In Landau, Lifshitz, "Quantum Mechanics, non-relativistic theory" in $\S18$ "The fundamental properties of Schrödinger's equation" the following is said about potential $U(x,y,z)$ in a footnote:



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> it must be mentioned that, for some particular mathematical forms of the function $U(x,y,z)$ (which have no physical significance), a discrete set of values may be absent from the otherwise continuous spectrum.
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For reference, in Russian version the wording is



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> надо, однако, оговориться, что при некоторых определенных видах функции $U(x,y,z)$ (не имеющих физического значения) из непрерывного спектра может выпадать дискретный набор значений.
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I wonder, what are the examples of such mathematical forms of potential?


I've previously [posted this question](https://math.stackexchange.com/questions/737976/) on Math.SE, even tried offering a bounty, but apparently no one knows the answer there.



Examples of potentials for which Schrödinger equation lacks discrete points in continuous spectrum
The spectrum of an operator is always a closed set. But perhaps they are defining the "continuous spectrum" to be all points of the spectrum that are not in the point spectrum (i.e. not eigenvalues). Then there can be eigenvalues surrounded by continuous spectrum. The classic example of this in a Schrödinger operator is due to Wigner and von Neumann. See e.g. [this recent paper](http://arxiv.org/abs/1201.4840) of Milivoje Lukic.


