  A characterization for the Fourier multipliers and eigenvalues of linear
peridynamic operators is provided. The analysis is presented for state-based
peridynamic operators for isotropic homogeneous media in any spatial dimension.
We provide explicit formulas for the eigenvalues in terms of the space
dimension, the nonlocal parameters, and the material properties.
  The approach we follow is based on the Fourier multiplier analysis developed
for the nonlocal Laplacian. The Fourier multipliers of linear peridynamic
operators are second-order tensor fields, which are given through integral
representations. It is shown that the eigenvalues of the peridynamic operators
can be derived directly from the eigenvalues of the Fourier multiplier tensors.
  We reveal a simple structure for the Fourier multipliers in terms of
hypergeometric functions, which allows for providing integral representations
as well as hypergeometric representations of the eigenvalues. These
representations are utilized to show the convergence of the eigenvalues of
linear peridynamics to the eigenvalues of the Navier operator of linear
elasticity in the limit of vanishing nonlocality. Moreover, the hypergeometric
representation of the eigenvalues is utilized to compute the spectrum of linear
peridynamic operators.
