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On Fermi's model for the scattering of a slow neutron from a bound proton

Domenico Finco, Raffaele Scandone, Alessandro Teta

Abstract

We consider a model Hamiltonian, introduced by Fermi in 1936, describing a two-particle system made of a neutron and a harmonically bound proton, where the neutron-proton interaction has the form of a $δ$-potential. For such Hamiltonian we prove the Limiting Absorption Principle and describe the stationary scattering theory. Finally, we derive Fermi's formula for the scattering cross-section valid in the Born approximation.

On Fermi's model for the scattering of a slow neutron from a bound proton

Abstract

We consider a model Hamiltonian, introduced by Fermi in 1936, describing a two-particle system made of a neutron and a harmonically bound proton, where the neutron-proton interaction has the form of a -potential. For such Hamiltonian we prove the Limiting Absorption Principle and describe the stationary scattering theory. Finally, we derive Fermi's formula for the scattering cross-section valid in the Born approximation.
Paper Structure (5 sections, 9 theorems, 110 equations)

This paper contains 5 sections, 9 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Model Hamiltonian and main results
  3. Limiting Absorption Principle for $H$
  4. Stationary scattering theory
  5. Born Approximation

Key Result

Proposition 2.1

The resolvent $R (z)$ is a meromorphic operator valued function on $\mathbb{C} \setminus \overline{\mathbb{R}^+}$ with values in $\mathcal{B}(L^2 (\mathbb{R}^6) )$ and where $\mathcal{G}(z) \in \mathcal{B}(L^2(\mathbb{R}^3), L^2(\mathbb{R}^6))$, and $(\Gamma (z) +\alpha )^{-1} \in \mathcal{B}(L^2(\mathbb{R}^3))\,$ for $\,z\in \mathbb{C}\setminus (\mathbb{R}^+ \cup \sigma_p (H))$.

Theorems & Definitions (19)

  • Proposition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 9 more