hysteresis or bistability between symmetric and asymmetric modes.

Generic examples of unstable symmetric and stable asymmetric states of both trapped ($k>0$) the leaky ($k_{\mathrm{bif}}<k<0$) types are displayed in Figs. \[symmetric and asymmetric modes\](a) and (b), respectively. In the latter case, the delocalized tails of the leaky mode are extremely small, with amplitude $$u_{\mathrm{rad}}^{(\max )}\approx 1.2\times 10^{-4}.  \label{rad}$$Taking into regard the value of the amplitude of the delocalized mode at its center, $U_{0}\approx 0.40$, tunneling coefficient (\[T\]) predicts amplitude $u_{\mathrm{rad}}^{(\max )}\sim TU_{0}\simeq 1.8\times 10^{-4}$, in reasonable agreement with its numerically found counterpart (\[rad\]). For given $k$, the spatial period of the tail in the free space is expected to be $\lambda =\pi \sqrt{2/|k|}\approx \allowbreak 15$ for $k=-0.085$ in Fig. \[symmetric and asymmetric modes\](b), while the numerical solutions demonstrates a close value, $\lambda \approx 13.5$ (a small deviation from the predicted value may be explained by the proximity of the tail to the outer barrier).

 

The formally diverging contribution of the tails to the total norm of the leaky mode is negligible, $N_{\mathrm{rad}}\simeq \left( L/2\right) \left[
u_{\mathrm{rad}}^{(\max )}\right] ^{2}$, where $L\approx 20$ is the total size of the free-space part of the configuration displayed in Fig. [symmetric and asymmetric modes]{}(b). Indeed, using 