Absence of non-threshold embedded eigenvalues for nonlinear Schr\"odinger equations

Abstract

We consider the focusing nonlinear Schr\"odinger equation on $\mathbb{R}^d$ \begin{equation*} i\partial_tu+\Delta u+|u|^{p-1}u=0. \end{equation*} We prove the absence of non-threshold embedded eigenvalues for the linearized operator around the positive ground state when $d=1,2$ and $p>1$, or when $d\ge3$ and $1<p\le1+\frac4{d-2}$. Starting from the fluid variables of NLS, we derive a biharmonic wave equation for the phase gradient after linearization around the ground state. Seeking a virial identity for this equation, we identify a mechanism for establishing a repulsive structure, which guides the subsequent ODE analysis to exclude embedded eigenvalues.