Frame 3: Algebraic Transformation & Symmetry-Induced Cancellation Removal

Frame 3 begins with the change of variable:

and writes the function in the form:

where:

This separates the function into two parts: the oscillatory factor \(\sin u\), and the envelope \(g(u)\) which reweights different parts of the domain. The envelope is not symmetric under reflection, and this property is the basis of the symmetry test.

Define:

then the operator:

The key result is:

This is a differential identity, valid on the interior domain where \(g\) is positive and smooth and \(p\) is defined. It says that applying \(H_g\) to the product of the envelope and a function is equivalent to applying the operator to \(Y\) and multiplying the result by \(g\). If \(Y_n\) satisfies:

then:

But the identity alone does not produce a complete eigenvalue problem: that would require specifying the domain of the operator and the boundary conditions. Moreover, the values \(\lambda_n\) follow directly from the choice of \(Y_n\), and are not an independent spectral result.

Define the transition operator:

and use the reflection about the midpoint of the domain:

In the reference model (without envelope), the states are:

They satisfy:

At the same time:

When computing the transition element between the first and third states, the product inside the integral becomes odd under reflection, so the integral vanishes:

The element between the first and second states does not vanish; its integral using the unnormalized states is:

After normalizing the reference states \(Y_1, Y_2\) to norm \(\sqrt{\pi}\) in \(L^2(0,2\pi)\), the reference transition element becomes \(\mathcal{A}_{12}^{(0)} = \tfrac12\).

So the vanishing in the reference case is due to symmetry, not to numerical approximation. Introducing the envelope \(g\) weights the product asymmetrically, so the symmetry no longer forces the first–third transition element to vanish.

Define the states normalized under the unweighted inner product:

Then compute the transition elements:

Results at the chosen value of the parameter:

• \(\mathcal A_{12}\approx 0.62207485\)
• \(\mathcal A_{13}\approx -0.48326669\)
• \(R_{13/12}=|\mathcal A_{13}|^2/|\mathcal A_{12}|^2 \approx 0.60351552\) (about \(60.35\%\))

That is, the first–third element, which was zero in the symmetric reference model, becomes nonzero with the envelope under the chosen definitions.

Scope of the result: The reference model establishes that the vanishing of the transition element is due to the reflection symmetry under the stated definitions. Introducing the asymmetric envelope \(g\), and using the unweighted inner product, produces numerically a non-zero element and a ratio at the chosen value. That value depends on the choice of envelope, states, transition operator, and inner product; it is therefore presented as a result of the specified mathematical model, while its physical interpretation and the determination of the appropriate state space remain subjects for future investigation.

With the unweighted inner product, the envelope factors enter the integral, so the asymmetry of \(g\) affects the transition elements. With the weighted inner product:

the envelope factors reduce algebraically in every transition element (for every pair \((m,n)\)); after re-normalizing the states under this inner product, the symmetry again forces \(\mathcal{A}_{13}\) specifically to vanish — without setting \(\mathcal{A}_{12}\) to zero.

Qualification. This reduction is an algebraic consequence conditional on the states being in a domain that is square-integrable and normalizable in the weighted space. Since \(g\) is singular near the endpoints of the domain, the weight \(1/g(u)^2\) becomes singular there, and treating this inner product as a valid Hilbert space for the problem requires verifying integrability, specifying the appropriate domain, and imposing boundary conditions.

Frame 3 uses the envelope to build the states \(\Psi_n\), and studies its effect on transition elements and symmetry. Frame 4 uses the same function to build the potential and operator, and then poses the study of its spectrum. Both frames use the same envelope, but study different properties: Frame 3 studies transition elements, and Frame 4 studies the differential operator and the candidate spectrum.

Frame 3 produces a specific mathematical model for how a symmetry-imposed cancellation is lifted when the states are re-weighted by an asymmetric envelope, together with numerical computations and checkable code. What it preserves most carefully is the separation between the symmetry proof in the reference case, the numerical computation in the weighted case, and any possible physical interpretation.