn−1(z)−2πiγn−1C(b Pn−1W)(z)! where C(F)(z) = 1 2πiRR F(t) t−zdt and γn=κ∗ nκn. Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples Solution of the RHP for MOP Yn:C→C2N×2Nsuch that 1Ynis analytic in C\R 2Yn +(x) = Yn −(x)INW(x) 0 INwhen x∈R 3Yn(z)=(I2N+O(1/z)) znIN0 0z−nINas z→ ∞ For n≥1 the unique solution of the RH problem above is given by Yn(z) = b Pn(z)C(b PnW)(z) −2πiγn−1b Pn−1(z)−2πiγn−1C(b Pn−1W)(z)! where C(F)(z) = 1 2πiRR F(t) t−zdt and γn=κ∗ nκn. Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples The Lax pair I We look for a pair of first-order difference/differential equations of the form Yn+1(z) = En(z)Yn(z),d dz Yn(z) = Fn(z)Yn(z) Goal: obtain an invertible transformation Yn→Xnsuch that Xnhas a constant jump across R. Consider Xn(z) = Yn(z)V(z) where V(z) = T(z)0 0 T−∗(z) where Tis an invertible N×Nsmooth matrix function. This motivates to consider a factorization of the weight in the form W(x) = T(x)T∗(x),x∈R. This factorization is not unique since T(x) = b T(x)S(x),x∈R b T(x) is upper triangular and S(x) is an arbitrary smooth and unitary matrix. Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples The Lax pair I We look for a pair of first-order difference/differential equations of the form Yn+1(z) = En(z)Yn(z),d dz Yn(z) = Fn(z)Yn(z) Goal: obtain an invertible transformation Yn→Xnsuch that Xnhas a constant jump across R. Consider Xn(z) = Yn(z)V(z) where V(z) = T(z)0 0 T−∗(z) where Tis an invertible N×Nsmooth matrix function. This motivates to consider a factorization of the weight in the form W(x) = T(x)T∗(x),x∈R. This factorization is not unique since T(x) = b T(x)S(x),x∈R b T(x) is upper triangular and S(x) is an arbitrary smooth and unitary matrix. Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples The Lax pair I We look for a pair of first-order difference/differential equations of the form Yn+1(z) = En(z)Yn(z),d dz Yn(z) = Fn(z)Yn(z) Goal: obtain an invertible transformation Yn→Xnsuch that Xnhas a constant jump across R. Consider Xn(z) = Yn(z)V(z) where V(z) = T(z)0 0 T−∗(z) where Tis an invertible N×Nsmooth matrix function. This motivates to consider a factorization of the weight in the form W(x) = T(x)T∗(x),x∈R. This factorization is not unique since T(x) = b T(x)S(x),x∈R b T(x) is upper triangular and S(x) is an arbitrary smooth and unitary matrix. Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples The Lax pair I We look for a pair of first-order difference/differential equations of the form Yn+1(z) = En(z)Yn(z),d dz Yn(z) = Fn(z)Yn(z) Goal: obtain an invertible transformation Yn→Xnsuch that Xnhas a constant jump across R. Consider Xn(z) = Yn(z)V(z) where V(z) = T(z)0 0 T−∗(z) where Tis an invertible N×Nsmooth matrix function. This motivates to consider a factorization of the weight in the form W(x) = T(x)T∗(x),x∈R. This factorization is not unique since T(x) = b T(x)S(x),x∈R b T(x) is upper triangular and S(x) is an arbitrary smooth and unitary matrix. Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples The Lax pair II We additionally assume T0(z) = G(z)T(z), where Gis a matrix polynomial of degree m(most of our examples) Yn+1(z) = z−αn1 2πiγ−1 n −2πiγn0 | {z } En(z;G) Yn(z) d dz Yn(z) = −Bn(z;G)−1 2πiγ−1 nAn(z;G) 2πiAn−1(z;G)γn−1B∗ n(z;G) | {z } Fn(z;G) Yn(z) where Anand Bnare matrix polynomials of degree m−1 and mrespectively. Cross-differentiating the Lax pair yield the compatibility conditions E0 n(z;G) + En(z;G)Fn(z;G) = Fn+1(z;G)En(z;G) Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples The Lax pair II We additionally assume T0(z) = G(z)T(z), where Gis a matrix polynomial of degree m(most of our examples) Yn+1(z) = z−αn1 2πiγ−1 n −2πiγn0 | {z } En(z;G) Yn(z) d dz Yn(z) = −Bn(z;G)−1 2πiγ−1 nAn(z;G) 2πiAn−1(z;G)γn−1B∗ n(z;G) | {z } Fn(z;G) Yn(z) where Anand Bnare matrix polynomials of degree m−1 and mrespectively. Cross-differentiating the Lax pair yield the compatibility conditions E0 n(z;G) + En(z;G)Fn(z;G) = Fn+1(z;G)En(z;G) Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples The Lax pair II We additionally assume T0(z) = G(z)T(z), where Gis a matrix polynomial of degree m(most of our examples) Yn+1(z) = z−αn1 2πiγ−1 n −2πiγn0 | {z } En(z;G) Yn(z) d dz Yn(z) = −Bn(z;G)−1 2πiγ−1 nAn(z;G) 2πiAn−1(z;G)γn−1B∗ n(z;G) | {z } Fn(z;G) Yn(z) where Anand Bnare matrix polynomials of degree m−1 and mrespectively. Cross-differentiating the Lax pair yield the compatibility conditions E0 n(z;G) + En(z;G)Fn(z;G) = Fn+1(z;G)En(z;G) Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples The Lax pair III If there exists a non-trivial matrix-valued function S, non-singular on C, smooth and unitary on R, s.t. H(z) = T(z)S0(z)S∗(z)T−1(z) is also a polynomial, then e T=TS satisfies W(x) = e T(x)e T ∗(x),x∈R,e T 0(z) = e G(z)e T(z),z∈C, with e G(z) = G(z) + H(z) and the matrix Xnsatisfies d dz Xn(z) = Fn(z;G)Xn(z) + Fn(z;H)Xn(z)−Xn(z)χ(z)0 0−χ∗(z) with χ(z) = S0(z)S∗(z). Consequences: We have a class of ladder operators. Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples In order to use the freedom in the matrix case by a unitary matrix function Swe have to impose additional constraints on the weight W. The matrix Hcan be written as H(x) = eAxχe−Ax=χ+ adA(χ)x+ ad2 A(χ)x2 2+· · · , where χ(x) = S0(x)S∗(x) is skew-Hermitian on R. This matrix equation was considered already by Dur´an-Gr¨unbaum (2004), when χis a constant matrix. If deg H= 0 then χ=iaIN,a∈R⇒No new ladder operators. If deg H= 1 then 1A=L= N X i=1 νiEi,i+1,and χ=iJ=i N X i=1 (N−i)Ei,i ⇒adA(χ) = −Aand S(x) = eiJx 2A=L(IN+L)−1,and χ=iJ ⇒adA(χ) = −A+A2and S(x) = eiJx Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples In order to use the freedom in the matrix case by a unitary matrix function Swe have to impose additional constraints on the weight W. The matrix Hcan be written as H(x) = eAxχe−Ax=χ+ adA(χ)x+ ad2 A(χ)x2 2+· · · , where χ(x) = S0(x)S∗(x) is skew-Hermitian on R. This matrix equation was considered already by Dur´an-Gr¨unbaum (2004), when χis a constant matrix. If deg H= 0 then χ=iaIN,a∈R⇒No new ladder operators. If deg H= 1 then 1A=L= N X i=1 νiEi,i+1,and χ=iJ=i N X i=1 (N−i)Ei,i ⇒adA(χ) = −Aand S(x) = eiJx 2A=L(IN+L)−1,and χ=iJ ⇒adA(χ) = −A+A2and S(x) = eiJx Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples In order to use the freedom in the matrix case by a unitary matrix function Swe have to impose additional constraints on the weight W. The matrix Hcan be written as H(x) = eAxχe−Ax=χ+ adA(χ)x+ ad2 A(χ)x2 2+· · · , where χ(x) = S0(x)S∗(x) is skew-Hermitian on R. This matrix equation was considered already by Dur´an-Gr¨unbaum (2004), when χis a constant matrix. If deg H= 0 then χ=iaIN,a∈R⇒No new ladder operators. If deg H= 1 then 1A=L= N X i=1 νiEi,i+1,and χ=iJ=i N X i=1 (N−i)Ei,i ⇒adA(χ) = −Aand S(x) = eiJx 2A=L(IN+L)−1,and χ=iJ ⇒adA(χ) = −A+A2and S(x) = eiJx Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples In order to use the freedom in the matrix case by a unitary matrix function Swe have to impose additional constraints on the weight W. The matrix Hcan be written as H(x) = eAxχe−Ax=χ+ adA(χ)x+ ad2 A(χ)x2 2+· · · , where χ(x) = S0(x)S∗(x) is skew-Hermitian on R. This matrix equation was considered already by Dur´an-Gr¨unbaum (2004), when χis a constant matrix. If deg H= 0 then χ=iaIN,a∈R⇒No new ladder operators. If deg H= 1 then 1A=L= N X i=1 νiEi,i+1,and χ=iJ=i N X i=1 (N−i)Ei,i ⇒adA(χ) = −Aand S(x) = eiJx 2A=L(IN+L)−1,and χ=iJ ⇒adA(χ) = −A+A2and S(x) = eiJx Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples In order to use the freedom in the matrix case by a unitary matrix function Swe have to impose additional constraints on the weight W. The matrix Hcan be written as H(x) = eAxχe−Ax=χ+ adA(χ)x+ ad2 A(χ)x2 2+· · · , where χ(x) = S0(x)S∗(x) is skew-Hermitian on R. This matrix equation was considered already by Dur´an-Gr¨unbaum (2004), when χis a constant matrix. If deg H= 0 then χ=iaIN,a∈R⇒No new ladder operators. If deg H= 1 then 1A=L= N X i=1 νiEi,i+1,and χ=iJ=i N X i=1 (N−i)Ei,i ⇒adA(χ) = −Aand S(x) = eiJx 2A=L(IN+L)−1,and χ=iJ ⇒adA(χ) = −A+A2and S(x) = eiJx Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples In order to use the freedom in the matrix case by a unitary matrix function Swe have to impose additional constraints on the weight W. The matrix Hcan be written as H(x) = eAxχe−Ax=χ+ adA(χ)x+ ad2 A(χ)x2 2+· · · , where χ(x) = S0(x)S∗(x) is skew-Hermitian on R. This matrix equation was considered already by Dur´an-Gr¨unbaum (2004), when χis a constant matrix. If deg H= 0 then χ=iaIN,a∈R⇒No new ladder operators. If deg H= 1 then 1A=L= N X i=1 νiEi,i+1,and χ=iJ=i N X i=1 (N−i)Ei,i ⇒adA(χ) = −Aand S(x) = eiJx 2A=L(IN+L)−1,and χ=iJ ⇒adA(χ) = −A+A2and S(x) = eiJx Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples In order to use the freedom in the matrix case by a unitary matrix function Swe have to impose additional constraints on the weight W. The matrix Hcan be written as H(x) = eAxχe−Ax=χ+ adA(χ)x+ ad2 A(χ)x2 2+· · · , where χ(x) = S0(x)S∗(x) is skew-Hermitian on R. This matrix equation was considered already by Dur´an-Gr¨unbaum (2004), when χis a constant matrix. If deg H= 0 then χ=iaIN,a∈R⇒No new ladder operators. If deg H= 1 then 1A=L= N X i=1 νiEi,i+1,and χ=iJ=i N X i=1 (N−i)Ei,i ⇒adA(χ) = −Aand S(x) = eiJx 2A=L(IN+L)−1,and χ=iJ ⇒adA(χ) = −A+A2and S(x) = eiJx Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples First case A=L New compatibility conditions Jαn−αnJ+αn=L+1 2(L2αn−αnL2),J−γ−1 nJγn=Lαn+αnL−2α2 n New ladder operators (0-th order) b Pn(x)J−J b Pn(x)−x( b Pn(x)L−L b Pn(x)) + 2βn b Pn(x)−n b Pn(x) = 2(L−αn)βn b Pn−1(x) b Pn(x)(J−xL)−γ−1 n(J−xL∗)γn b Pn(x)+2βn+1 b Pn(x)−(n+1) b Pn(x) = 2(αn−L) b Pn+1(x) First-order differential equation (L−αn) b P 0 n(x)+(L−αn+xIN)( b Pn(x)L−L b Pn(x))−2βn b Pn(x) = b Pn(x)J−J b Pn(x)−n b Pn(x) Sturm-Liouville type differential equation (Dur´an-Gr¨unbaum, 2004) b P 00 n(x) + 2 b P 0 n(x)(L−xIN) + b Pn(x)(L2 −2J) = (−2nIN+L2 −2J) b Pn(x) Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples First case A=L New compatibility conditions Jαn−αnJ+αn=L+1 2(L2αn−αnL2),J−γ−1 nJγn=Lαn+αnL−2α2 n New ladder operators (0-th order) b Pn(x)J−J b Pn(x)−x( b Pn(x)L−L b Pn(x)) + 2βn b Pn(x)−n b Pn(x) = 2(L−αn)βn b Pn−1(x) b Pn(x)(J−xL)−γ−1 n(J−xL∗)γn b Pn(x)+2βn+1 b Pn(x)−(n+1) b Pn(x) = 2(αn−L) b Pn+1(x) First-order differential equation (L−αn) b P 0 n(x)+(L−αn+xIN)( b Pn(x)L−L b Pn(x))−2βn b Pn(x) = b Pn(x)J−J b Pn(x)−n b Pn(x) Sturm-Liouville type differential equation (Dur´an-Gr¨unbaum, 2004) b P 00 n(x) + 2 b P 0 n(x)(L−xIN) + b Pn(x)(L2 −2J) = (−2nIN+L2 −2J) b Pn(x) Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples First case A=L New compatibility conditions Jαn−αnJ+αn=L+1 2(L2αn−αnL2),J−γ−1 nJγn=Lαn+αnL−2α2 n New ladder operators (0-th order) b Pn(x)J−J b Pn(x)−x( b Pn(x)L−L b Pn(x)) + 2βn b Pn(x)−n b Pn(x) = 2(L−αn)βn b Pn−1(x) b Pn(x)(J−xL)−γ−1 n(J−xL∗)γn b Pn(x)+2βn+1 b Pn(x)−(n+1) b Pn(x) = 2(αn−L) b Pn+1(x) First-order differential equation (L−αn) b P 0 n(x)+(L−αn+xIN)( b Pn(x)L−L b Pn(x))−2βn b Pn(x) = b Pn(x)J−J b Pn(x)−n b Pn(x) Sturm-Liouville type differential equation (Dur´an-Gr¨unbaum, 2004) b P 00 n(x) + 2 b P 0 n(x)(L−xIN) + b Pn(x)(L2 −2J) = (−2nIN+L2 −2J) b Pn(x) Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples Final remarks Conclusions 1The ladder operators method gives more insight about the differential properties of MOP and new phenomena 2This method works for every weight matrix W. The corresponding MOP satisfy differential equations, but not necessarily of Sturm-Liouville type Future directions 1Examples when supp(W)⊂[0,+∞) or supp(W)⊂[−1,1] 2Uniform asymptotics: steepest descent analysis for RHP (Deift-Zhou,1993) extended to MOPRL Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP The RH problem for OP The RH problem for MOP The RHP for MOPs The Lax pair Examples Final remarks Conclusions 1The ladder operators method gives more insight about the differential properties of MOP and new phenomena 2This method works for every weight matrix W. The corresponding MOP satisfy differential equations, but not necessarily of Sturm-Liouville type Future directions 1Examples when supp(W)⊂[0,+∞) or supp(W)⊂[−1,1] 2Uniform asymptotics: steepest descent analysis for RHP (Deift-Zhou,1993) extended to MOPRL Manuel Dom´ınguez de la Iglesia Algebraic aspects of the RHP for MOP