Standing waves for the NLS on the double-bridge graph and a rational-irrational dichotomy

We study a boundary value problem related to the search of standing waves for the nonlinear Schr\"odinger equation (NLS) on graphs. Precisely we are interested in characterizing the standing waves of NLS posed on the {\it double-bridge graph}, in which two semi-infinite half-lines are attached at a circle at different vertices. At the two vertices the so-called Kirchhoff boundary conditions are imposed. The configuration of the graph is characterized by two lengths, $L_1$ and $L_2$, and we are interested in the existence and properties of standing waves of given frequency $\omega$. For every $\omega>0$ only solutions supported on the circle exist (cnoidal solutions), and only for a rational value of $L_1/L_2$; they can be extended to every $\omega\in \mathbb{R}$. We study, for $\omega<0$, the solutions periodic on the circle but with nontrivial components on the half-lines. The problem turns out to be equivalent to a nonlinear boundary value problem in which the boundary condition depends on the spectral parameter $\omega$. After classifying the solutions with rational $L_1/L_2$, we turn to $L_1/L_2$ irrational showing that there exist standing waves only in correspondence to a countable set of frequencies $\omega_n$. Moreover we show that the frequency sequence $\{\omega_n\}_{n \geq 1}$ has a cluster point at $-\infty$ and it admits at least a finite limit point, in general non-zero. Finally, any negative real number can be a limit point of a set of admitted frequencies up to the choice of a suitable irrational geometry $L_1/L_2$ for the graph. These results depend on basic properties of diophantine approximation of real numbers.