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Name:
Miloslav Znojil
Reviewer number:
9689
Email:
znojil@ujf.cas.cz
Item's zbl-Number:
DE 018 207 684
Author(s):
Remling, Christian:
Shorttitle:
Schroedinger operators with decaying potentials: some counterexamples
Source:
Duke Math. J. 105, No. 3, 463 - 496 (2000)
Classification:
34L40Particular operators Dirac, one-dimensional Schroedinger, etc.
Primary Classification:
81Q10Selfadjoint operator theory in quantum theory, including spectral analysis
Secondary Classification:
47A25Spectral sets
Keywords:
Schroedinger operator on half-line; slowly decaying potentials; absolutely continuous spectrum; embedded singular spectrum; Hausdorff dimension of the exceptional set; contructive proof of optimality of its estimate; sum rules for asymptoticaly Coulomb potentials
Review:

Schropedinger equation is studied on half-axis, with an
asymptoticaly slowly decreasing potential V(x) which is
asymptotically bounded by a negative power (between minus one and
- not included - minus one half) of the coordinate. A mixed boundary
condition in the origin is admitted. A review is offered of the
results concerning the essential spectrum (on the positive
energy half-line) and its possible (i.e., point or absolutely
continuous or singular continuous) more detailed character. The
purpose is to construct examples which show that the results
available in the literature are, in fact, optimal.

An extremely well readable presentation of rigorous results (well
motivated and complemented by heuristic arguments, e.g., in secton 3)
constructs its``counterexample" sample potentials, mostly, by pasting
together carefully chosen periodic pieces. Then, the bound on the
embedded singular spectrum can only be show optimal (in Theorem 1.3)
for the asymptotic decay of V(x) betwen two thirds and one, which is
a not entirely pleasant natural limitation of the constructive
approach. Its strength, on the other hand, manifests itself at the
Coulombic end (especially via sum rules of Theorem 1.5 and its
Corollary 1.6). The proofs pay due attention to the effects of
pasting (non-smoothness) and make ample use of periodicity of the
pasted pieces, say, for related asymptotic estimates of the so called
Ljapunov function (= trace of the transfer matrix), with
considerations closely related to some recent Kiselev's results [1].
Remarks to the editors:

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