There are asymptotic formulas for eigenvalues $\lambda_n$ and eigenfunctions $\phi(x, \lambda_n)$ of a Sturm-Liouville boundary value problem: $$ \rho_n = \sqrt{\lambda_n} = n + \frac{\omega}{\pi n} +\frac{\varkappa_n}{n} $$ $$ \phi(x, \lambda_n) = \cos nx + \frac{\xi_n(x)}{n} $$ where $$ \omega = h + H + \frac{1}{2}\int_{0}^{\pi}q(t)dt; \quad \{\varkappa_n\} \in l_2; \quad |\xi_n(x)|\leq C \quad (C>0) $$ The boundary value problem itself is: $$ -y''+q(x)y=\lambda y \quad 0<x<\pi $$ $$ y'(0)-hy(0)=0 \quad y'(\pi)+Hy(\pi)=0 $$
Question: What is the meaning of and the logic behind the above-mentioned asymptotic formulas? Why are they useful?