  Given a sparse Hermitian matrix $A$ and a real number $\mu$, we construct a
set of sparse vectors, each approximately spanned only by eigenvectors of $A$
corresponding to eigenvalues near $\mu$. This set of vectors spans the column
space of a localized spectrum slicing (LSS) operator, and is called an LSS
basis set. The sparsity of the LSS basis set is related to the decay properties
of matrix Gaussian functions. We present a divide-and-conquer strategy with
controllable error to construct the LSS basis set. This is a purely algebraic
process using only submatrices of $A$, and can therefore be applied to general
sparse Hermitian matrices. The LSS basis set leads to sparse projected matrices
with reduced sizes, which allows the projected problems to be solved
efficiently with techniques using sparse linear algebra. As an example, we
demonstrate that the LSS basis set can be used to solve interior eigenvalue
problems for a discretized second order partial differential operator in
one-dimensional and two-dimensional domains, as well as for a matrix of general
sparsity pattern.
