observability results for inverse problems for parabolic equations. Inverse problems for ultrahyperbolic equations were considered in [2, 26, 29], where the unique continuation and stability were proved by using the Carleman estimates. G¨ olgeleyen and Yamamoto [12] proved conditional H¨ older stability for some inverse problems for ultrahyperbolic equation. If n = 0 then (1.1) is a classical Schr¨ odinger equation which describes the evalution of wave function of a charged particle under the influence of electrical potential p. As for the classical Schr¨ odinger equation, Baudouin and Puel in [3] established a global Carleman estimate and proved the uniqueness and Lipshchitz stability based on the idea by Imanuvilov and Yamamoto [16]. This result was improved by Mercado et al. [27] under a relaxed pseudoconvexity condition. In [3, 27], the main assumption is that the part of the boundary where the measurement is made satisfies a geometric condition related to geometric optics condition for the observability. This geometric condition was relaxed in Bellassoued and Choulli [4] under the assumption that the potential is known in a neigborhood of the boundary of the spatial domain. Yuan and Yamamoto [33] obtained a Carleman estimate with a regular weight function in Sobolev spaces of negative orders. They proved the uniqueness in the inverse problem of determining L p potentials and obtained an L 2 level observability inequality and unique continuation results for the Schr¨ odinger equation. Cristofol and Soccorsi [9] considered the inverse problem of determining time-dependent coefficient of the Schr¨ odinger equation from a finite number of Neumann data. Kian et al. [22] extended the stability result of [3] to the case of unbounded domains. This paper consists of four sections. The rest of the paper is organized as follows. In Section 2, the main result of this paper (Theorem 1) is presented. In Section 3, a Carleman estimate (Proposition 1) which will be used in the proof of our main result is established. Finally, Section 4 is devoted to the proof of Theorem 1. 3