Critical coupling constants and eigenvalue asymptotics of perturbed periodic Sturm-Liouville operators. (English) Zbl 0953.34069

Schmidt, Karl Michael

Commun. Math. Phys. 211, No. 2, 465-485 (2000).

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.)

Keywords:

perturbed periodic Sturm-Liouville operators; rotationally symmetric partial differential operators; critical coupling constant; eigenvalues