The Deflation of Eigenvalue Problems with Second-Order Ordinary Differential Operators

Abstract

Suppose that one is faced with an eigenvalue problem with a denumerable set of eigenvalucs X1, 2*X . Suppose also that one has obtained a number of these eigenvalues, say X1 , , Xk-. It is then desirable to be able to state a problem which has as solutions the eigenvalues Xk , ,k+ X , but not the eige nvalues X1 , * , Xk-1 . This is what we mean by the of the eigenvalue problem. There are at least two reasons for wishing to deflate an eigenvalue problem. One might be solving for the eigenvalues by a numerical iterative process; it is then helpful to deflate the problem in order to avoid converging to an eigenvalue which has already been found. Again, one might be proving inductively a theorem about eigenvalues; it is then helpful to be able to deflate the problem in order to make a statement about Xk , Xk+1, * which is not true for X1, ,XkOne method for deflating an eigenvalue problem is the use of the minimization of a Rayleigh quotient under the side condition that the admissible functions must be orthogonal to previously found eigenfunctions. If the problem is finite-dimensional, one might also use standard techniques for the deflation of matrices or polynomials. The method described in the present paper has an advantage over the Rayleigh quotient deflation process in that we do not assume that the eigenvalues are found in order of magnitude. The technique is most similar to the technique of deflating polynomials. The arguments used are primarily formal algebraic manipulations. We will describe a sequence of problems P1, P2, * . . The first problem P1 is the original eigenvalue problem. Suppose we have found a particular eigelnvalue X1 and its eigenfunction ul . Then the second problem P2 , which depends on u1, has as solutions the eigenvalues X2 >X~ , . . . , with corresponding functions (not eigenfunctions) U22 I U23 y * X X . Having found a solution to P2, say X2 and u2 = u22 , then we may solve P3 , which depends on u2, etc. We obtain in this way the eigenvalues of the original problem; the eigenfunctions are not obtained by our process. We must prove a sort of existence and uniqueness theorem. First we will show that our sequence of problems has a solution; then we will show that anly sequence of solutions to the problems yields eigenvalues of the original problem.