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

It is proved that a nucleus of charge Z can bind at most Z 4-0(Z α ) electrons, with a = 47/56.Consider the Hamiltonian for a nucleus of charge Z and N quantized electrons, H zw =ΣΓ(-4< )-ZΊ

No abstract

We present a simple argument which gives a bound on the ionization energy of large atoms that implies the bound on the excess charge of Fefferman and Seco [2].

Let E(Z, N) be the ground-state energy of N quantized electrons and a single nucleus of charge Z. For fixed Z, E(Z, N) is independent of N for N a Ncrtica(Z). Physically, this means that at most NCtw electrons can bind to the nucleus. We prove that N,=, ' Z + CZa with a = 0.84.