Goal
Validate the hypothesis T(E) ∝ σ_min²(E) for 1D quantum scattering by analyzing the singular-value spectrum of the gluing matrix. The repo provides stabilized matrix builders, robust SVD wrappers, a gap protocol for multiplicity detection, and runnable examples.
Instead of thinking of tunneling as a mysterious quantum effect, we propose a much more intuitive view:
Transmission = how well local solutions fit together into a global wave
In a piecewise potential, the wavefunction is built from local solutions in each region.
To get a physical solution, these pieces must match perfectly at the boundaries.
- If they match well → wave passes through → high transmission
- If they mismatch → wave is blocked → low transmission
We encode all matching conditions into a single matrix:
the gluing matrix (A(E))
Then we analyze it using Singular Value Decomposition (SVD).
The smallest singular value measures how well everything fits together
And remarkably:
T(E) ∝ σ²ₘᵢₙ(E)
- small σₘᵢₙ → poor fit → low transmission
- large σₘᵢₙ → strong coherence → high transmission
We turn tunneling into a coherence problem:
- Physics view: particle crosses a barrier
- Our view: local waves successfully glue into a global solution
SVD answers:
“How close is this system to having a perfectly consistent global solution?”
- near-zero singular value → almost perfect solution exists
- multiple small singular values → multiple coherent modes (resonances)
- Gives an intuitive explanation of tunneling
- Provides a numerically stable diagnostic tool
- Reveals hidden structure (resonances, multiplicity)
- Connects physics with geometry / sheaf-like thinking
Tunneling is not magic — it’s the degree of global consistency of local wave solutions, and SVD measures exactly that.
Author: AI / Drifting 03.2026