The essential spectrum of Neumann Laplacians on some bounded singular domains

Abstract

In the present paper we consider Neumann Laplacians on singular domains of the type "rooms and passages" or "combs" and we show that, in typical situations, the essential spectrum can be determined from the geometric data. Moreover, given an arbitrary closed subset S of the non-negative reals, we construct domains Q = Q(S) such that the essential spectrum of the Neumann Laplacian on R is just this set S. 'PI 1991 Academnc PEW, Inc

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“…The results of [13] indicate that such domains might not have discrete spectrum. The exclusion of these domains from our consideration does not seem to be an essential limitation because lip domains with shapes significantly different from the square are not likely to have multiple second eigenvalues.…”

“…However, by pathwise uniqueness of solutions to (13), under P , Z(u, T, ·) and Z(u, 0, ·) are equal in law, and therefore so are U (u, T, ·) and U (u, 0, ·). Consequently, g(r) = Ef (U (r, 0, t)) = P t (r, f ).…”

Rates of Eigenvalues on a Dumbbell Domain. Simple Eigenvalue Case

“…He also obtained nice estimates on the L°° behavior of the eigenfunctions. The work in [13] includes results in this case also.…”

Universal local parametrizations via heat kernels and eigenfunctions of the Laplacian

“…This in turn only occurs if there are boundary points where the Wiener series (for the boundary) converges [55,28]. For the Neumann case the situation is more complicated [34,26,33]. In particular, there are domains with arbitrary closed continuous Neumann spectrum [26].…”

“…For the Neumann case the situation is more complicated [34,26,33]. In particular, there are domains with arbitrary closed continuous Neumann spectrum [26]. We therefore restrict ourselves in this paper to domains (and, later, manifolds) where conditions (2.1.1) and (2.1.2) are valid.…”