On the Absorption of Eigenvalues by Continuous Spectrum in Regular Perturbation Problems

for all 1,4 E Q(A). Under this hypothesis A + hB is an entire analytic family of type (B) in the sense of Kato [3]. In particular, the theory of Rellich [6] and Kato [2, 31 is applicable: If CL,, is an eigenvalue of A + X,,B which is discrete (i.e., an isolated point of spec(A + &,B)) an o multiplicity K and either k = 1 d f or h, is real, then for h near &, , the only spectrum of A + /\B near pa is discrete, of total multiplicity k, and given by one or more functions analytic in X near & . Let us restrict X to be real henceforth and suppose pFLo is a discrete eigenvalue of A + &,B (for simplicity, suppose k = 1). As /\ varies, the eigenvalue p,, varies being given by a real analytic function p(X). The Kato-Rellich theory described above continues to be applicable so long as r(x) stays away from the nondiscrete spectrum of A + XB. The questions which will concern us in this note involve the situation which occurs when p(X) approaches the nondiscrete spectrum as X approaches some critical value of X, . A typical phenomenon that