 ![The wavefunctions $\Psi_0$ (solid line) and $\Psi_0$ (dashed line) are shown for $\omega=0$ for (a) the first ($n=0$) mode and (b) the second ($n=0$) mode. A potential profile is provided as a guide for the eye (dotted line). The insets show the electron density profile for the corresponding modes.[]{data-label="fig:Cosh_0/0"}](fig0a "fig:"){width="0.0cm"} ![The wavefunctions $\Psi_0$ (solid line) and $\Psi_0$ (dashed line) are shown for $\omega=0$ for (a) the first ($n=0$) mode and (b) the second ($n=0$) mode. A potential profile is provided as a guide for the eye (dotted line). The insets show the electron density profile for the corresponding modes.[]{data-label="fig:Cosh_0/0"}](fig0b "fig:"){width="0.0cm"}  