Convergence theorems for intermediate problems. II

The method of intermediate problems of Weinstein and Aronszajn (cf. [19, 20]) provides a means of calculation for bounds to eigenvalues complementary to the Rayleigh{Ritz bounds. Convergence theorems for the method of intermediate problems in a generality that includes problems with essential spectra can be found in [5, 7{9, 12]. This paper supplements the general convergence theorems of [7, 9] in two respects. In each of [7, 9], convergence theorems for the T ¤ T method of Bazley and Fox require an alignment of projections hypothesis that is both inconvenient and ignored in computational practice. By examining this method directly and not relating the Bazley{Fox projections, commonly called inner projections by physicists, to the earlier Aronszajn or outer projections, we are able to remove the alignment hypothesis. The basic tool, as in [5, 7,9], is a monotone convergence theorem of Weidmann [18]. We then turn our attention to the so-called eigenvector-free (EVF) methods, which, like truncation methods (cf. [5, 12, 19, 20]), provide a practical means of dealing with the Weinstein{Aronszajn determinant without severely constraining the choice vectors. These methods can be viewed as extensions of both Bazley’s method of special choice (cf. [19,20]) and Temple{Lehmann methods (cf. [16,17,19]). They originate in Gay [10], and like Rayleigh{Ritz calculations, the choice vectors are selected as approximations to the unknown eigenvectors rather than to add a perturbation to a base operator. But precise means of relating the approximate eigenvalues to those desired awaited the work of Goerisch [11], who coined the phrase EVF (cf. also [4]).