Higher Eigenvalues and Topological Perturbations

Some fundamental estimates for higher eigenvalues of standard Laplacians have already been derived in Sect. 4.6 . The goal of this chapter is twofold: on the one hand considering the standard Laplacian we derive explicit fundamental estimates for higher eigenvalues and describe the behaviour of such eigenvalues under topological perturbations. Here techniques developed in the previous chapter are used. On the other hand, considering Schrödinger operators with most general vertex conditions we analyse the behaviour of the spectrum under topological perturbations and show that intuition gained during our studies of standard Laplacians cannot always be applied: the eigenvalues may depend on topological perturbations in a completely opposite way.