The Quantum Mechanical Problem of a Particle on a Ring with Delta Well

The problem of a spin-free electron with mass $m$, charge $e$ confined onto a ring of radius $R_0$ and with an attractive Dirac delta potential with scaling factor (depth) $\kappa$ in non-relativistic theory has closed form analytical solutions. The single bound state function is of the form of a hyperbolic cosine that however contains a parameter $d>0$ which is the single positive real solution of the transcendental equation $\coth (d)=\lambda d$ for non zero real $\lambda =\frac{2}{\pi \kappa }$. The energy eigenvalue of the bound state $\epsilon =-\frac{d^2}{2{\pi }^2}\approx \frac{qem{R}_0}{2{\mathrm{\hslash }}^2}$. In addition a discretly infinite set of unbounded solutions exists, formally these solutions are obtained from the terms for the bound solution by substituting $d\to id$ yielding $\cot (d)=\lambda d$ as characteristic equation with the corresponding set of solutions $d_k,k\in \mathbb{N}$, the respective state functions can be obtained via $\cosh (x)\cos (x)$.