Half-inverse problems on the finite interval

The spectrum of the Sturm-Liouville problem is given. The potential on the half-interval is known. It is necessary to reconstruct the potential on the whole interval. The uniqueness of the solution of the formulated problem is proved (see Hochstadt H and Lieberman B 1978 SIAM J. Appl. Math. 34 676-80, Hald O 1984 Commun. Pure Appl. Math. 37 539-77, Gesztesy F and Simon B 2000 Trans. AMS 352 2765-89). Under some conditions in this paper we prove the existence of the solution and give a method of reconstructing this solution.