  In this paper, we discuss problems arising when computing resonances with a
finite element method. In the pre-asymptotic regime, we detect for the one
dimensional case, spurious solutions in finite element computations of
resonances when the computational domain is truncated with a perfectly matched
layer (PML) as well as with a Dirichlet-to-Neumann map (DtN). The new test is
based on the Lippmann-Schwinger equation and we use computations of the
pseudospectrum to show that this is a suitable choice. Numerical simulations
indicate that the presented test can distinguish between spurious eigenvalues
and true eigenvalues also in difficult cases. Keywords: scattering resonances,
Lippmann-Schwinger equation, nonlinear eigenvalue problems, acoustic resonator,
dielectric resonator, Bragg resonator
