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 \(\lambda_0\).


34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34B24 Sturm-Liouville theory
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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