Inverse Quantum Potential Reconstruction via Generalized Bertlmann-Martin Inequalities

TL;DR

This work tackles the inverse problem of reconstructing a 1D quantum potential from limited bound-state data by proposing a Laplace-moment reconstruction pipeline that leverages generalized Bertlmann--Martin inequalities to constrain even moments, supplements odd moments with a principled interpolation, and stabilizes the transform via Padé continuation before inverting to obtain $\rho(r)$ and $V(r)$. The method assembles a reproducible pipeline with four stages: GBM-based moment ladders, Laplace representation of $r^2\rho(r)$, Padé-based analytic continuation with model averaging, and inverse Laplace inversion to recover the density and potential, including a practical density-to-potential mapping on a regulated grid. Benchmark studies across Coulomb, harmonic oscillator, Hulthén, Kratzer, and a hyperbolic molecular well demonstrate accurate reconstruction with minimal input (often 7–22 spectral inputs) and show that GBM–Laplace outperforms a least-squares baseline in data efficiency and stability. The approach emphasizes transparency, reproducibility, and tunable diagnostics, offering a viable path for experimental contexts where only a handful of bound-state energies are accessible. Overall, the paper advances a data-efficient, transform-based inverse method for quantum potentials with broad applicability to bound-state systems and provides a detailed reproducibility framework for future validation.

Abstract

Reconstructing a 1D quantum potential V(r) from a few bound-state energies is a long-standing inverse problem. We present a Laplace-moment reconstruction pipeline that ties the Bertlmann-Martin gap bound to generalized Bertlmann-Martin (GBM) even-moment ladders, continues the Laplace image with Pade approximants, and inverts the transform to recover rho(r) and V(r). Odd moments are supplied by a physically consistent interpolation scheme. The paper emphasizes mode ladders that isolate each approximation layer and a reproducibility spine that records benchmark settings and diagnostics. All numerical results are tied to archived configurations, and conclusions are reported empirically under the stated benchmark settings.

Figures (13)

• Figure 1: Coulomb benchmark sheet ($Z=1$): reconstructed versus exact $V(r)$, $r^{2}\rho_{0,0}(r)$, $L(q)$, and $\chi_{0,0}(r)$.
• Figure 1: Fit for eigenvalues using the Dirichlet-Dirichlet boundary conditions.
• Figure 2: Harmonic oscillator benchmark sheet ($\omega=1$): reconstructed versus exact $V(r)$, $r^{2}\rho_{0,0}(r)$, $L(q)$, and $\chi_{0,0}(r)$.
• Figure 2: Fit for eigenvalues using the Dirichlet-Neumann boundary conditions.
• Figure 3: Hulthén benchmark sheet ($V_{0}=\lambda=0.5$): reconstructed versus exact $V(r)$, $r^{2}\rho_{0,0}(r)$, $L(q)$, and $\chi_{0,0}(r)$.

Theorems & Definitions (3)

• Theorem Appendix A.1
• Theorem Appendix A.2
• Theorem Appendix A.3