Half inverse problem and interior inverse problem for the Dirac operators with discontinuity

In this paper, the half inverse problem and interior inverse problems for Dirac operators with discontinuity inside the interval (0, T) is considered. It is shown that (i) if the potential is given on \(\Big (0,\frac{(1+\alpha )T}{4}\Big )\), then one spectrum can uniquely determine the potential on the whole interval; (ii) one spectrum and a set of values of eigenfunctions at some internal points can also uniquely determine the potential on the whole interval.