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Asymptotic properties of polynomials orthogonal with respect to varying weights, and related topics of spectral theory. (English. Russian original) Zbl 1304.37054

St. Petersbg. Math. J. 25, No. 2, 223-240 (2014); translation from Algebra Anal. 25, No. 2, 101-124 (2013).
The paper deals with the asymptotic properties as \(n\to\infty\) of the coefficients of the three-term relations \[ a_{l-1}^{(n)}p_{l-1}^{(n)} (x)+ b_l^{(n)}p_l^{(n)}(x) + a_l^{(n)}p_{l+1}^{(n)}(x)=\lambda p_l^{(n)}(x) \] for polynomials orthogonal with respect to a varying weight \(e^{-n V(x)}\) , where \(V\) is a continuous function such that \(V(x)\geq (1+\varepsilon)\log (1+x^2)\). The support of the corresponding equilibrium measure in this case consists of a finite number \(g\) of finite intervals. The asymptotic formulas for the coefficients \(a_{n-1}^{(n)}\) and \(b_{n-1}^{(n)}\) were obtained by Deift, Kriecherbauer, McLaughlin and Zhou via a Riemann-Hilbert problem approach. Using the same approach, the authors establish the asymptotics of the coefficients \(a_{n+k}^{(n)}\) and \(b_{n+k}^{(n)}\) as \(n\to \infty\) for \(k\in\mathbb Z\). The following property of these coefficients is obtained: Let \(\mathbf x\in [0,1]^g\) be an arbitrary vector; then, there exists a subsequence \(n_i(\mathbf x)\to \infty\) such that \[ a_{n_i(\mathbf x)+k}^{(n_i(\mathbf x))}\to a_k(\mathbf x), \quad b_{n_i(\mathbf x)+k}^{(n_i(\mathbf x))}\to b_k(\mathbf x), \] where \(\{a_k(\mathbf x), b_k(\mathbf x)\}\) are the coefficients of a finite gap Jacobi operator. This operator has the continuous spectrum on the support of an equilibrium measure. The formula for the initial divisor depending on \(\mathbf x\) is given.

MSC:

37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35Q53 KdV equations (Korteweg-de Vries equations)
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
35Q15 Riemann-Hilbert problems in context of PDEs
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