Upper and Lower Bounds of Eigenvalues for Composite-Type Regions

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Abstract
A survey is made to compare the essential features of different variational methods. Schwinger's method is found to have special merits for boundary value and eigenvalue problems concerning composite-type regions. It can give both an upper bound and a lower bound for the eigenvalue of any mode, one bound being as good as the other. It closely resembles the Courant-Trefftz method and is simpler. In both methods the trial functions are so chosen as to satisfy the differential equation while the boundary or continuity conditions are perturbed. The applicability and power of Schwinger's method, being hitherto demonstrated only for particular examples, are exhibited for a whole class of problems by a general formulation of the method in precise terms. The formulation for the upper-bound case presents no difficulty, and a simple proof is given in this paper. A similar proof for the lower-bound case is possible but has a more restricted field of application. The general formulation places the two different cases on equal footing. A way of avoiding the difficulty of having no exact knowledge of the lower eigenfunctions in higher mode cases is discussed in detail.

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