  Consider the eigenvalue problem generated by a fixed differential operator
with a sign-changing weight on the eigenvalue term. We prove that as the
negative part of the weight is rescaled towards negative infinity on some
subregion, the spectrum converges to that of the original problem restricted to
the complementary region. On the interface between the regions the limiting
problem acquires Dirichlet-type boundary conditions. Our main theorem concerns
eigenvalue problems for sign-changing bilinear forms on Hilbert spaces. We
apply our results to a wide range of PDEs: second and fourth order equations
with both Dirichlet and Neumann-type boundary conditions, and a problem where
the eigenvalue appears in both the equation and the boundary condition.
