The Riemann-Hilbert problem for matrix-valued orthogonal polynomials

The Riemann-Hilbe p oblem o ma ix- alued o hogonal polynomials1 Manuel Domínguez de la Iglesia Depa men o Ma hema ics, K. U. Leu en Semina classical analysis Leu en, No embe 5, 2008 1join wo k wi h And ei Ma ínez Finkelsh ein P elimina ies The RH p oblem o OMP An example Ou line 1P elimina ies 2The RH p oblem o OMP 3An example Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example Ou line 1P elimina ies 2The RH p oblem o OMP 3An example Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example Le Wbe a N×Naweigh ma ix such ha dW (x) = W(x)dx. We can cons uc a amily o OMP such ha ZR Pn(x)W(x)P∗ m(x)dx =δn,mI,n,m⩾0 Pn(x) = γn(xn+an,n−1xn−1+···) = γnb Pn(x) The ma ix- alued polynomials o he second kind, de ined by Qn(x) = ZR Pn( )W( ) −xd ,n⩾0 (Pn)nand (Qn)nsa is y a h ee e m ecu ence ela ion Pn( ) = An+1Pn+1( ) + BnPn( ) + A∗ nPn−1( ),n⩾0 de (An+1)6=0,Bn=B∗ n Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example Le Wbe a N×Naweigh ma ix such ha dW (x) = W(x)dx. We can cons uc a amily o OMP such ha ZR Pn(x)W(x)P∗ m(x)dx =δn,mI,n,m⩾0 Pn(x) = γn(xn+an,n−1xn−1+···) = γnb Pn(x) The ma ix- alued polynomials o he second kind, de ined by Qn(x) = ZR Pn( )W( ) −xd ,n⩾0 (Pn)nand (Qn)nsa is y a h ee e m ecu ence ela ion Pn( ) = An+1Pn+1( ) + BnPn( ) + A∗ nPn−1( ),n⩾0 de (An+1)6=0,Bn=B∗ n Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example Le Wbe a N×Naweigh ma ix such ha dW (x) = W(x)dx. We can cons uc a amily o OMP such ha ZR Pn(x)W(x)P∗ m(x)dx =δn,mI,n,m⩾0 Pn(x) = γn(xn+an,n−1xn−1+···) = γnb Pn(x) The ma ix- alued polynomials o he second kind, de ined by Qn(x) = ZR Pn( )W( ) −xd ,n⩾0 (Pn)nand (Qn)nsa is y a h ee e m ecu ence ela ion Pn( ) = An+1Pn+1( ) + BnPn( ) + A∗ nPn−1( ),n⩾0 de (An+1)6=0,Bn=B∗ n Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example Le Wbe a N×Naweigh ma ix such ha dW (x) = W(x)dx. We can cons uc a amily o OMP such ha ZR Pn(x)W(x)P∗ m(x)dx =δn,mI,n,m⩾0 Pn(x) = γn(xn+an,n−1xn−1+···) = γnb Pn(x) The ma ix- alued polynomials o he second kind, de ined by Qn(x) = ZR Pn( )W( ) −xd ,n⩾0 (Pn)nand (Qn)nsa is y a h ee e m ecu ence ela ion Pn( ) = An+1Pn+1( ) + BnPn( ) + A∗ nPn−1( ),n⩾0 de (An+1)6=0,Bn=B∗ n Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example The coe icien s o he TTRR sa is y An=γn−1γ−1 n,Bn=γn(an,n−1−an+1,n)γ−1 n The TTRR o monic OMP xb Pn(x) = b Pn+1(x) + αnb Pn(x) + βnb Pn−1(x),n⩾0 αn=an,n−1−an+1,n,βn= (γ∗ nγn)−1(γ∗ n−1γn−1) Second-o de di e en ial equa ions o hype geome ic ype P00 n(x)F2(x) + P0 n(x)F1(x) + Pn(x)F0(x) = ΛnPn(x),n⩾0 deg Fi⩽i,ΛnHe mi ian Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example The coe icien s o he TTRR sa is y An=γn−1γ−1 n,Bn=γn(an,n−1−an+1,n)γ−1 n The TTRR o monic OMP xb Pn(x) = b Pn+1(x) + αnb Pn(x) + βnb Pn−1(x),n⩾0 αn=an,n−1−an+1,n,βn= (γ∗ nγn)−1(γ∗ n−1γn−1) Second-o de di e en ial equa ions o hype geome ic ype P00 n(x)F2(x) + P0 n(x)F1(x) + Pn(x)F0(x) = ΛnPn(x),n⩾0 deg Fi⩽i,ΛnHe mi ian Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example Also we ind a solu ion o he in e se (Yn)−1=  −ZR W( )b P∗ n−1( ) −zd γ∗ n−1γn−1−1 2πiZR W( )b P∗ n( ) −zd 2πib P∗ n−1(z)γ∗ n−1γn−1b P∗ n(z)   The Liou ille-Os og adski o mula Qn(z)P∗ n−1(z) − Pn(z)Q∗ n−1(z) = A−1 n The He mi ian p ope y Qn(z)P∗ n(z) = Pn(z)Q∗ n(z) Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example The h ee- e m ecu ence ela ion I we call R=Yn+1(Yn)−1and deno ing Yn(z) = I+1 zYn 1+On(1/z2)znI0 0z−nI,z→∞ Yn+1(z) = zI +Yn+1 111 −Yn 111 −Yn 112 Yn+1 121 0Yn(z) γ∗ n−1γn−1= − 1 2πiYn 121 = − 1 2πiYn−1 1−1 12 αn=Yn 111 −Yn+1 111,βn=Yn 112Yn 121 Bn=γnYn 111 −Yn+1 111γ−1 n,A∗ n=γnYn 112Yn 121γ−1 n−1 Yn+1 121Yn 112 =Yn 112Yn+1 121 =I,Yn 111 +Yn 1∗ 22 =0 Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example The h ee- e m ecu ence ela ion I we call R=Yn+1(Yn)−1and deno ing Yn(z) = I+1 zYn 1+On(1/z2)znI0 0z−nI,z→∞ Yn+1(z) = zI +Yn+1 111 −Yn 111 −Yn 112 Yn+1 121 0Yn(z) γ∗ n−1γn−1= − 1 2πiYn 121 = − 1 2πiYn−1 1−1 12 αn=Yn 111 −Yn+1 111,βn=Yn 112Yn 121 Bn=γnYn 111 −Yn+1 111γ−1 n,A∗ n=γnYn 112Yn 121γ−1 n−1 Yn+1 121Yn 112 =Yn 112Yn+1 121 =I,Yn 111 +Yn 1∗ 22 =0 Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example The h ee- e m ecu ence ela ion I we call R=Yn+1(Yn)−1and deno ing Yn(z) = I+1 zYn 1+On(1/z2)znI0 0z−nI,z→∞ Yn+1(z) = zI +Yn+1 111 −Yn 111 −Yn 112 Yn+1 121 0Yn(z) γ∗ n−1γn−1= − 1 2πiYn 121 = − 1 2πiYn−1 1−1 12 αn=Yn 111 −Yn+1 111,βn=Yn 112Yn 121 Bn=γnYn 111 −Yn+1 111γ−1 n,A∗ n=γnYn 112Yn 121γ−1 n−1 Yn+1 121Yn 112 =Yn 112Yn+1 121 =I,Yn 111 +Yn 1∗ 22 =0 Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example The h ee- e m ecu ence ela ion I we call R=Yn+1(Yn)−1and deno ing Yn(z) = I+1 zYn 1+On(1/z2)znI0 0z−nI,z→∞ Yn+1(z) = zI +Yn+1 111 −Yn 111 −Yn 112 Yn+1 121 0Yn(z) γ∗ n−1γn−1= − 1 2πiYn 121 = − 1 2πiYn−1 1−1 12 αn=Yn 111 −Yn+1 111,βn=Yn 112Yn 121 Bn=γnYn 111 −Yn+1 111γ−1 n,A∗ n=γnYn 112Yn 121γ−1 n−1 Yn+1 121Yn 112 =Yn 112Yn+1 121 =I,Yn 111 +Yn 1∗ 22 =0 Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example The h ee- e m ecu ence ela ion I we call R=Yn+1(Yn)−1and deno ing Yn(z) = I+1 zYn 1+On(1/z2)znI0 0z−nI,z→∞ Yn+1(z) = zI +Yn+1 111 −Yn 111 −Yn 112 Yn+1 121 0Yn(z) γ∗ n−1γn−1= − 1 2πiYn 121 = − 1 2πiYn−1 1−1 12 αn=Yn 111 −Yn+1 111,βn=Yn 112Yn 121 Bn=γnYn 111 −Yn+1 111γ−1 n,A∗ n=γnYn 112Yn 121γ−1 n−1 Yn+1 121Yn 112 =Yn 112Yn+1 121 =I,Yn 111 +Yn 1∗ 22 =0 Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example The ke nel Le (Pn)nan o hono mal amily. Then we ha e Kn(x,y) = n−1 X j=0 P∗ j(y)Pj(x) = P∗ n−1(y)AnPn(x) − P∗ n(y)A∗ nPn−1(x) x−y This ke nel has he ollowing p ope ies 1Kn(x,y) = K∗ n(y,x) 2Kn(x,y) = RRKn(s,y)W(s)Kn(x,s)ds We also ha e ha Kn(x,y) = 1 2πi(x−y)0I(Yn)−1 +(y)(Yn)+(x)I 0 Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example The ke nel Le (Pn)nan o hono mal amily. Then we ha e Kn(x,y) = n−1 X j=0 P∗ j(y)Pj(x) = P∗ n−1(y)AnPn(x) − P∗ n(y)A∗ nPn−1(x) x−y This ke nel has he ollowing p ope ies 1Kn(x,y) = K∗ n(y,x) 2Kn(x,y) = RRKn(s,y)W(s)Kn(x,s)ds We also ha e ha Kn(x,y) = 1 2πi(x−y)0I(Yn)−1 +(y)(Yn)+(x)I 0 Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example The ke nel Le (Pn)nan o hono mal amily. Then we ha e Kn(x,y) = n−1 X j=0 P∗ j(y)Pj(x) = P∗ n−1(y)AnPn(x) − P∗ n(y)A∗ nPn−1(x) x−y This ke nel has he ollowing p ope ies 1Kn(x,y) = K∗ n(y,x) 2Kn(x,y) = RRKn(s,y)W(s)Kn(x,s)ds We also ha e ha Kn(x,y) = 1 2πi(x−y)0I(Yn)−1 +(y)(Yn)+(x)I 0 Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example A di e en ial equa ion We conside weigh ma ices o he o m W(x) = ρ(x)T(x)T∗(x), whe e Tsa is ies T0(x) = G(x)T(x). Conside Xn(z) = Yn(z)J(z) = Yn(z)ρ(z)1/2T(z)0 0ρ(z)−1/2(T(z))−∗ Then d dz Xn(z)Xn(z)−1is en i e and nea in ini y i beha es like I+Y1 z+O(z−2) 1 2 ρ0(z) ρ(z)+G(z)0 0−1 2 ρ0(z) ρ(z)−G(z)∗!I−Y1 z+O(z−2) Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example The Lax pai Le Ynbe he solu ion o he RH o Wand conside Xn(z) = Yn(z)e−z2/2eAz 0 0ez2/2e−A∗z Xn+1(z) = zI +Yn+1 111 −Yn 111 −Yn 112 Yn+1 121 0Xn(z) d dz Xn(z) = −zI +A2Yn 112 −2Yn 121 zI −A∗Xn(z) Compa ibili y condi ions 2(βn+1−βn) = Aαn−αnA+I αn=1 2(A+ (γ∗ nγn)−1A∗(γ∗ nγn)) Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example The Lax pai Le Ynbe he solu ion o he RH o Wand conside Xn(z) = Yn(z)e−z2/2eAz 0 0ez2/2e−A∗z Xn+1(z) = zI +Yn+1 111 −Yn 111 −Yn 112 Yn+1 121 0Xn(z) d dz Xn(z) = −zI +A2Yn 112 −2Yn 121 zI −A∗Xn(z) Compa ibili y condi ions 2(βn+1−βn) = Aαn−αnA+I αn=1 2(A+ (γ∗ nγn)−1A∗(γ∗ nγn)) Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example Ladde ope a o s Lowe ing ope a o b P0 n(z) = Ab Pn(z) − b Pn(z)A+2βnb Pn−1(z) The e o e βn=1 2(nI +an,n−1A−Aan,n−1) Raising ope a o b P0 n(z)=−2b Pn+1(z) + 2zb Pn(z) + Ab Pn(z) − b Pn(z)A−2αnb Pn(z) Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example Ladde ope a o s Lowe ing ope a o b P0 n(z) = Ab Pn(z) − b Pn(z)A+2βnb Pn−1(z) The e o e βn=1 2(nI +an,n−1A−Aan,n−1) Raising ope a o b P0 n(z)=−2b Pn+1(z) + 2zb Pn(z) + Ab Pn(z) − b Pn(z)A−2αnb Pn(z) Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example Second-o de di e en ial equa ions In oduce he ollowing di e en ial/di e ence ope a o s R1=∂1+∂0A,L1=2βnE−1+AE0 R2=∂1+∂0(A−2zI),L2= −2E1+ (A−2αn)E0 ∂k=dk dzk,Ek (n) = (n+k) The ladde ope a o s a e equi alen o b Pn(z)R1=L1b Pn(z),and b Pn(z)R2=L2b Pn(z) The e o e, he OMP b Pnsa is y wo second-o de di e en ial equa ions b Pn(z)R1R2=L1b Pn(z)R2=L1L2b Pn(z), b Pn(z)R2R1=L2b Pn(z)R1=L2L1b Pn(z) Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example Second-o de di e en ial equa ions In oduce he ollowing di e en ial/di e ence ope a o s R1=∂1+∂0A,L1=2βnE−1+AE0 R2=∂1+∂0(A−2zI),L2= −2E1+ (A−2αn)E0 ∂k=dk dzk,Ek (n) = (n+k) The ladde ope a o s a e equi alen o b Pn(z)R1=L1b Pn(z),and b Pn(z)R2=L2b Pn(z) The e o e, he OMP b Pnsa is y wo second-o de di e en ial equa ions b Pn(z)R1R2=L1b Pn(z)R2=L1L2b Pn(z), b Pn(z)R2R1=L2b Pn(z)R1=L2L1b Pn(z) Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example Second-o de di e en ial equa ions In oduce he ollowing di e en ial/di e ence ope a o s R1=∂1+∂0A,L1=2βnE−1+AE0 R2=∂1+∂0(A−2zI),L2= −2E1+ (A−2αn)E0 ∂k=dk dzk,Ek (n) = (n+k) The ladde ope a o s a e equi alen o b Pn(z)R1=L1b Pn(z),and b Pn(z)R2=L2b Pn(z) The e o e, he OMP b Pnsa is y wo second-o de di e en ial equa ions b Pn(z)R1R2=L1b Pn(z)R2=L1L2b Pn(z), b Pn(z)R2R1=L2b Pn(z)R1=L2L1b Pn(z) Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example We will p oo he ollowing: Bo h equa ions a e equi alen They a e also equi alen o he second-o de di e en ial equa ion o hype geome ic ype b P00 n(z) + b P0 n(z)(2A−2zI) + b Pn(z)(A2−2J) = (−2nI +A2−2J)b Pn(z) The i s one is b P00 n(z) + 2b P0 n(z)(A−zI) + b Pn(z)A(A−2zI) = −4βnb Pn(z) + 2βn(A−2αn−1)b Pn−1(z) − 2Ab Pn+1(z) + A(A−2αn)b Pn(z) And he second b P00 n(z) + 2b P0 n(z)(A−zI) + b Pn(z)(A−2zI)A−2b Pn(z) = −4βn+1b Pn(z) + 2(A−2αn)βnb Pn−1(z) − 2Ab Pn+1(z) + (A−2αn)Ab Pn(z) Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example We will p oo he ollowing: Bo h equa ions a e equi alen They a e also equi alen o he second-o de di e en ial equa ion o hype geome ic ype b P00 n(z) + b P0 n(z)(2A−2zI) + b Pn(z)(A2−2J) = (−2nI +A2−2J)b Pn(z) The i s one is b P00 n(z) + 2b P0 n(z)(A−zI) + b Pn(z)A(A−2zI) = −4βnb Pn(z) + 2βn(A−2αn−1)b Pn−1(z) − 2Ab Pn+1(z) + A(A−2αn)b Pn(z) And he second b P00 n(z) + 2b P0 n(z)(A−zI) + b Pn(z)(A−2zI)A−2b Pn(z) = −4βn+1b Pn(z) + 2(A−2αn)βnb Pn−1(z) − 2Ab Pn+1(z) + (A−2αn)Ab Pn(z) Manuel Domínguez de la Iglesia The RH p oblem o OMP P elimina ies The RH p oblem o OMP An example We will p oo he ollowing: Bo h equa ions a e equi alen They a e also equi alen o he second-o de di e en ial equa ion o hype geome ic ype b P00 n(z) + b P0 n(z)(2A−2zI) + b Pn(z)(A2−2J) = (−2nI +A2−2J)b Pn(z) The i s one is b P00 n(z) + 2b P0 n(z)(A−zI) + b Pn(z)A(A−2zI) = −4βnb Pn(z) + 2βn(A−2αn−1)b Pn−1(z) − 2Ab Pn+1(z) + A(A−2αn)b Pn(z) And he second b P00 n(z) + 2b P0 n(z)(A−zI) + b Pn(z)(A−2zI)A−2b Pn(z) = −4βn+1b Pn(z) + 2(A−2αn)βnb Pn−1(z) − 2Ab Pn+1(z) + (A−2αn)Ab Pn(z) Manuel Domínguez de la Iglesia The RH p oblem o OMP