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The problem of a spin-free electron with mass $m$, charge $e$ confined onto aring of radius $R_0$ and with an attractive Dirac delta potential with scalingfactor (depth) $\kappa$ in non-relativistic theory has closed form analyticalsolutions. The single bound state function is of the form of a hyperboliccosine that however contains a parameter $d>0$ which is the single positivereal solution of the transcendental equation $\coth(d) = \lambda d$ for nonzero real $\lambda=\frac{2}{\pi\kappa}$. The energy eigenvalue of the boundstate $\varepsilon=-\frac{d^2}{2\pi^2}\approx \frac{q e m R_0}{2 \hbar^2}$. Inaddition a discrete infinite set of unbounded solutions exists, formally theyare obtained from the terms for the bound solution by substituting $d \to i d $yielding $\cot(d) = \lambda d$ as characteristic equation with thecorresponding set of solutions $d_k, k\in\mathbb{N}$, the respective statefunctions obtain via $\cosh(x)\overset{x \to i x}{\longrightarrow}\cos(x)$ ofthe form of cosine functions.

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