Class Note for PHYS 142 at UA
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Date Created: 02/06/15
LECTURE 26 THE AIRY FUNCTIONS AND THEIR CONNECTION TO RIEMANNiHILBERT PROBLEMS Lecture plan We will summarize how we have completed the asymptotic calculation followed by an explanation of how we can build the local parametrix out of Airy functions SUMMARY OF THE ASYMPTOTIC CALCULATION We began with Az solving the original RiemanniHilbert problem for orthogonal polynomials The rst transformation was 1 Ba e 3Aze N9Z 3 The second transformation was as follows So we then de ned Dz as follows 0 For 2 outside the lens shaped region surrounding the interval 711 Dz o For 2 within the upper lens shaped region we set Dz BltZgtUltZgt71 o For 2 within the lower lens shaped region we set Dz Bzv E1 24 22 E5 23 The matrix D now solves a new RiemanniHilbert problem RiemannHilbert Problem 1 Find Dz satisfying the following three conditions Analyticity Dz is analytic for z E C E and takes continuous boundary values Dz Dz with x E 2 Jump Condition The boundary values are connected by the relation 0 DowD4awxazez Normalization The matrix Dz is normalized at z 00 as follows 3 lim Dz l The jump matrix VD was de ned as follows For 2 E 24 U 25 we have VDz VBz For 2 E 22 we have a lt315 o Forz E 21 we have VDz vz o Forz E 23 we have VDz v And it is clear that the new unknown D is analytic of the more complicated union of contours shown above Moreover given the above considerations the jump matrices satisfy the following important property For any 6 gt 0 the jump matrix VDz is exponentially Close to l for all values of 2 whose distance from 711 is greater than 6 1 global approximation to D We then began building a global approximation to D by asking if we could nd D solving the following reduced RiemanniHilbert problem RiemannHilbert Problem 2 Find satisfying the fallmuing thiee conditions 1 analyticity The matrix is analytic in C 711 2 Normalization Dz l O as z gt 00 3 Boundary values and jump relation 0 1 lt5 Dltxgt D7ltxgt 1 0 The function is explicitly known 6 DzlF W 3 117 1 0 W where F is the following explicit matrix 7 1Fltll The intuition which we have developed indicates that 8 132 Dz 1 1 should be a new unknown which has no jump acmss 711 and has jumps that are exponentially near to l for z in the contour E but bounded away from i1 and so maybe this quantity satis es the guiding principle we have spoken about in Lecture 21 and see the discussion in the RHPSurVey lecture posted on the website But we are not quite home yet the jump matrices are not uniformly near to l ln Lecture 25 we began discussing the local behavior of the jump matrices in a Vicinity of z 1 The end result was that under the transformation 9 342 27riN1x13sds We more or less Veri ed that z is analytic for z in a neighborhood of z 1 and maps a disc of xed size centered at z 1 to a neighborhood of 4 0 The neighborhood in the plane is Very large roughly 0N23 The point to this transformation is that in the new plane the jump matrices VD take on the following form gZ 0 1 1 argco 01 lt1 0 a argcna NC I l 1 0 72 egg2 1 a a 1 0 4n 171 eggs2 1 argCZ Now the idea here is as o11ows cau we cuustcuct a matuc ualuea funatw39h whwh has ecactlu these jumps tn the 4 plaue Ifsa cau we puat 2t hach uuec tu the splaue aua use 215 as a pammetmas M a uewhlwahuua 0f 1 1 Let s ca11 this function 1314 Similar calculations may be carried out in a vicinity of z 71 and for those calculations we will let the analogous function be referred to as 1114 So now we have built three separate matrix va1ued functions which together may be used as a global approximation to Dzt Here is the de nition 0 For an 1 e C but uutswle discs of radius a gt 0 centered at z i1 we de ne lt10 Dem Dltzgtv o For 12711 lt a we de ne 11 A12P1 2v For 1211 lt a we de ne 12 Daz A711P71 2v NOTE The matrices A1 and A4 are matrices which are still yet to be determined They will be aualutw in a vicinity of 1 and 71 respectively HOW GOOD IS THE GLOBAL APPROXIMATION As we have seen in previous 1ectures the only way to assess the approximation 13a is to consider the ratio 13 132 DO Daz 1a which solves yet another RiemanrrHjlbert problemv RiemannHilbert Problem 3 Ma 132 satwtwup the tulluwtup thaee cuuahtwus Analyticity 132 us analytta far 1 e C 2E aua tahes aantz huaus huuuaaau ualues 1312 Ez wtth x e Z Jump Condition The Imu hdmy ualues ate cuuuecteal by the rel11mm 14 1312 EsVEs 1 e 212 Normalization The matmc 132 Ls uarmaltzeal at z oo as fallawsx 15 dime Es 1 25 211 27 The jump matwc VE us ae uea as fallawsx Far as e 2 u 2395 we haue 1492 VBs 1 o E QWWK Wants FM 2 e 23 we haue Vps n1 11 0 54 Fur z e 23 we haue 1422 us 7 1 0 E CN Far as e 27 we haue Vps A12P12D2 1 Far as e 25 we haue Vps A12P12Dz 1 The c1aim left to the 1ecture today is that we can hui1d the 1oca1 parametrjces P1 and 131 along with the ana1ytic prefactors A1 and AA so that the jumps on the contours 25 and 27 are na11y 11 0 N41 3 BUILDING P1 VIA AIRY FUNCTIONS So we consider a full edged RiemanniHilbert problem for 1315 in the Cplane RiemannHilbert Problem 4 Find 1315 satisfying the following three conditions Analyticity P1 is analytic for C 6 CZp1 and takes continuous boundary values 131r C P17 with z 6 2 Jump Condition The boundary values are connected by the relation 16gt Pl c P1gtltcgtalt1gtltcgt c 6 2p Normalization The matrix 1315 is normalized at C 00 as follows PCgt l l6 3lt10ltc32gtgt we The jump matrix 73lt1gt is de ned as follows 1 1 e C32 0W2 0 1 argc0 A 1 1 0 afgczn 1 0 27 MIME egg2 1 7 mg 2 g 1 0 4313 91Ce c32 1 argczi We rst de ne the matrixvalued function as follows A A 72i1r3 17 1 lt 72i1r231i672i 123C gt v for 0 lt mg lt 7B A 7 72i7r3A 2i7r3 18gt M lt eAieigo C gt for 7 w lt argltltgt lt o where w eZiquot3 Using 11 we now de ne P1 as follows 19gt Pm CeCSZ 3 for l arm lt 2 20 1315 Me eCSLUS lt 761432 1 gt for 2g lt arg lt 7r7 21 1315 Me eCSLUS lt egg2 1 gt for 7r lt arg lt 22 We ll verify in the lecture that P1 so de ned satis es the above RiemanniHilbert problem The veri cation is not extremely illuminating it is straightforward to check that P1 satis es the jump relationships on the contours argC 27r3 and argC 47r3 The veri cation on the real axis requires the following basic property of the Airy function 23 Ai wAiwC wZAiw2 0 4 The Veri cation of the asymptotic behavior as Q 4r 0 requires knowledge of the asymptotic behavior of the Airy function as 5 4r 00 no k Auoi pcir Etd u m wgdltwr k0 so 7k 23 29 1ka forargCl lt7ri Mm 1quot3k f 6k1 54kklrk 7 6k 1 But what is mysterious from the outset is the arrival on the scene of the infamous Airy function Whence came this connection to special functions As we have seen there is a general uniqueness theorem for a wide class of matrix RiemannrHilbert problems and the RiemannrHilbert problem we are considering in this section falls within that class So we have built the only solution to this RiemanneHilbert problem Let s understand this connection more completely Consider the following new matrix valued function lt24 mo P1 e 432 3i 4 e C social it is straightforward to verify that 1 is analytic in the same domain as 1315 and moreover 4 possesses mstont jumps indeed the jumps for 1 are as follows soro1 xk ii39k forkZl 25 1 407140 26 mg lt5 1 for gem 27 M4 lt i g for argg 27r3 and arg447r3 28 mg lt 31 a forgellL Now in addition to assuming that 1 exists let us also assume that it is differentiable in 4 and that the derivative IJ possesses nice boundary values it then follows that IJ possesses the same jumps as 11 itself As we have seen before an immediate consequence is that the ratio IJ mgri is same However if we in addition assume that the asymptotic behavior of IJ may be obtained by differentiating the asymptotic behavior of 4 we may deduce that the ratio IJ mgri is not only entire but also a polynomial in 4 Skipping the calculations for now we will do them in class we nd I 71 7 0 1 lt29 m lt4 M e lt 4 0 This is a differential equation and some further jacking around leads to the Airy equation We ll see this in class REFERENCES 1 P Deift T Kriecherbauer K TeR McLaughlin s Venalndes and x Zhou Uniform asymptotics for polynomials orthogonal With respect to varying exponential weights and applications to universality questions in random matrix theoryquot Comm Pu raAppl Math521885714251999 2 P Deift T Kriecherbauer K TVR McLaughlin s Venalndes andx Zhou Strong asymptotics of orthogonal polyno mials with respect to exponential Weightsquot Comm Pars Appl Math 52 149171552 1999 a A Fokas A Its and A V Kitaev Discrete Painleve equations and their appearance in quantum gravityquot Commun Math Phys 142 3137344 1991 5 4 TV Kriecherbauer and KY TrRV McLaughlin Strong Asymptotics of Polynomials Orthogonal With Respect to Freud Weights Int Math Res Not Nov 6 pp 299333 1999
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