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Convergent expansions of eigenvalues of the generalized Friedrichs model with a rank-one perturbation. (English) Zbl 1480.81047

Summary: We study analytic behavior of eigenvalues of the generalized Friedrichs model \(H_\mu(p)\), with a rank-one perturbation, depending on parameters \(\mu>0\) and \(p\in\mathbb{T}^2\). Under certain conditions, the existence of a unique eigenvalue lying below the essential spectrum has been shown in thw first 2 authors et al. [Abstr. Appl. Anal. 2012, Article ID 180953, 14 p. (2012; Zbl 1247.81120)]. Here, we obtain an absolutely convergent expansion for that eigenvalue at \(\mu(p)\), the coupling constant threshold. The expansion is dependent to a large extent on whether the lower bound of the essential spectrum is a threshold resonance, a threshold eigenvalue or neither of them.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
47A10 Spectrum, resolvent
81U30 Dispersion theory, dispersion relations arising in quantum theory
35B34 Resonance in context of PDEs
35P05 General topics in linear spectral theory for PDEs
40A05 Convergence and divergence of series and sequences

Citations:

Zbl 1247.81120
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References:

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