  This paper introduces the sparsifying preconditioner for the pseudospectral
approximation of highly indefinite systems on periodic structures, which
include the frequency-domain response problems of the Helmholtz equation and
the Schr\"odinger equation as examples. This approach transforms the dense
system of the pseudospectral discretization approximately into an sparse system
via an equivalent integral reformulation and a specially-designed sparsifying
operator. The resulting sparse system is then solved efficiently with sparse
linear algebra algorithms and serves as a reasonably accurate preconditioner.
When combined with standard iterative methods, this new preconditioner results
in small iteration counts. Numerical results are provided for the Helmholtz
equation and the Schr\"odinger in both 2D and 3D to demonstrate the
effectiveness of this new preconditioner.
