Examples of potentials for which Schrödinger equation lacks discrete points in continuous spectrum 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.
 A: 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 of Milivoje Lukic.
A: Consider the (stationary) Schrödinger equation
$$
-\Delta u+ Vu=0,
$$
or the differential inequality
$\vert\Delta u \vert\le \vert V u\vert$, where $V$ is some "potential" function. The following unique continuation property holds and is optimal with respect to the exponents involved. Take $$\vert V(x)\vert \le \frac{C}{\vert x\vert^2}.
$$
Then if $\vert\Delta u \vert\le \vert V u\vert$ and $u$ is flat at $0$, i.e. is such that 
$$
\int_{\vert x\vert\le R}\vert u(x)\vert^{2} dx= O(R^N),\quad\text{for all $N\in \mathbb N$},
$$
then $u$ vanishes identically. You can prove using that unique continuation property that no eigenvalue is embedded in the continuous spectrum for the operator $-\Delta + V$. Various other conditions could replace our example above such that $L^{dimension/2}_{loc}$.
