Coulomb potential in one dimension with minimal length: A path integral approach

We solve the path integral in momentum space for a particle in the field of the Coulomb potential in one dimension in the framework of quantum mechanics with the minimal length given by $(\mathrm{\Delta }X{)}_0=\mathrm{\hslash }\sqrt{\beta }$, where $\beta$ is a small positive parameter. From the spectral decomposition of the fixed energy transition amplitude we obtain the exact energy eigenvalues and momentum space eigenfunctions.