Half inverse problem for the impulsive diffusion operator with discontinuous coefficient

The half inverse problem is to construct coefficients of the operator in a whole interval by using one spectrum and potential known in a semi interval. In this paper, by using the Hocstadt-Lieberman and Yang-Zettl?s methods we show that if p(x) and q(x) are known on the interval (?/2,?), then only one spectrum suffices to determine p (x),q(x) functions and ?,h coefficients on the interval (0,?) for impulsive diffusion operator with discontinuous coefficient.