On the internal modes of the ground state for 3D cubic--quintic equations

Motivated by the problem of asymptotic stability for ground states of the Klein–Gordon and Schrödinger equations in three spatial dimensions in the presence of internal modes, we study the corresponding models with cubic–quintic nonlinearities. These equations arise in several physical contexts; in particular, the cubic–quintic Klein–Gordon equation appears naturally in the study of spin-$0$ particles in quantum field theory. More precisely, let $Q_{\omega }$ be the positive radial ground state of the three-dimensional cubic–quintic elliptic equation $$-\mathrm{\Delta }{Q}_{\omega }+\omega Q_{\omega }-Q_{\omega }^3+Q_{\omega }^5=0.$$ We study two related, but logically distinct, spectral problems as $\omega$ approaches the endpoint $3\mathrm{/}16$ of the ground-state branch. First, we determine the complete discrete spectrum of the scalar Hessian $$L_{+}=-\mathrm{\Delta }+\omega -3Q_{\omega }^2+5Q_{\omega }^4\mathrm{.}$$ This operator is also the exact linearized spatial operator around the static state $Q_{\omega }$ for the real scalar cubic–quintic Klein–Gordon equation with mass parameter $\omega$. We prove that the number of its discrete angular-momentum sectors tends to infinity, and obtain sharp asymptotics for their locations and multiplicities. Second, for the cubic–quintic nonlinear Schrödinger equation, we analyze the full Hamiltonian matrix linearization and prove the existence of internal modes in precisely an interval of angular-momentum sectors of length comparable to $(3\mathrm{/}16-\omega {)}^{-1}$. %The rotating complex Klein–Gordon problem leads %to a different gyroscopically coupled pencil; since that pencil is not %analyzed here, no identification of its internal spectrum is asserted. These results provide a rigorous linear foundation for future nonlinear stability and radiation-damping analysis in three-dimensional cubic–quintic models.