[Paper Review] Zero-range potentials with Inner structure: fitting parameters for resonance scattering

Vladimir I. Kruglov, B. S. Pavlov|ArXiv.org|Jun 23, 2003

Cold Atom Physics and Bose-Einstein Condensates7 references3 citations

TL;DR

This paper introduces a solvable zero-range potential model with inner structure to describe low-energy neutron-nucleus scattering, incorporating a non-trivial inner Hamiltonian and indefinite metric space. By enforcing analyticity of the Cayley transform of the S-matrix in the wave number $k$, the method uniquely determines all model parameters—including scattering length, effective radius, and indefinite metric tensor—enabling exact analytical solutions for resonance scattering in the s-channel.

ABSTRACT

The solution of the classical Fermi problem of low-energy neutron scattering by nuclei, when the excitations of the nuclei in scattering processes are taken into account, is found by the method of zero-range potentials with inner structure. This model is a generalization of the Fermi zero-range potential obtained by adding a non-trivial inner Hamiltonian and inner space with indefinite metric. We propose a general principle of analyticity of the Caley-transform of the S-scattering matrix, written as a function of wave number. This permits us to evaluate all parameters of the model, including the indefinite metric tensor of the inner space, once the spectrum of the inner Hamiltonian, the scattering length and the effective radious are chosen.

Motivation & Objective

• To develop a solvable model for low-energy neutron scattering by nuclei that accounts for internal nuclear excitations.
• To generalize the Fermi zero-range potential by introducing an inner Hamiltonian and indefinite metric space to describe multiparticle interactions.
• To provide a unitary S-matrix with correct pole and zero placement in the complex $k$-plane, avoiding non-Hermitian Hamiltonians used in the optical model.
• To establish a general principle for fitting all model parameters—scattering length, effective radius, and indefinite metric tensor—based on physical observables.
• To resolve the limitations of singular $δ$-potentials in zero-range approximations, particularly their failure in the $a<0$ and $a>0$ cases under the $\varepsilon \to 0$ limit.

Proposed method

• Formalize the zero-range potential with inner structure as an extension of the self-adjoint operator via a rigged Hilbert space with indefinite metric.
• Introduce an inner Hamiltonian acting on a finite-dimensional inner space, coupled to the external scattering space via a boundary condition.
• Apply the Cayley transform of the S-matrix and impose analyticity in the complex $k$-plane as a fundamental principle to constrain model parameters.
• Use the Kreín formula to derive the resolvent of the extended operator, linking spectral data to the inner space structure.
• Derive explicit analytical expressions for the scattering phase shift $\delta(k)$ and $k\cot\delta(k)$ in terms of the scattering length $a$, effective radius $r_0$, and bound state energy $\kappa$.
• Demonstrate that the standard $\delta$-potential approximation fails in both attractive ($a<0$) and repulsive ($a>0$) cases under the $\varepsilon \to 0$ limit, invalidating its use in zero-range models.

Experimental results

Research questions

• RQ1 How can a zero-range potential model be generalized to include internal nuclear structure while preserving unitarity and correct analytic structure of the S-matrix?
• RQ2 What principle allows for the unique determination of all model parameters—including the indefinite metric tensor—given physical inputs like scattering length and effective radius?
• RQ3 Why does the standard $\delta$-potential approximation fail in both attractive and repulsive scattering regimes when the range is taken to zero?
• RQ4 How does the analyticity of the Cayley transform of the S-matrix constrain the parameter space of the zero-range model with inner structure?
• RQ5 Can the Fermi zero-range model be consistently extended to include bound states and resonance scattering with exact analytical solutions?

Key findings

• The model provides an exact analytical solution for s-wave resonance scattering in the s-channel, with $k\cot\delta(k) = -\frac{1}{a} + \frac{r_0}{2}k^2$, valid in the low-energy limit.
• The scattering length $a$ and effective radius $r_0$ are directly related to the bound state energy $\kappa$ via $\frac{1}{a} = \kappa - \frac{r_0}{2}\kappa^2$ in the weakly bound limit.
• The singular $\delta$-potential $V(r) = U_0\delta({\bf r})$ fails as a zero-range approximation: for $a>0$, the total cross-section vanishes as $\varepsilon \to 0$; for $a<0$, the limit does not exist.
• The model avoids non-Hermitian Hamiltonians, ensuring a unitary S-matrix with correct pole and zero structure in the complex $k$-plane.
• The indefinite metric tensor of the inner space is uniquely determined by the analyticity of the Cayley transform of the S-matrix, enabling full parameter fitting from physical observables.
• The method successfully generalizes the Fermi model to include internal structure, providing a solvable framework for resonance scattering with exact parameterization.