Continuous Domain Dependence of the Eigenvalues of the Dirichlet Laplacian and Related Operators in Hilbert Space

In a Hilbert space (H, ‖·‖) is given a dense subspace W and a closed positive semidefinite quadratic form Q on W×W. Thus W is a Hilbert space with the norm ‖u‖1=(‖u‖2+Q(u))1/2. For any closed subspace D of (W, ‖·‖1) let A(D) denote the selfadjoint operator in the closure of D in H such that ‖A(D)1/2 u‖2=Q(u) for every u∈D. For any decreasing sequence of closed subspaces Di of W with intersection ∩ Di=D such that each A(Di) has compact resolvent it is shown that, for every n, the nth eigenvalue λn(Di) of A(Di) converges to that of A(D), and that A(Di)−1→A(D)−1 in operator norm. Similar results are obtained for any order convergent sequence in the conditionally complete lattice of all closed subspaces D of W such that A(D)−1 has compact resolvent. Next, these results are applied to the Dirichlet laplacian, and more generally to the Dirichlet poly-laplacian, on a sequence of bounded open subsets of RN.