We consider an inverse source problem for the Schrödinger equation with variable coefficients. We prove the uniqueness of solution of the problem by data on a flat subboundary over time interval, under a certain condition of the coefficients of the principal terms. We first reduce the inverse problem to a Cauchy problem for a system of integro-differential equations by using Fourier transform. Next, we establish a pointwise Carleman type inequality which is the key tool in the proof of our main result.

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