  We study finite quantum wires and rings in the presence of a charge density wave gap induced by a periodic modulation of the chemical potential. We show that the Tamm-Shockley bound states emerging at the ends of the wire are stable against weak disorder and interactions, for discrete open chains and for continuum systems. The low-energy physics can be mapped onto the Jackiw-Rebbi equations describing massive Dirac fermions and bound end states. We treat interactions via the continuum model and show that they increase the charge gap and further localize the end states. In an Aharonov-Bohm ring with weak link, the bound states give rise to an unusual $4\pi$-peridodicity in the spectrum and persistent current as function of an external flux. The electrons placed in the two localized states on the opposite ends of the wire can interact via exchange interactions and this setup can be used as a double quantum dot hosting spin-qubits. 
Localized end states in density modulated quantum wires and rings