AMATH 731: Applied Functional Analysis Fall 2009 Weak convergence in a Hilbert space and the Ritz approximation method

Strictly positive operators are often encountered in applications, e.g., vibrational modes and associated energies of a mechanical system (e.g., a vibrating rod or plate), Hamiltonian operators for quantum mechanical systems (the eigenvalues correspond to energies). In these cases, the Hilbert space H of concern is infinite dimensional and therefore difficult to work with practically. A goal of many studies is to estimate the eigenvalues and eigenvectors of such systems. A natural starting-point is to consider finite-dimensional representations, or “truncations” of the infinite-dimensional problem, with the hope that as the dimension N of the finitedimensional representation is increased, better estimates of the “true” energies are obtained. This is the essence of the so-called “Rayleigh-Ritz” methods. The usual formulation of such methods is in terms of a variational problem involving the minimization of a functional, e.g., energy. The following discussion is intended to give a very rough idea of the philosophy of this approach. A slighly more detailed description is provided in the later sections. One may consider the truncations of our infinite-dimensional system as being the result of working with a finite collection of of linearly independent basis functions, say, {v1, v2, · · · vN}. In many cases, the minimization of the functional of interest – call it J – involves finding the optimal linear combination of these basis functions, i.e. c1v1 + c2v2 + · · ·+ cNvN . (2)