FURTHER DEVELOPMENTS IN THE APPROXIMATION THEORY OF EIGENVALUES**This research has been sponsored in part by the Aerospace Research Laboratories through the European Office of Aerospace Research, OAR, United States Air Force, under Grant EOOAR 69-0066.

Publisher Summary This chapter describes developments in the approximation theory of eigenvalues. It chapter an earlier a method for getting lower bounds for the eigenvalues of a class of positive linear operators. The application of the method to elliptic differential operators relies on a structure theorem for the Green operator of the boundary value problem under consideration. On the other hand this structure theorem is founded on the explicit knowledge of a fundamental solution operator. That, in practice, restricts the application of the method to operators with constant coefficients. This chapter discusses the way by which the method can be applied to the general case of operators with variable coefficients. It also describes eigenvalue problems for linear elliptic systems.