Singular limits of sign-changing weighted eigenproblems

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