If the resolvent of a (not
necessarily bounded) self-adjoint operator Hκ converges strongly to the resolvent of a
selfadjoint operator H, and if λ is an isolated eigenvalue of H of multiplicity m < ∞,
then although Hκ need not have an eigenvalue near λ, the spectrum of Hκ will in
some cases become “concentrated” near λ as κ is reduced. In fact, there exist sets
Cκ with Lebesgue measure o(κp),p ≧ 0, such that the spectral projection
assigned by Hκ to Cκ converges strongly as κ → 0 to the projection on the
λ-eigenspace of H, if and only if there exist m pairs (λjκ,φjκ),j = 1,⋯,m, where
λjκ→ λ, the φjκ are nearly-orthogonal unit vectors converging strongly to the
λ-eigenspace, and ∥(Hκ− λjκ)φjκ∥ = o(κp). In this case, Cκ may be taken as
the union of intervals about the λjκ, and the λjκ are essentially the only
numbers associated in this way with “pseudoeigenvectors” φjκ of Hκ. The
result is applied to the weak-quantization problem in the theory of the Stark
effect, where H is the Hamiltonian operator for the hydrogen atom, and
Hκ is the same for the atom in a uniform electric field which vanishes with
κ.
