Vol. 55, No. 1, 1974

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Puiseux series for resonances at an embedded eigenvalue

James Secord Howland

Vol. 55 (1974), No. 1, 157–176

Let H(κ) = T + κBA be a self-adjoint perturbation of the self-adjoint operator T, and suppose that T has an eigenvalue λ0 of finite multiplicity m embedded in its continuous spectrum. If the operator

Q (z) = A(T − z)− 1B ∗

is bounded and can be continued meromorphically across the axis at λ0, the asymptotic spectral concentration of the family H(κ) at λ0 is determined by the poles of

κA(H (κ)− z)−1B∗ = I − [I + κQ(z)]−1

These “resonances” can be expanded in a series of fractional powers of κ, and therefore have a unitarily invariant significance for the family H(κ). An example shows that nonanalytic series may indeed occur; however, if a resonance is an actual eigenvalue of H(κ) for all sufficiently small real κ, its series is analytic. Because the resonances cannot lie on the first sheet when κ is real, these series must have a special form. In the generic case, they yield, as the lowest order approximation to the imaginary parts of the resonances, the famous Fermi’s Golden Rule. The case when λ0 is embedded at a branch point of (1) is studied by means of a simple example.

Mathematical Subject Classification 2000
Primary: 47A40
Received: 23 March 1973
Published: 1 November 1974
James Secord Howland