The extraordinary spectral properties of radially periodic Schrödinger operators

Since it became clear that the band structure of the spectrum of periodic Sturm–Liouville operators t = − (d/dr2) + q(r) does not survive a spherically symmetric extension to Schrödinger operators T = −1+V with V (x) = q(|x|) for x ∈ R , 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,∞[ of T with exponentially decaying eigenfunctions provided evidence for the existence of intervals of dense point spectrum, eventually proved by spherical separation into perturbed Sturm–Liouville operators tc = t + (c/r2). Subsequently, a numerical approach was employed to investigate the distribution of eigenvalues of T more closely. An eigenvalue was discovered below the essential spectrum in the case d = 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 of tc contributing to an interval of dense point spectrum of T . We gained evidence that an asymptotic formula, valid for c → ∞, 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.