The extraordinary spectral properties of radially periodic Schrödinger operators

• Published in: Proceedings of the Indian Academy of Sciences. Mathematical sciences Vol. 112; no. 1; pp. 85 - 98
• Main Author: Hinz, Andreas M.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Nature B.V 01.02.2002
• Subjects: Eigenvalues; Eigenvectors; Operators (mathematics)
• ISSN: 0253-4142; 0973-7685
• DOI: 10.1007/BF02829642

Since it became clear that the band structure of the spectrum of periodic Sturm-Liouville operatorst = - (d2/dr2) +q(r) does not survive a spherically symmetric extension to Schrödinger operatorsT =- Δ+ V with V(x) =q(¦x¦) for x ∈ ℝd,d ∈ ℕ 1, a wealth of detailed information about the spectrum of such operators has been acquired. The observation of eigenvalues embedded in the essential spectrum [μ0, ∞[ ofT with exponentially decaying eigenfunctions provided evidence for the existence of intervals of dense point spectrum, eventually proved by spherical separation into perturbed Sturm-Liouville operatorstc = t +(c/r2). Subsequently, a numerical approach was employed to investigate the distribution of eigenvalues ofT more closely. An eigenvalue was discovered below the essential spectrum in the cased = 2, and it turned out that there are in fact infinitely many, accumulating at μ0. Moreover, a method based on oscillation theory made it possible to count eigenvalues oftc contributing to an interval of dense point spectrum ofT. We gained evidence that an asymptotic formula, valid forc → ∞, does in fact produce correct numbers even for small values of the coupling constant, such that a rather precise picture of the spectrum of radially periodic Schrödinger operators has now been obtained.