Large degree asymptotics of orthogonal polynomials with respect to an oscillatory weight on a bounded interval (Q401430)

The paper studies polynomials \(p_n^{\omega}(x)\) (degree \(n\geq 0\)), orthogonal with respect to the oscillatory weight NEWLINE\[NEWLINEw(x)=e^{i\omega x}NEWLINE\]NEWLINE on the interval \([-1,1],\;\omega>0\) being a real parameter.NEWLINENEWLINEThe layout of the paper is as follows.NEWLINENEWLINE{ \S1. Introduction}NEWLINENEWLINESome historical references, followed by rewriting the weight as NEWLINE\[NEWLINEw_n(x)=e^{-nV(x)},\;V(x)=-{i\omega_n x\over n}=-i\lambda x.NEWLINE\]NEWLINE It is indicated that the results on the large \(n\) asymptotic behavior of \(p_n^{\omega}(x)\) will be obtained using a Riemann-Hilbert formulation of the problem and the Deift-Zhou method of steepest descent.NEWLINENEWLINE{ \S2. Statement of main results}NEWLINENEWLINEWith \(\lambda\) as above, introduce NEWLINE\[NEWLINEh(\lambda)=2\log{\left({2+\sqrt{\lambda^2+4}\over \lambda}\right)}-\sqrt{\lambda^2+4},NEWLINE\]NEWLINE and the unique solution \(\lambda_0\) of \(h(\lambda)=0\) (the value is given as \(\approx 1.325486839\)).NEWLINENEWLINEUsing NEWLINE\[NEWLINE\varphi(z)=z+(z^2-1)^2,NEWLINE\]NEWLINE the main results are then formulated asNEWLINENEWLINE{ Theorem 2.1.} Let \(V(z)=-i \lambda z\) with \(0\leq \lambda<\lambda_0\), thenNEWLINENEWLINE1. there exists a smooth curve \(\gamma_{\lambda}\) joining \(z=1\) and \(z=-1\) that is part of the level set \(\text{Re}\,\phi(z)=0\), where \(\phi(z)=2\log{\varphi(z)}+i\lambda (z^2-1)^{1/2}\), and the cut of the square root is taken on \(\gamma_{\lambda}\),NEWLINENEWLINE2. the measure NEWLINE\[NEWLINE\text{d}\mu_{\lambda}(z)=\psi_{\lambda}(z)=-\,{1\over 2\pi i}\,{2+i\lambda z\over (z^2-1)^{1/2}}\text{d}z,NEWLINE\]NEWLINE with a branch cut taken on \(\gamma_{\lambda}\), is the equilibrium measure on \(\gamma_{\lambda}\) in the external field \(\text{Re}\,V(z)\),NEWLINENEWLINE3. the curve \(\gamma_{\lambda}\) joining \(z=-1\) and \(z=1\) has the \(S\)-property in the external field \(\text{Re}\,V(z)\),,NEWLINENEWLINE4. if we consider the normalized zero counting measure of \(p_n^{\omega}(x)\), then NEWLINE\[NEWLINE\mu_n={1\over n}\,\sum_{p_n^{\omega}(\zeta)=0}\;\delta(\zeta)\;\buildrel \ast\over \rightarrow\mu_{\lambda},NEWLINE\]NEWLINE as \(n\rightarrow\infty\) in the sense of weak convergence of measures.NEWLINENEWLINE{ Theorem 2.2.} Let \(0\leq \lambda<\lambda_0\), then the following holds true:NEWLINENEWLINE1. For large enough \(n\) the orthogonal polynomial \(p_n^{\omega}(x)\) exists uniquely and its zeros accumulate on \(\gamma_{\lambda}\) as \(n\rightarrow\infty\).NEWLINENEWLINE2. For \(z\in\mathbb C\setminus\gamma_{\lambda}\), the monic polynomial \(p_n^{\omega}(x)\) has the asymptotic behavior NEWLINE\[NEWLINEp_n^{\omega}(x)={\varphi(z)^{n+1/2}\over 2^{n+1/2}(z^2-1)^{1/4}}\exp{\left(-\,{in\lambda\over 2\varphi(z)}\right)}\left(1+{\mathcal O}\left({1\over n}\right)\right),\;n\rightarrow\infty.NEWLINE\]NEWLINENEWLINENEWLINENEWLINE3. Fix a neighbourhood \(U\) of \(\gamma_{\lambda}\) in the complex plane, and two discs NEWLINE\[NEWLINED(\pm 1,\delta)=\{z\in\mathbb C\,:\, |z\mp 1|<\delta\},NEWLINE\]NEWLINE with \(\delta>0\). For \(z\in U\setminus (D(1,\delta)\cup D(-1,,\delta))\), we have as \(n\rightarrow\infty\) NEWLINE\[NEWLINEp_n^{\omega}(x)={2^{1/2-n}e^{-in\lambda z/2}\over (1-z^2)^{1/4}}\times \left[\cos{\left(\left(n+{1\over 2}\right)\arccos{z} +{n\lambda\over 2}(z^2-1)^{1/2}-{\pi\over 4}\right)}+{\mathcal O}(1/n)\right].NEWLINE\]NEWLINENEWLINENEWLINE4. For \(z\in D(1,\delta)\) we have NEWLINE\[NEWLINEp_n^{\omega}(x)=2^{-n}(2n\pi )^{1/2}f(z)^{1/4}e^{-in\lambda z/2}\times \left[\beta(z)^{-1}J_0\left(-{in\phi(z)\over 2}\right)-i\beta(z)J_0'\left(-{in\phi(z)\over 2}\right) +{\mathcal O}(1/n)\right],NEWLINE\]NEWLINE as \(n\rightarrow\infty\), in terms of standard Bessel functions, with \(\phi(z)\) given as before and \(f(z)=\phi(z)^2/16\). Here \(\beta(z)=\left({z-1\over z+1}\right)^{1/4}\), with a branch cut taken on \(\gamma_{\lambda}\).NEWLINENEWLINE{ Theorem 2.5.} Let \(0\leq \lambda<\lambda_0\), then the coefficients \(a_n^2\) and \(b_n\) in the three term recurrence relation NEWLINE\[NEWLINExp_n^{\omega}(x)=p_{n+1}^{\omega}(x)+b_np_n^{\omega}(x)+a_n^2p_{n-1}^{\omega}(x),NEWLINE\]NEWLINE exist for large enough \(n\),and they satisfy NEWLINENEWLINE\[NEWLINEa_n^2={1\over 4}+{4-\lambda^2\over 4(4+\lambda^2)}\,{1\over n^2}+{\mathcal O}\left({1\over n^3}\right),\;b_n=-\,{2i\lambda\over (4+\lambda^2)^2}\,\left({1\over n^3}\right).NEWLINE\]NEWLINENEWLINENEWLINE{ \S3. Proof of Theorem 2.1} (\(7{1\over 2}\) pages)NEWLINENEWLINE{ \S4. Proof of Theorem 2.2} (\(9{1\over 2}\) pages)NEWLINENEWLINE{ \S5. Proof of Theorem 2.5} (\(4\) pages)NEWLINENEWLINE{ Acknowledgements}NEWLINENEWLINE{ Appendix. Steepest descent analysis for fixed \(\omega\)} (\(2\) pages)NEWLINENEWLINE{ References} (\(28\) items)