By Hans F. Weinberger

Presents a typical atmosphere for varied equipment of bounding the eigenvalues of a self-adjoint linear operator and emphasizes their relationships. A mapping precept is gifted to attach the various tools. The eigenvalue difficulties studied are linear, and linearization is proven to offer very important information regarding nonlinear difficulties. Linear vector areas and their houses are used to uniformly describe the eigenvalue difficulties offered that contain matrices, traditional or partial differential operators, and integro-differential operators.

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DEFINITION. The ratio ^(u, M)/J^(M, u) is called the Rayleigh quotient. 4) with the convention that if \in is not attained, /in = /^ n + 1 = fin +2 = • • • , are called the eigenvalues of the Rayleigh quotient $l(u,u)/s/(u,u). (w, U)/J/(M, u) is bounded above. 5 as a criterion to establish the existence of eigenvalues, we need a way of generating linear functionals with the needed properties. The most important result along these lines is the following. 7 (Poincare's inequality). Let iKx^ • • • , XN) be a square integrable function with square integrable first partial derivatives on the N -dimensional cube Ka of side a(ve Hl(Ka)).

Hence its definition can be extended by continuity to V^. We thus obtain a linear operator T on V^. 12). 3) on V with the problem on the completion V^. 1. 13). 3) corresponding to the same eigenvalue. Proof. 12), for all v in V. Since V is dense in V^ we can take limits of v in V to find that this equation still holds for a v in V^. fi/(TM — uu, Tu — uu) = 0. 13). 13) is satisfied with u in V. l(a), Bu — uAu = 0, which completes the proof. 1). 13) has eigenvectors in the completion V^ which are not in the original space V.

A sesquilinear functional on F x Fis said to be a Hermitian functional on F if for all u and u in F. If F is the field of real numbers, the bilinear form B(u, v) is more commonly said to be symmetric if B(u, v) = B(v, u). 3. 1 is Hermitian if and only ifm = nandbjk = Bkj for all; and k. The matrix (fe y ) is then said to be Hermitian. DEFINITION. A mapping from a linear vector space Fto the real numbers is said to be a quadratic functional (or form) if there exists a Hermitian (or, if F is over the field of real numbers, a symmetric) functional B(u, v) on F x F in terms of which the mapping has the form u -> B(u, u).

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