On amplitude zeros at threshold

The occurrence of zeros of 2 to n amplitudes at threshold in scalar theories is studied. We find a differential equation for the scalar potential, which incorporates all known cases where the 2 to n amplitudes at threshold vanish for all sufficiently large $n$, in all space-time dimensions, $d\ge 1$. This equation is related to the reflectionless potentials of Quantum Mechanics and to integrable theories in 1+1 dimensions. As an application, we find that the sine-Gordon potential and its hyperbolic version, the sinh-Gordon potential, also have amplitude zeros at threshold, ${\cal A}(2\to n)=0$, for $n\ge 4$ and $d\ge 2$, independently of the mass and the coupling constant.