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 Question: What is the meaning of and the logic behind the above-mentioned asymptotic formulas? Why are they useful?
Asymptotic formulas for eigenvalues and eigenfunctions
1
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functional-analysis
asymptotics
spectral-theory
sturm-liouville
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1You probably have an error in $\phi(x,\lambda_n)$ because $\lambda_n$ is missing on the right. The idea is that, for large $\lambda$, the function $q$ is overwhelmed by $\lambda$ in $-y''=(\lambda-q)y$ and the eigenfunctions/eigenvalues behave very much like those for the problem for $q=0$ with the same endpoint conditions. The pointwise convergence of the Fourier series in the new eigenfunctions is then related to classical cases of trigonometric expansions associated with $q=0$ (they're equivalent.) – 2017-01-15