A bound for the difference Laplacian

by R. F. Streater; Bull. London Math. Soc., 11, 354-357, 1979.

ABSTRACT

Let D\subseteq Z² be a subset of the two-dimensional lattice, and let d be the length of the side of the largest square, with sides parallel to the axes, whose intersection with Z² lies in D. Let T_i be the translation operator on l²(Z²), i=1 or 2. Let Delta_D be the Dirichlet difference operator with zero boundary conditions at the boundary of D, so that Delta_D=E_D Delta E_D, where E_D is the indicator function of D, and -Delta=delta_1 delta_1^*+ delta_2 delta_2^*, with delta_i=T_i-1. We show that

-Delta_D geq frac{2}{9}{1-\cos\frac{\pi}{8d+8}}

under a condition on D called local simply-connectedness. In the continuum limit this gives a new best bound for the lowest eigenvalue of the Laplacian in a multipliconnected region with thin corridors.

This paper uses a clever idea due to Walter Hayman, for the case of harmonic operator in the plane, adapted here to the lattice. The results are no longer anywhere near the best bounds for simply-connected domains, but do remain the best for domains which are the complements of the union of very many thin corridors.