Resonance matching of 2-$\delta$ and 3-$\delta$ potentials in 1D Quantum Scattering

TL;DR

This paper explores the limits of resonance matching in one-dimensional quantum scattering involving 2- and 3-$\delta$ potentials, revealing theoretical impossibilities and practical constraints for approximating spectra.

Contribution

It provides a theoretical proof that exact isospectrality is impossible for non-trivial configurations and identifies practical constraints for resonance matching in quantum scattering.

Findings

Exact isospectrality is impossible for non-trivial 2- and 3-$\delta$ potentials.

Numerical experiments determine minimal constraints for practical resonance matching.

Results have implications for spectral non-uniqueness in quantum scattering.

Abstract

We investigate whether a 3-$\delta$ system with positive coupling strengths can approximate the transmission spectrum of a 2-$\delta$ resonance system with opposite-sign couplings for $k <3$. Theoretical analysis establishes exact isospectrality -- perfectly matched transmission spectrum -- is impossible for physically non-trivial configurations, while numerical experiments identify the minimal constraint set for practicability. These results establish both the practical limits and achievable accuracy of resonance matching under sign constraints, with implications for understanding spectral non-uniqueness in quantum scattering problems.