Resonances in Models of Spin Dependent Point Interactions

Abstract

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.

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

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• Perturbations of eigenvalues embedded at threshold: one, two and three dimensional solvable models
• A quantum hybrid with a thin antenna at the vertex of a wedge
• Spectrum of a family of spin-orbit coupled Hamiltonians with singular perturbation
• A Quantum Model of Feshbach Resonances
• Computation of unitary group for the Rashba spin-orbit coupled operator, with application to point-interactions