Second Order Spectral Estimates and Symmetry Breaking for Rotating Wave Solutions

We consider rotating wave solutions of the nonlinear wave equation $$\{\begin{array}{rlrl}{\displaystyle {\mathrm{\partial }}_t^{2}v-\mathrm{\Delta }v+mv}& {\displaystyle =\mathrm{\mid }v{\mathrm{\mid }}^{p-2}v}& {\displaystyle }& {\displaystyle \text{in }\mathbb{R}\times \mathbf{\text{B}}}\\ {\displaystyle v}& {\displaystyle =0}& {\displaystyle }& {\displaystyle \text{on }\mathbb{R}\times \mathrm{\partial }\mathbf{\text{B}}}\end{array}$$ for $2<p<\mathrm{\infty }$, $m\in \mathbb{R}$ on the unit disk $\mathbf{\text{B}}\subset {\mathbb{R}}^2$. This leads to the study of a reduced equation involving the elliptic-hyperbolic operator $L_{\alpha }=-\mathrm{\Delta }+{\alpha }^2{\mathrm{\partial }}_{\theta }^2$ with $\alpha >1$. We find that the structure of the spectrum of $L_{\alpha }$ strongly depends on the quantity $$\sigma =\frac{\pi }{\sqrt{{\alpha }^2-1}-\arccos \frac{1}{\alpha }}>0.$$ By giving precise estimates for certain sequences of Bessel function zeros, we can classify the spectrum for all $\alpha >1$ such that $\sigma$ is rational and further find that the existence of accumulation points explicitly depends on arithmetic properties of $\sigma$. Using these characterizations, we deduce existence and symmetry breaking results for ground state solutions of the reduced equation, extending known results.