  We study an initial value problem for the one-dimensional non-stationary
linear Schr\"odinger equation with a point singular potential. In our approach,
the problem is considered as a system of coupled initial-boundary value (IBV)
problems on two half-lines, to which we apply the unified approach to IBV
problems for linear and integrable nonlinear equations, also known as the Fokas
unified transform method. Following the ideas of this method, we obtain the
integral representation of the solution of the initial value problem. Since the
unified approach is known as providing efficient solutions to both linear and
nonlinear problems, the present paper can be viewed as a step in solving the
initial value problem for the non-stationary {\em nonlinear} Schr\"odinger
equation with a point singular potential.
