Two-sided approximations to single eigenvalues of the Laplace operator over bounded Lipschitzian Graph domains are obtainable by the method of point solutions. Each eigenvalue must be estimated separately, and to tell which of the ordered eigenvalues is approximated, additional a priori information, such as the Weyl asymptotics, must be invoked. This method uses solutions of the homogeneous Helmholtz equation as trial functions, but does not require satisfaction of any boundary conditions. A generalization, which also estimates one eigenvalue at a time, is the method of a posteriori/a priori inequalities.

To estimate several lower eigenvalues with one calculation, the common method is the finite element variant of the Rayleigh-Ritz procedure. This provides upper bounds for which the error estimates are asymptotic rather than exact. For complementary bounds over domains of irregular shape, procedures due to Weinstein and Weinberger apply in the case of Dirichlet boundary conditions. These involve the construction of finite dimensional projections, which is fairly straightforward for the Laplacian, but not so for variable coefficients operators. By modifying their ideas in the spirit of techniques originating from Aronszajn we obtain two convergent lower bound procedures for the Dirichlet Laplacian.

Our technique is a presumably new means of setting up a method known as truncation including the remainder (Weinstein school), or as Aronszajns method with a truncated base problem (Aronszajn school). One of the procedures extends readily to Dirichlet problems for self- adjoint variable coefficient operators, to the clamped bending and buckling problems for the biharmonic operator, to potential well problems with non-trivial essential spectrum, and to domains with slits. In very special cases the boundary conditions are not required to be Dirichlet on a portion of the boundary.

This is joint work with Lotfi Hermi.