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:

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.

For reference, in Russian version the wording is

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.

I wonder, what are the examples of such mathematical forms of potential?

I've previously posted this question on Math.SE, even tried offering a bounty, but apparently no one knows the answer there.