Constructing the Volterra Integral Equation

With the Jost transformation established, the next objective was to reformulate the scattering problem once again.

While the transformed differential equation was considerably easier to analyze than the original radial equation, an integral formulation offered several important advantages.

Integral equations naturally separate known quantities from unknown corrections, making them particularly well suited for perturbative analysis and asymptotic expansions.

For this reason, the investigation turned toward constructing the corresponding Volterra integral equation.

As with every previous stage of the project, the derivation was reconstructed independently before being compared against the published literature.

The objective was not simply to reproduce an established equation.

It was to understand how the integral representation emerged from the underlying differential equation and what physical assumptions entered the derivation.

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Objective

Construct a Volterra integral equation equivalent to the transformed scattering equation.

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Motivation

Integral equations provide a framework that is often better suited to perturbation theory and asymptotic analysis than differential equations.

Developing the equation independently also ensured that later comparisons with the literature would rest on an independently verified mathematical foundation.

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Work Performed

• Derived the Volterra integral equation from the transformed differential equation.
• Verified the mathematical consistency of the formulation.
• Investigated the zero-energy limit.
• Checked the derivation repeatedly for algebraic consistency.

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Results

🟢 Verified

A Volterra integral formulation of the scattering problem was successfully obtained.

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🟢 Verified

The integral equation reproduced the expected zero-energy behavior.

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🟢 Verified

The formulation established a suitable basis for the perturbative analysis that followed.

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Research Notes

This stage completed the mathematical framework needed for the remainder of the theoretical investigation.

The project now possessed multiple equivalent descriptions of the same physical problem:

• the original radial differential equation,
• the Jost-transformed equation,
• and the Volterra integral equation.

Having several mathematically equivalent formulations proved valuable later, particularly when checking derivations and comparing different approaches in the published literature.

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Next Steps

Investigate the exact zero-energy solution and verify that the integral formulation reproduces the expected physical behavior.