  We consider boundary conditions of self-adjoint banded Toeplitz matrices. We
ask if boundary conditions exist for banded self-adjoint Toeplitz matrices
which satisfy operator inequalities of Dirichlet-Neumann bracketing type. For a
special class of banded Toeplitz matrices including integer powers of the
discrete Laplacian we find such boundary conditions. Moreover, for this class
we give a lower bound on the spectral gap above the lowest eigenvalue.
