Universal relations in asymptotic formulas for orthogonal polynomials

Orthogonal polynomials $P_{n}(\lambda)$ are oscillating functions of $n$ as $n\to\infty$ for $\lambda$ in the absolutely continuous spectrum of the corresponding Jacobi operator $J$. We show that, irrespective of any specific assumptions on the coefficients of the operator $J$, the amplitude and phase factors in asymptotic formulas for $P_{n}(\lambda)$ are linked by certain universal relations found in the paper. Our proofs rely on the study of a time-dependent evolution generated by suitable functions of the operator $J$.