The Extraordinary Spectral Properties of Radially Periodic Schrödinger Operators

Proceedings Mathematical Sciences

Andreas M Hinz¹ 

[1] Zentrum Mathematik, Technische Universität München, 00 München, Germany$$

关键词: Schrödinger operator;    self-adjointness;    embedded eigenvalue;    exponential decay;    dense point spectrum.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Since it became clear that the band structure of the spectrum of periodic Sturm Liouville operators 𝑡 = - (d2/d𝑟2) + 𝑞(𝑟) does not survive a spherically symmetric extension to Schrödinger operators 𝑇 = - 𝛥 + 𝑉 with 𝑉(𝑥) = 𝑞(|𝑥|) for $x in mathbb{R}^d, d in mathbb{N}ackslash{1}$, 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.

【 授权许可】

Unknown