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Perturbation series for Jacobi matrices and the quantum Rabi model

Charif Mirna, Zielinski Lech

Opuscula Mathematica

2021

Vol. 41, no. 3

303--333

We investigate eigenvalue perturbations for a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. In particular we obtain explicit estimates for the convergence radius of the perturbation series and error estimates for the Quantum Rabi Model including the resonance case. We also give expressions for coefficients near resonance in order to evaluate the quality of the rotating wave approximation due to Jaynes and Cummings. .

Notes on the Krupa-Zawisza ultrapower of self-adjoint operators

Ando H., Saigo H., Ojima I.

Probability and Mathematical Statistics

2014

Vol. 34, Fasc. 1

147--159

Let ω ∈ βN \ N be a free ultrafilter on N. It is known that there is a difficulty in constructing the ultrapower of unbounded operators. Krupa and Zawisza gave a rigorous definition of the ultrapower Aω of a self-adjoint operator A. In this note, we give an alternative description of Aω and the Hilbert space H(A) on which Aω is densely defined. This provides a criterion to determine a representing sequence (ξn)n of a given vector ξ ∈ dom(Aω) which has the property that Aωξ = (Aξn)ω holds. An explicit core for Aω is also described.