Resonances in Models of Spin Dependent Point Interactions

Details Resonances in Models of Spin Dependent Point Interactions Resonances in Models of Spin Dependent Point Interactions

2008-07-24

arxiv.org/abs/0807.3924v3

J. Phys. A: Math. Theor. 42 (2009) 035202

Physics

Mathematical Physics

Changes in the proof of theorem 3, few misprints corrected, 21 pages

Scientific paper

10.1088/1751-8113/42/3/035202

In dimension $d=1,2,3$ we define a family of two-channel Hamiltonians obtained as point perturbations of the generator of the free decoupled dynamics. Within the family we choose two Hamiltonians, $\hat H_0$ and $\hat H_\ve$, giving rise respectively to the unperturbed and to the perturbed evolution. The Hamiltonian $\hat H_0$ does not couple the channels and has an eigenvalue embedded in the continuous spectrum. The Hamiltonian $\hat H_\ve$ is a small perturbation, in resolvent sense, of $\hat H_0$ and exhibits a small coupling between the channels. We take advantage of the complete solvability of our model to prove with simple arguments that the embedded eigenvalue of $\hat H_0$ shifts into a resonance for $\hat H_\ve$. In dimension three we analyze details of the time behavior of the projection onto the region of the spectrum close to the resonance.

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