Our goal is to find asymptotic formulas for orthonormal polynomials $$P_{n}(z)$$ P n ( z ) with the recurrence coefficients slowly stabilizing as $$n\rightarrow \infty $$ n → ∞ . To that end, we develop scattering theory of Jacobi operators with long-range coefficients and study the corresponding second-order difference equation. We introduce the Jost solutions $$f_{n}(z)$$ f n ( z ) of this equation by a condition for $$n\rightarrow \infty $$ n → ∞ and suggest an Ansatz for them playing the role of the semiclassical Liouville–Green Ansatz for the corresponding solutions of the Schrödinger equation. This allows us to study Jacobi operators and their eigenfunctions $$P_{n}(z)$$ P n ( z ) by traditional methods of spectral theory developed for differential equations. In particular, we express all coefficients in asymptotic formulas for $$P_{n}(z)$$ P n ( z ) as $$\rightarrow \infty $$ → ∞ in terms of the Wronskian of the solutions $$\{P_{n} (z)\}$$ { P n ( z ) } and $$\{f_{n} (z)\}$$ { f n ( z ) } .

中文翻译：

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正交多项式的半经典渐近行为

我们的目标是找到正交多项式 $$P_{n}(z)$$ P n ( z ) 的渐近公式，其中递归系数缓慢稳定为 $$n\rightarrow \infty $$ n → ∞ 。为此，我们开发了具有长程系数的雅可比算子的散射理论，并研究了相应的二阶差分方程。我们通过 $$n\rightarrow \infty $$ n → ∞ 的条件引入这个方程的 Jost 解 $$f_{n}(z)$$fn ( z ) 并建议一个 Ansatz 为他们扮演薛定谔方程对应解的半经典 Liouville-Green Ansatz。这使我们能够通过为微分方程开发的传统谱理论方法研究雅可比算子及其本征函数 $$P_{n}(z)$$P n ( z )。特别是，