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Critical coupling constants and eigenvalue asymptotics of perturbed periodic Sturm-Liouville operators. (English) Zbl 0953.34069
Summary: Perturbations of asymptotic decay $c/r^2$ arise in the partial-wave analysis of rotationally symmetric partial differential operators. The author shows that for each end-point $\lambda_0$ of the spectral bands of a perturbed periodic Sturm-Liouville operator, there is a critical coupling constant $c_{\text{crit}}$ such that eigenvalues in the spectral gap accumulate at $\lambda_0$ if and only if $c/c_{\text{crit}}>1$. The oscillation theoretic method used in the proof also yields the asymptotic distribution of the eigenvalues near $\lambda_0$.

34L20Asymptotic distribution of eigenvalues for OD operators
34B24Sturm-Liouville theory
34L40Particular ordinary differential operators
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