Examples of potentials for which Schrödinger equation lacks discrete points in continuous spectrum

Question

In Landau, Lifshitz, "Quantum Mechanics, non-relativistic theory" in $\S18$ "The fundamental properties of Schrödinger's equation" the following is said 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.

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

Answer 1

This question has been satisfactorily answered on MathOverflow. Here's it, for reference:

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.

For completeness, here's the Wigner–von Neumann example (description taken from here and reshaped a bit).

Consider 3D radial Schrödinger equation

$$-\frac1{r^2}\frac d{dr}\left(r^2\frac d\psi(r){dr}\right)+q(r)\psi(r)=\lambda\psi(r),$$

where

$$q(r)=\frac{-32\sin(r)\left(g(r)^3\cos r-3g(r)^2\sin^3 r+g(r)\cos r+\sin^3 r\right)}{(1+g(r)^2)^2}$$

with $g(r)=2r-\sin(2r).$

The Hamiltonian here obviously has continuous spectrum from $0$ to $+\infty$. But there's also an eigenvalue $\lambda=1$ with the following eigenfunction:

$$\psi(r)=\frac{\sin r}{r\left(1+g(r)^2\right)},$$

which it's easy to see is square integrable with weight of $r^2$.