By John Fortier
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L) The matrix Aˆ n (z) is obtained from An by replacing the un−lq+s−1 and vn−lq+s (s) (s) by xn and yn respectively. One may ask whether the uN and vN functions must have such periodic behaviors. The explanation is that the string equations are applied to reorganize the wave functions, not the particles. In the momentum aspect, the parameters and the corresponding functions such as uN and vN control the wave functions of the random variables, so that the asymptotics of these functions are not directly related to the behaviors of the particles.
In this section, we discuss how to get an analytic function ω(η) with the asymptotics 12 W (η) − η1 as η → ∞ in the complex plane, where = ∂/∂η. 20) > 0. We have the following asymptotics e−V (z)/2 × − 12 V (z)+n ln z pn (z) ∼ e as z → ∞. This asymptotics gives a hint that the differential equation for the orthogonal polynomials may help us to find the ω(η). In the following, we introduce the basic Lax pair theory to construct the coefficient matrix An (z). The orthogonal polynomials satisfy a recursion formula , pn+1 (z) + un pn (z) + vn pn−1 (z) = zpn (z).
We finally have 1 Re π θ ω(η)dη γ3 j −1 2m = jgj j =2 q=1 j − 1 j −q−1 q+1 b a q [q/2]−μq r=0 q Θ2r+μq +1 . 10), the lemma is proved. The Θl in the above lemma can be further simplified by the recursion for some elementary integrals as described in the following. 3 For k = 0, 1, 2, . . , there are π θ e2iθ 1 − e2iθ k+ 12 0 π 0 θ eiθ 1 − e2iθ k+ 12 dθ = π , (2k + 3)i 1 dθ = −2 0 1 0 where B(·, ·) is the Euler beta function. 1 Free Energy for the One-Interval Case 49 Proof The first equation in this lemma can be easily verified by using integration by parts, π k+ 12 θ e2iθ 1 − e2iθ dθ = 0 π 1 (2k + 3)i 1 − e2iθ k+ 32 dθ = 0 π .
15th Virginia Cavalry by John Fortier