The Airy function is a Fredholm determinant

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1 Journal of Dynamics and Differential Equations manuscript No. (will e inserted y the editor) The Airy function is a Fredholm determinant Govind Menon Received: date / Accepted: date Astract Let G e the Green s function for the Airy operator Lϕ := ϕ + ϕ, < <, ϕ() =. We show that the integral operator defined y G is Hilert-Schmidt and that the 2-modified Fredholm determinant det 2 (1 + zg) = Ai(z) Ai(), z C. Keywords Airy function Fredholm determinant Hilert-Schmidt operators Mathematics Suject Classification (2) MSC 47G1 MSC 33C1 1 Introduction Let L denote the Airy operator on the half-line R + with Dirichlet oundary condition Lϕ := ϕ + ϕ, < <, ϕ() =. (1) Recall that the differential equation ϕ = ϕ admits two linearly independent solutions, denoted Ai and Bi [1]. As, Ai() decays, and Bi() grows, at the super-eponential rate 1/4 e 23/2 /3. For z C, oth Ai(z) and Bi(z) are entire functions of eponential type 3/2. The zeros of Ai lie on the negative real ais and are denoted Supported in part y NSF grant G. Menon Division of Applied Mathematics Brown University, Providence, RI, USA Tel.: Fa: menon@dam.rown.edu <... < a n < a n 1 <... < a 1 <. (2)

2 2 Govind Menon For each n, Ai( a n ) is an eigenfunction of L with eigenvalue a n. Further, these eigenfunctions form a complete asis for L 2 (R + ) (see e.g. [7, 4.12]). The spectral prolem for L may also e approached via the Hilert-Schmidt theory of integral equations. Let f ± denote linearly independent solutions to the Airy equation y y =, with f + () = Ai(), and Then f () = and f () = The Green s function for L is and the integral equation π (Ai()Bi() Ai()Bi()). (3) Ai() W (f +, f ) := f + ()f () f ()f +() = 1. (4) G(, y) = f (min(, y))f + (ma(, y)),, y R +, (5) ϕ() = z is equivalent to the spectral prolem G(, y)ϕ(y) dy (6) Lϕ = zϕ, z C. (7) Theorem 1 The kernel G defines a Hilert-Schmidt operator (also denoted G) on L 2 (R + ). The 2-modified Fredholm determinant det 2 (1 + zg) = Ai(z) «Ai() = Γ Ai(z), z C. (8) 3 Let L α, α ( a 1, ), denote the Airy operator on [α, ) with Dirichlet oundary condition at α, and let G α e the associated Green s function analogous to (5). Corollary 1 det 2 (1 + zg α ) = Ai(z + α), z C. (9) Ai(α) The point here is the eplicit epression for det 2 (1+zG). The asymptotics of Ai() and Bi() imply that G is Hilert-Schmidt, ut not trace class, on L 2 (R + ). As a consequence, the 2-modified Fredholm determinant det 2 (1 + zg) eists, and is an entire function whose zeros coincide with minus the eigenvalues of L. However, this only tells us that det 2 (1 + zg)/ai(z) is an entire function with no zeros, not that it is a constant. Thus, (8) is not ovious. Much of the current interest in Airy functions and Fredholm determinants stems from their importance in proaility theory; for eample, in the description of the Tracy-Widom distriution [8], and of Brownian motion with paraolic drift [4]. Theorem 1 was motivated y a surprising formula in Burgers turulence [5, equation (19)]. While these proailistic connections will e presented elsewhere, the identity (8) strikes me as a calculation of independent interest. In an area this classical, the odds that it is new would seem to e rather low. However, I could not find (8) in the literature, even after a thorough search. Thus, Theorem 1, if not new, is certainly hard to find, and I hope the reader will find

3 The Airy function is a Fredholm determinant 3 some value in this eposition. It is rare that one can compute Fredholm determinants eplicitly, and despite the fact that there are no new methods here, the calculation is quite pleasing. It is a pleasure to dedicate this article to John Mallet-Paret, in respect for his deep knowledge of differential equations and functional analysis. 2 Proof of Theorem A remark on trace ideals Recall that the notion of a Fredholm determinant may e generalized to the trace ideals I p, 1 p [6]. The trace-class operators form the ideal I 1 and the Hilert-Schmidt operators form I 2. The 2-modified determinant for Hilert- Schmidt operators is defined y etending the following formula det 2 (1 + A) = det 1 (1 + A)e Tr(A), A I 1, (1) to operators in I 2. The factor e Tr(A) provides a suitale renormalization of the divergent factor Tr(A) when A I 2 is not trace-class [6, Ch. 9]. The proof of Theorem 1 proceeds as follows. First, we show that G I 2. Net we approimate G y a sequence of trace-class operators G otained y restricting the Airy operator to the interval [, ]. The Fredholm determinant det 1 (1 + zg ) may e computed y the methods of the Goherg-Krein school. Finally, the identity (8) is otained y applying (1) with A = zg and passage to the it. 2.2 The operator G is Hilert-Schmidt In all that follows, the asymptotics of Ai(z) and Bi(z) as z play an important role. As z in the sector arg(z) < π/3 we have [1] Ai(z) e ζ 2 πz, Bi(z) e ζ, ζ = 2 1/4 πz 1/4 3 z3/2. (11) These asymptotics imply rather easily that G is Hilert-Schmidt, ut not traceclass. If G were trace-class, its trace would e G(, ) d = f ()f + () d. (12) Since R Ai2 (s) ds <, it is clear that R G(, ) d is finite if and only if Ai()Bi() d <. But as a consequence of (11) R Ai()Bi() 1 2π,, (13) and Ai()Bi() is not summale. Thus, G is not trace-class. However, a similar calculation shows that G is Hilert-Schmidt. By definition, G 2 HS(L 2 (R + )) = G 2 (, y) d dy.

4 4 Govind Menon In order to show that the aove integral is finite, it suffices to show that = (Ai()Bi(y)1 y< + Bi()Ai(y)1 <y ) 2 d dy (14) Ai 2 () Z Bi 2 (y) dy + Bi 2 () Ai 2 (y) dy <. The asymptotic formulae (11) imply that these integrals are finite if and only if e 43/2 /3 Z e 4y3/2 /3 y dy < and e 43/2 /3 e 4y3/2 /3 y dy <. (15) This is easily checked y making the change of variales 3/2 = u, y 3/2 = v. (A similar calculation is presented in section 2.5). 2.3 The Fredholm determinant on a finite interval [, ], < Fi < < and consider the operator L defined y L ϕ := ϕ + ϕ, < <, ϕ() =, ϕ() =. (16) The Green s function for L is given y G (, y) = f (min(, y))f +(ma(, y)) (17) where f () denotes the solution to ϕ +ϕ = given y (3) (it does not depend on ), and f+() is the solution that satisfies the oundary condition f+() =, as well as the normalization W (f, f ) = 1. We find f+() = c Ai() Bi() Ai() Bi() «, c = Ai()Bi() Bi()Ai(). (18) Both f and f + lie in L 2 (, ) and G is trace-class. Its Fredholm determinant may e computed eplicitly. Proposition 1 det 1 (1 + zg ) = (19) 1 zπ Ai() Bi() Ai() «(Bi(z)Ai( + z) Ai(z)Bi( + z)) d. Ai() Bi() Proof There are two steps in the proof. 1. Theorem 3.1 [3, Ch. IX] yields the following formula for the Fredholm determinant: det 1 (1 + zg ) = U 22 (, z), (2) where the 2 2 matri U(, z) is the fundamental solution to the canonical system J = U = zja()u, U(, z) = I, (21) «1 f «, A() = + () f 1 f () +() f (). (22)

5 The Airy function is a Fredholm determinant 5 2. Canonical systems such as (21) may e solved as follows (we follow [2]). Let fˆ + (, z) and ˆf (, z) denote the solutions to the Volterra equations fˆ + (, z) = f +() z ˆf (, z) = f () z We may then check y sustitution that U(, z) = I z Z Z R f (y) f ˆ R + (y, z) dy R f +(y) f ˆ + (y, z) dy G (, y) f ˆ + (y, z) dy, (23) G (, y) ˆf (y, z) dy. f (y) ˆf! R (y, z) dy f +(y) ˆf (y, z) dy is the unique solution to (21). The calculation so far applies to all semi-separale kernels. Since G is the Green s function of L, we may go further. We apply L to (23) to find that (24) L ˆ f + = z ˆ f +, L ˆf = z ˆf. (25) Thus, f ˆ and ˆf are linear cominations of Ai( + z) and Bi( + z). Moreover, the initial conditions on f ˆ + and ˆf are determined y fˆ + ()! fˆ + ()! = f +() f+, ()! ˆf () ˆf () = «f () f (). (26) We solve for f ± and sustitute in (24) to find that U 22 (, z) is given y (19). 2.4 The it and the proof of Theorem 1 The kernel G may e trivially etended to an integral operator on L 2 (R) y setting G (, y) = if either or y is greater than. We ause notation and continue to denote this etension y G. The proof of Theorem 1 now rests on the following assertions. Lemma 1 Lemma 2 det 1(1 + zg )e ztrg Ai(z) =, Re(z) >. (27) Ai() G G HS(L 2 (R + )) =. (28) Proof (of Theorem 1) For each z in the right-half plane Ai(z) Ai() = det 1(1 + zg )e ztrg = det 2(1 + zg ) = det 2 (1 + zg). The first equality follows from Lemma 1, the second equality from Lemma 2. This estalished the identity (8) for Re(z) >. Since det 2 (1 + zg) is an entire function of z, the identity holds for all z C.

6 6 Govind Menon 2.5 Proof of Lemma 1 Lemma 3 Tr(G ) = 1. (29) Proof We must consider the asymptotics of the integral Tr(G ) = R f ()f () d. It is clear from formulas (3) and (18) for f and f + that the dominant term in this it must involve Bi() as. Moreover, c 1. Thus, Tr(G ) π Ai() Bi() Ai() «Bi(). Bi() Equation (13) shows that the integral R Ai()Bi() is divergent, whereas as we show elow, as a consequence of (11) and Lemma 4 Hence we find Ai() Bi() Tr(G ) = Bi 2 () d =. (3) Z π Ai()Bi() d = 1. (31) Proof (of Lemma 1) Fi z such that Re(z) >. Lemma 3 allows us to ignore all the terms in (19) that remain finite as, since these are weighted y the decaying factor e z as. An inspection of the terms in (19), equation (3) and Lemma 3 imply that det 2(1 + zg ) = πzai(z) Ai() e z Ai()Bi( + z) d. (32) As, equation (11) yields the leading order asymptotics Thus, to leading order Ai()Bi( + z) 1 2π ez. Ai()Bi( + z) d 1 e d = ez 2π πz, which comines with the right hand side of (32) to yield (27). These asymptotics can e rigorously justified without much effort (for eample, y rescaling so that the integrals are over a fied domain [, 1] and using the dominated convergence theorem to justify taking pointwise its).

7 The Airy function is a Fredholm determinant Proof of Lemma 2 Since G vanishes when > or y > we have G G 2 HS = Since G HS <, G (, y) G(, y) 2 d dy+ G 2 HS G 2 HS G 2 (, y) d dy =. Therefore, to prove Lemma 2, it is enough to show that G 2 (, y) d dy. G (, y) G(, y) 2 d dy =. (33) We use the definition of G and G in equations (5) and (17) to otain G (, y) G(, y) = f (min(, y)) f+(ma(, y) f + (ma(, y). We use the definition of f + in (18) and recall that f + (s) = Ai(s) to otain f+(s) 2 f + (s) = (1 c ) 2 Ai 2 (s) + 2c (1 c ) Ai() Bi() Bi(s) + c2 Ai 2 () Bi 2 () Bi2 (s). We must show that the contriution of each of these terms vanishes in the it. Since c 1 it is easy to see that the first term gives a vanishing contriution. The calculation for the second and third term is similar, and we present the calculation only for the third term Ai 2 () Bi 2 () f 2 (, y)bi 2 (ma(, y)) d dy. Since f is a linear comination of Ai() and Bi(), and Ai() decays fast, it will suffice to show that the worst term Ai 2 () Bi 2 () Z Bi 2 (min(, y))bi 2 (ma(, y)) d dy = Ai2 () «2 Bi 2 Bi 2 () d () vanishes in the it. Applying the asymptotic relations (11), we see that we must prove that Ai() Bi() Bi 2 () d = e 4 2/3 /3 1/2 e 42/3 /3 d =. We define u = 4 2/3 /3 and note that the it aove is (upto a constant) Z 4 2/3 2/3 /3 e 4 /3 which vanishes y the following lemma. u 2/3 e u du,

8 8 Govind Menon Lemma 4 Fi < α < 1. Then M e M Z M u α e u du =. (34) Proof Fi a >. We separate the integral into an integral over two intervals: (, M a) and (M a, M). First, Similarly, e M Z M a e M Z M M a u α e u du 1 1 α e a M 1 α. u α e u du a (M a) α. We now choose a(m) in a such a way that e a M 1 α and am α (a = M α/2 will do). Acknowledgements Supported y NSF grant References 1. Aramowitz, M., Stegun, I.A.: Handook of mathematical functions with formulas, graphs, and mathematical tales, National Bureau of Standards Applied Mathematics Series, vol. 55. For sale y the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. (1964) 2. Gesztesy, F., Makarov, K.A.: (Modified) Fredholm determinants for operators with matrivalued semi-separale integral kernels revisited. Integral Equations Operator Theory 47(4), (23). DOI 1.17/s y. URL 3. Goherg, I., Golderg, S., Kaashoek, M.A.: Classes of linear operators. Vol. I, Operator Theory: Advances and Applications, vol. 49. Birkhäuser Verlag, Basel (199). DOI 1.17/ URL 4. Groeneoom, P.: Brownian motion with a paraolic drift and Airy functions. Proa. Theory Related Fields 81(1), (1989). DOI 1.17/BF URL 5. Menon, G., Srinivasan, R.: Kinetic theory and La equations for shock clustering and Burgers turulence. J. Stat. Phys. 14(6), (21). DOI 1.17/s URL 6. Simon, B.: Trace ideals and their applications, Mathematical Surveys and Monographs, vol. 12, second edn. American Mathematical Society, Providence, RI (25) 7. Titchmarsh, E.C.: Eigenfunction epansions associated with second-order differential equations. Part I. Second Edition. Clarendon Press, Oford (1962) 8. Tracy, C.A., Widom, H.: Level-spacing distriutions and the Airy kernel. Comm. Math. Phys. 159(1), (1994). URL