  We propose a novel and simple spectral method based on the semi-discrete
Fourier transforms to discretize the fractional Laplacian
$(-\Delta)^\frac{\alpha}{2}$. Numerical analysis and experiments are provided
to study its performance. Our method has the same symbol
$|\boldsymbol\xi|^\alpha$ as the fractional Laplacian
$(-\Delta)^\frac{\alpha}{2}$ at the discrete level, and thus it can be viewed
as the exact discrete analogue of the fractional Laplacian. This {\it unique
feature} distinguishes our method from other existing methods for the
fractional Laplacian. Note that our method is different from the Fourier
pseudospectral methods in the literature which are usually limited to periodic
boundary conditions (see Remark \ref{remark0}). Numerical analysis shows that
our method can achieve a spectral accuracy. The stability and convergence of
our method in solving the fractional Poisson equations were analyzed. Our
scheme yields a multilevel Toeplitz stiffness matrix, and thus fast algorithms
can be developed for efficient matrix-vector multiplications. The computational
complexity is ${\mathcal O}(2N\log(2N))$, and the memory storage is ${\mathcal
O}(N)$ with $N$ the total number of points. Extensive numerical experiments
verify our analytical results and demonstrate the effectiveness of our method
in solving various problems.
