Class Note for PHYS 142 at UA
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Date Created: 02/06/15
LECTURE 20 THE CONNECTION BETWEEN ORTHOGONAL POLYNOMIALS AND RIEMANNiHILBERT PROBLEMS PROOFS AND OUTLINE OF ASYMPTOTIC CALCULATION Lecture plan We will carry out the proof that orthogonal polynomials can be characterized in terms of a RiemanniHilbert problem and then we will begin discussing the asymptotic analysis of RiemanniHilbert problems ORTHOGONAL POLYNOMIALS AND RIEMANNiHILBERT PROBLEMS The following RiemannHilbert problem I is known to characterize the polynomials p orthogonal with va N J respect to e RiemannHilbert Problem 1 Find a 2 X 2 matrix Az Az nN with the properties Analyticity Az is analytic for 2 E C R and takes continuous boundary values A I A as 2 tends to z with z E R andz 6 3 2 E 3 Jump Condition The boundary values are connected by the relation 1 67 Nvm lt1 Altzgt A z 0 I Normalization The matrix Az is normalized at 2 00 as follows 2 0 2 lim Az ll zaoo It was discovered in I that RiemannHilbert Problem I characterizes polynomials orthogonal with respect to dlz I e NW dz The connection between these orthogonal polynomials and the solution of Riemann Hilbert Problem I is the following 1 l pnse NV5 7107i 2 7 ds 532 lt 2mm R s 7 z 3 152 7NV 5 27ri illii nilp 1z Hllii nil pnilltsgte d5 R S i Z This relationship provides a useful avenue for asymptotic analysis of the orthogonal polynomials in the limit n A 00 it is suf cient to carry out a rigorous asymptotic analysis of RiemannHilbert Problem I UNIQUENESS The fact that there is at most one solution to RiemanniHilbert problem I may be seen as follows First you observe that if you have a solution A it is invertible You verify that det A is entire and converges to l as 2 A 00 thus detA E l by Liouville s theorem Next you assume that there are two solutions 1412 and 1422 and de ne 4 192 A1ZA271ZA You may easily verify that is entire and converges to I as 2 A 00 and then Liouville s theorem again takes over This then implies that 5 AM AW and hence there is at most one solution to the RiemanniHilbert problem 1 EXISTENCE The proof that the matrix 3 solves the RiemanniHilbert problem 1 starts by observing that 142 de ned in 3 automatically satis es the analyticity condition and the jump condition To verify the jump condition you need to know the following fact if F2 1 fR fwds and if is a smooth and rapidly decaying 27m 572 function then F2 is analytic for 2 6 LF U C1 and possesses nice boundary values Fz and F for z E R which satisfy Fz 7 F This can be extended in a variety of different directions R can be replaced by a much more general contour the function f need not be smooth etc So what remains to verify is the behavior for 2 A 001 The condition 2 implies that as 2 A 00 1411 2n lower order terms 1421 62 1 lower order terms both being satis ed by There are two other entries in the matrix 14 and the condition 2 implies that as 2 A 00 14122 c27n 1 smaller terms 1422 2 smaller terms Now at rst glance these conditions are not satis ed by 1412 and 1422 as de ned in But upon closer inspection 7 1 7NVs 1 ampWdm4m e li z 7rmngtn2 n j 1 Sn1Zn1 7 eNvltsgt d 7 eNvltsgt d pn s e 3 pn s e 81 27riH LIQ2R 10 2 27riH Q2 1R 1 52 Now by orthogonality for j 011 1 1 n 7 l we know eNvltsgt 5 1 6 anse dsi and also 1 l 7 pn s e NWs 3 d3 pn s e NWs ipn 8 d3 71 R R So 14122 de ned in 3 satis es m mnee 1 2 27ri We leave you all to check that 1422 satis es the remaining asymptotic condition 9 14222 2 O 27 71 2 quot1 0 277172 1 REFERENCES 1 A Fokas A Its and A V Kitaev Discrete Painleve equations and their appearance in quantuln gravity Commun Math Phys 142 3137344 1991
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