Regular Perturbations

Abstract

Consider a family

$$ {H_\kappa } = {H_0} + \kappa V = \int {\lambda d} {E_\kappa }(\lambda ) $$

of selfadjoint operators on ℋ depending on a small parameter κ. Suppose that H₀ has an eigenvalue λ₀ of finite multiplicity m, and let P be the orthogonal projection onto the null space N(H₀ – λ₀) of H₀ – λ₀. If λ₀ is an isolated point of the spectrum of H₀, and V is mild, the resolvent

$$ R(z,\kappa ) = (H_\kappa - z)^{ - 1} $$

is analytic in κ, and hence, for small κ, there are (counting multiplicities) exactly m eigenvalues of H_κ near λ₀, which are analytic functions of κ [6, Chapter VII]. We shall then call H{inκ}a regular perturbation of λ₀. In other circumstances, however, H_κ may have only a continuous spectrum near λ₀ for non-zero κ Essentially two cases have been studied: 1) λ₀ isolated [1,4,6,10] and 2) λ₀ embedded in an absolutely continuous spectrum [2,5,11]. In the first case, the perturbation V must be exceedingly strong, as it is for Stark effect Hamiltonians [6,9,10], and R(z,κ) will not be analytic at κ = 0. In the second case, however, which is related to the Auger effect [11], the perturbation may be very mild, and R(z,κ) analytic.

Supported by DA-ARO-31-124-71-G182