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eigenvalues of matrices or operators

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Spectrum of perturbed differential operators

(i) Is there an analytical expression for the eigenvalues and eigenfunctions of $\mathcal{L_n}$? … .$$ Since $\kappa_n \rightarrow \kappa$ as $n \rightarrow \infty$, is there any chance, that also the eigenvalues/eigenfunctions of $\mathcal{L}_n$ converge to the eigenvalues/eigenfunctions of $\mathcal …
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