Resonances in Models of Spin Dependent Point Interactions

TL;DR

This paper studies how a small coupling perturbation in spin-dependent point interaction models causes embedded eigenvalues to turn into resonances, with detailed analysis in three dimensions.

Contribution

It introduces a solvable family of two-channel Hamiltonians for spin-dependent point interactions and demonstrates the eigenvalue-to-resonance transition under small perturbations.

Findings

Embedded eigenvalues shift into resonances under perturbation.

Explicit analysis of resonance behavior in three-dimensional models.

Complete solvability allows simple proofs of resonance formation.

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.