Consider a 1D zero-energy Schrödinger equation on the half-line,


$(-\partial\_x^2 + V(x))\psi(x)=0, \quad x \in (0, \infty)$


with a zero boundary condition $\psi(0) = 0$.


Is it true that if the zero-energy solution $\psi(x)$ has $N$ nodes in $(0, \infty)$, then $V(x)$ has exactly $N$ negative energy bound states?


If yes, may I ask for the reference?



A nodal theorem in 1D
Yes, this is true. This is essentially the Sturm oscillation theorem.


A pedagogical reference is 


B. Simon, in *Sturm-Liouville Theory* (Birkhäuser Basel, 2005), pp. 29—43.


Many thanks to Christian Remling for the right keyword.


