Class Note for PHYS 142 at UA
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Date Created: 02/06/15
LECTURE 21 RIEMANNiHILBERT PROBLEMS AND SINGULAR INTEGRAL EQUATIONS AND OUTLINE OF ASYMPTOTIC CALCULATIONI Lecture plan For the discussion of the asymptotic analysis of RiemanniHilbert problems we will need to go through the connection to singular integral equations and the theory of small norm RiemanniHilbert problems ORTHOGONAL POLYNOMIALS AND RIEMANNiHILBERT PROBLEMS Recall from the previous lecture that we have following RiemannHilbert problem which is known to N 7NV 1 j lt gt 1 i characterize the polynomials p orthogonal with 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 eina lt1 Altzgt A7ltzgt 0 l Normalization The matrix Az is normalized at 2 00 as follows 2 0 2 lim Az lli zaoo 0 Zn The connection between these orthogonal polynomials and the solution of RiemannHilbert Problem I is the following 1 I pnse NV5 7P7 2 7 ds i533 2mm R H 3 A z 7NV5 Zm ig nilpn 1z Hllii nil pnilltsgte d5 gt R s 7 2 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 Ii REPRESENTING THE KERNEL The kernel KN z y can be represented directly in terms of the solution of the RiemanniHilbert problem 4 Km y 7eiltvltwgtvltygtgt Y11Igt591ygt YWWHW 7T2 I i 9 J V V 1 1 1 5 2 TH 11 0 1 Y yYI 0 A 27ri I 7 y RIEMANN HILBERT PROBLEMS SINGULAR INTEGRAL EQUATIONS AND SMALL NORM THEORY The connection between RiemanniHilbert problems and singular integral equations as well as the related small norm theory77 is described in the lecture entitled RHP Survey77 available on the course website after Lecture 2L 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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