Vol. 23, No. 2, 1967

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Spectral concentration for self-adjoint operators

Ronald Cameron Riddell

Vol. 23 (1967), No. 2, 377–401

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 (λ),j = 1,,m, where λ λ, the φ are nearly-orthogonal unit vectors converging strongly to the λ-eigenspace, and (Hκ λ)φ= o(κp). In this case, Cκ may be taken as the union of intervals about the λ, and the λ are essentially the only numbers associated in this way with “pseudoeigenvectors” φ 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.

Mathematical Subject Classification
Primary: 47.30
Secondary: 81.00
Received: 30 April 1966
Published: 1 November 1967
Ronald Cameron Riddell