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

TL;DR

This work studies whether a 3-$\delta$ potential with all positive couplings can replicate the transmission spectrum of a 2-$\delta$ system with opposite-sign couplings in 1D quantum scattering for $k<3$. It proves that exact global isospectrality is impossible for non-trivial configurations and then develops a windowed differential-evolution optimization to approximate the 2-$\delta$ spectrum within individual resonance windows, under a strength constraint $0.5\le|\beta_i|\le 2|\alpha_1|$. Numerically, high-fidelity matches are demonstrated for configurations with 1–5 resonances, achieving mean-squared errors as low as $10^{-8}$ to $10^{-4}$ across windows, with consistent adherence to the strength bound. The results delineate practical limits and offer a scalable approach to resonance matching in quantum scattering, with potential implications for inverse scattering and engineered transmission devices.

Abstract

We investigate whether a 3-$δ$ system with positive coupling strengths can approximate the transmission spectrum of a 2-$δ$ 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.

Figures (4)

• Figure 1: Failed isospectrality attempt for global optimization over $k \in [0.01, 3.0]$ achieves only MSE = $4.01 \times 10^{-2}$ (Positions are not shown explicitly for the emphasis on transmission). This validates Proposition 2.1 and motivates the windowed optimization strategy (Section 3), which achieves MSE $\sim 10^{-10}$ locally with the same positive-only constraints.
• Figure 2: Two-resonance configuration with $\alpha = \pm 2$, $\Delta x = 2.65$. Resonances at $k_1 \approx 1.19$ and $k_2 \approx 2.37$ with MSE: $2.98 \times 10^{-10}$ (W1) and $4.94 \times 10^{-6}$ (W2). The strength distribution shows all $\beta_i$ values remain within the $2|\alpha_1| = 4$ bound.
• Figure 3: Five-resonance configuration with $\alpha = \pm 3$, $\Delta x = 5.97$. Resonances at $k_1 \approx 0.53$, $k_2 \approx 1.05$, $k_3 \approx 1.58$, $k_4 \approx 2.10$, $k_5 \approx 2.63$ with MSE values ranging from $1.61 \times 10^{-8}$ to $5.22 \times 10^{-5}$. All strengths satisfy $\beta_i \leq 2|\alpha_1| = 6$.
• Figure 4: Left: Average MSE versus number of resonances for different coupling strengths. Right: Average MSE organized by resonance position across all systems. Under the $2|\alpha_1|$ strength bound, performance remains consistently strong across all resonance counts and positions. Here, 'Average MSE' denotes the mean MSE computed over all windows within each 2-delta system.

Theorems & Definitions (4)

• Proposition 2.1: Necessary conditions for exact isospectrality
• Corollary 2.2: Triviality of exact matches
• proof
• Remark 1