Appendix C: Proof of Scattering Time Delay Function $\kappa (\omega )$

This appendix proves that the “residence time” of microscopic particles in interaction regions is equivalent to the spectral density of geometric paths, i.e., the mathematical construction of the $\kappa (\omega )$ function.

1. Scattering Matrix and Time Delay

Define the scattering matrix $S(\omega )$. The Eisenbud-Wigner time delay operator is defined as:

$$Q(\omega )=-i{S}^{†}(\omega )\frac{dS(\omega )}{d\omega }\text{(C.1)}$$

Its trace $\text{tr}Q(\omega )$ corresponds to the total time delay.

2. Spectral Shift Function and Birman-Krein Formula

According to the Birman-Krein formula in scattering theory, the determinant of the scattering matrix is related to the spectral shift function $\xi (\omega )$:

$$\det S(\omega )=e^{-2\pi i\xi (\omega )}$$

Taking logarithm and differentiating both sides, we obtain:

$$\frac{1}{2\pi }\text{tr}Q(\omega )=-\frac{d\xi }{d\omega }={\rho }_{rel}(\omega )$$

where ${\rho }_{rel}$ is the relative density of states.

3. Unified Time Scale Function

We define the unified function $\kappa (\omega )$:

$$\kappa (\omega ):=\frac{1}{2\pi }\text{tr}Q(\omega )=\frac{d\delta }{d\omega }\text{(C.2)}$$

This proves: the residence time (Delay) of particles in potential wells is strictly equivalent to the winding rate (Winding Rate) of geometric phase with energy in Hilbert space. This is the mathematical foundation of “residence is existence” in the main text.

4. Example: 1D $\delta$ Potential Barrier

For potential energy $V(x)=\Omega \delta (x)$, the scattering phase shift is $\tan \delta (k)=-m\Omega /k$.

Calculating its time delay:

$$\Delta t(\omega )=\frac{d\delta }{d\omega }=\frac{m^2\Omega }{k(k^2+m^2{\Omega }^2)}\text{(C.3)}$$

The result shows that near resonance points, time delay peaks, corresponding to “geometric knotting” of particles at the microscopic level.