Singular perturbation theory for semibounded operators

W. M. Greenlee
1976 Bulletin of the American Mathematical Society  
In this announcement an operator theoretic approach to singular perturbation expansions for simple eigenvalues is outlined. Corresponding results hold for eigenvectors, eigenvalues of finite multiplicity, and spectral concentration near eigenvalues of finite multiplicity. We seek first and second order approximations for problems to which the regular perturbation method does not apply (cf. [2] ). First presented are the abstract singular perturbation expansions, followed by mention of some
more » ... ts for differential problems. Application of the abstract results to concrete problems is involved, but our framework makes Lions' method of correctors [3], as well as boundary layer techniques (cf. [4] ), applicable to eigenvalue problems. Let H be a complex Hubert space with inner product (v, w) and norm |u|. Let b(v, w) be a Hermitian symmetric bilinear form defined on a linear manifold D(b) which is dense in H, We assume that the quadratic form corresponding to b(v, w) has a positive lower bound, and is closed. Then D(b), with inner product b(v, w), is a Hubert space. Further let a(v, w) be a Hermitian symmetric bilinear form defined on a linear manifold D(a) which is dense in D(b), and assume that the quadratic form corresponding to a(v 9 w) is nonnegative, and closed in D(b). Let B be the positive definite self adjoint operator in H defined by (Bv, w) the topology induced by ƒƒ}. Similarly let A € , e > 0, be the positive definite selfadjoint operator in H defined by (A € v, w) = ea(p, w) + b(v, w), and A the nonnegative selfadjoint operator in D(b) defined by b(Av, w) = a(v, w). Assume that X is an isolated simple eigenvalue of B with corresponding eigenvector u normalized in H. Assume further that X is stable under the above perturbation, i.e., that for e sufficiently small the intersection of any isolating interval for X and the spectrum of A € consists of a single simple eigenvalue X e ofi4 e , X e ->Xasel 0 (cf. [2]). Now let r\ = (A~lu, u). It is easily shown that 77 = X" 1 -eX"~2X' e , where X' e = i(A(eA + l)~lu 9 u) and that eX' e -* 0 as e I 0 (cf. [1]). THEOREM 1. (i) X e = X + 0(eX' e ) as e ~> 0. AMS (MOS) subject classifications (1970). Primary 47A55, 35B25.
doi:10.1090/s0002-9904-1976-14055-4 fatcat:f3wf3qcz3fekfka52t2smwi5gm