Abstract We study the spectral properties of a shallow donor confined in spirally rolled-up narrow quantum well, in the e˙ective mass approximation. By using the natural curvilinear coordinates, we reduce the problem for the actual system to a two-dimensional wave equation that describes the movement of the electron bound to donor inside the rolled-up well. Solving this equation by means of the double Fourier series expansion method, we found lower donor energies and the density of energy states for different donor positions. Our calculations show that donor energies are very sensible to the variation of the donor position and the geometry of the spiral.