Sharp thresholds for stability and instability of standing waves in a double power nonlinear Schr\"odinger equation

We study the stability/instability of standing waves for the one dimensional nonlinear Schr\"odinger equation with double power nonlinearities: \begin{align*} &i\partial_t u +\partial_x^2 u -|u|^{p-1}u +|u|^{q-1}u=0, \quad (t,x)\in \mathbb{R}\times\mathbb{R} ,~1<p<q. \end{align*} When $q<5$, the stability properties of standing waves $e^{i\omega t}\phi_{\omega}$ may change for the frequency $\omega$. A sufficient condition for yielding instability for small frequencies are obtained in previous results, but it has not been known what the sharp condition is. In this paper we completely calculate the explicit formula of $\lim_{\omega\to0}\partial_{\omega}\|\phi_{\omega}\|_{L^2}^2$, which is independent of interest, and establish the sharp thresholds for stability and instability of standing waves.