aiqbench analysis
This research demonstrates a fundamental property of the eigenstate quantum-mechanical bootstrap where convergence to physical moments occurs even though exactness is never achieved at any finite order. The authors prove this behavior for specific quasi-exactly solvable oscillators and polynomial Schrödinger operators, revealing persistent gaps in sum-of-squares certificates despite adding localizing constraints. These findings advance the theoretical understanding of moment problems and representational limits within quantum mechanical frameworks.
Convergence Without Finite Exactness in the Eigenstate Quantum Bootstrap
arXiv:2610.07775v1 Announce Type: cross Abstract: The eigenstate quantum-mechanical bootstrap can converge to the physical moment set while remaining nonexact at every finite order. We prove this for a one-dimensional quasi-exactly solvable sextic oscillator at explicitly known eigenvalues, including excited levels. At every finite order, the lower and upper bounds on $u^2$ remain strictly below and above its physical value, even after adjoining arbitrary finite polynomial Gram tests and all eigenstate-annihilator relations visible in the retained moment space. Hence neither side of the centered observable admits a finite sum-of-squares certificate modulo the eigenstate annihilator. The gap persists under any finite set of additional localizing constraints whose weights are strictly positive on polynomial vectors. We then prove convergence for polynomial Schr\"odinger operators satisfying explicit sum-of-squares confinement and derivative bounds. The eigenstate relations give moment estimates uniform in the truncation order; these yield a regular Schr\"odinger representation supported on the prescribed eigenspace. Finally, a four-mode extension combines this fixed-observable obstruction with feasible position moments having no positive representing measure at every finite order. At level three, this construction yields nonphysical feasible points with strictly positive definite moment matrices.
Abstract:The eigenstate quantum-mechanical bootstrap can converge to the physical moment set while remaining nonexact at every finite order. We prove this for a one-dimensional quasi-exactly solvable sextic oscillator at explicitly known eigenvalues, including excited levels. At every finite order, the lower and upper bounds on $u^2$ remain strictly below and above its physical value, even after adjoining arbitrary finite polynomial Gram tests and all eigenstate-annihilator relations visible in the retained moment space. Hence neither side of the centered observable admits a finite sum-of-squares certificate modulo the eigenstate annihilator. The gap persists under any finite set of additional localizing constraints whose weights are strictly positive on polynomial vectors. We then prove convergence for polynomial Schrödinger operators satisfying explicit sum-of-squares confinement and derivative bounds. The eigenstate relations give moment estimates uniform in the truncation order; these yield a regular Schrödinger representation supported on the prescribed eigenspace. Finally, a four-mode extension combines this fixed-observable obstruction with feasible position moments having no positive representing measure at every finite order. At level three, this construction yields nonphysical feasible points with strictly positive definite moment matrices.
36 pages. Numerical scripts and solver inputs and outputs included as ancillary files. Companion paper: arXiv:2610.03362
arXiv:2610.07775 [math-ph]
(or arXiv:2610.07775v1 [math-ph] for this version)
https://doi.org/10.48550/arXiv.2610.07775 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Full text under the publisher's open licence · 287 words
Related coverage
Physics-Informed Classical and Quantum Neural Networks for One-Dimensional Schrodinger Eigenvalue Problems
arXiv · Sep 22, 2026
Postselection-Free Reconstruction of Monitored SPT Flux-Charge Responses
arXiv · Aug 18, 2026
Researchers Bound Quantum System Complexity Using Exponentials
Quantum Zeitgeist · Oct 2, 2026
Utrecht University Cuts Simulation Times with New Symmetry Tool
Quantum Zeitgeist · Sep 10, 2026