Consider the operator $$ L = -\partial_x^2 + V(x),$$ for some bounded, decaying potential, i.e. $V(x)\to 0$ as $x\to \pm \infty$. I'm interested in the $L^2(\mathbb R)$ spectrum of $L$. We know that $L$ has continuous spectrum $[0,\infty)$, and in general, $L$ may or may not have negative discrete eigenvalues, depending on $V$.

My question: Are there any known conditions on $V$ that guarantee the existence or non-existence of a threshold resonance, i.e. a pole of the resolvent $(L-\lambda)^{-1}$, at $\lambda = 0$?