## 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 $$\lambda_0$$.

### MSC:

 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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