NDLI: The extraordinary spectral properties of radially periodic Schrödinger operators

Content Provider	Springer Nature Link
Author	Hinz, Andreas M.
Copyright Year	2002
Abstract	Since it became clear that the band structure of the spectrum of periodic Sturm-Liouville operatorst = - (d$^{2}$/dr$^{2}$) +q(r) does not survive a spherically symmetric extension to Schrödinger operatorsT =- Δ+ V with V(x) =q(¦x¦) for x ∈ ℝ$^{d}$,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 [μ$_{0}$, ∞[ 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 operatorst $_{c}$ = t +(c/r $^{2}$). 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 μ$_{0}$. Moreover, a method based on oscillation theory made it possible to count eigenvalues oft $_{c}$ 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.
Starting Page	85
Ending Page	98
Page Count	14
File Format	PDF
ISSN	02534142
Journal	Proceedings - Mathematical Sciences
Volume Number	112
Issue Number	1
e-ISSN	09737685
Language	English
Publisher	Springer India
Publisher Date	2002-01-01
Publisher Institution	Indian Academy of Sciences
Publisher Place	New Delhi
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Schrödinger operator self-adjointness embedded eigenvalue exponential decay dense point spectrum Mathematics
Content Type	Text
Resource Type	Article
Subject	Mathematics

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