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

✍ Scribed by Bent Fuglede

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 156 KB
• Volume: 167
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1999.3442

✦ Synopsis

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)) 1Â2 . 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) 1Â2 u& 2 =Q(u) for every u # D. For any decreasing sequence of closed subspaces D i of W with intersection D i =D such that each A(D i ) has compact resolvent it is shown that, for every n, the n th eigenvalue * n (D i ) of A(D i ) converges to that of A(D), and that A(D i ) &1 Ä 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 R N . 1999 Academic Press * n (D i )=* n (D), n # N (1) For a decreasing sequence of bounded open sets D i one would like (1) to hold, now with D taken as the interior of the intersection: D=int i D i .