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Strict non-tightness of the level-3 quantum bootstrap for a natural three-dimensional Hamiltonian, at an excited level

  • Daniel Keren
  • Ye Zhou
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Abstract

For the coupled three-dimensional double well H=1/2|p|2+2∑i(qi2−1)2−∑i<jqi2qj2 we exhibit, at the first excited energy level (and also at the ground state), a linear functional on polynomials of degree ≤ 6 that satisfies every constraint of the standard level-3 eigenstate bootstrap, has a \emph{strictly} positive definite moment matrix, and nevertheless assigns a negative value to a nonnegative polynomial. It is therefore an interior point of the level-3 relaxation that is realized by no quantum state. The construction reflects a fact that holds at every level: the eigenstate rows never reach the top-degree momentum moments. At level 3, where strictly feasible points exist, this means that the corresponding block of the moment matrix carries no information about which energies admit one. Every algebraic ingredient is an exact rational certificate, and the only numerical ingredients (spectral enclosures of an eigenfunction) are carried out in rigorous ball arithmetic.

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Cite

BibTeXRIS
BibTeX
@misc{keren2026strict,
  title = {{Strict non-tightness of the level-3 quantum bootstrap for a natural three-dimensional Hamiltonian, at an excited level}},
  author = {Daniel Keren and Ye Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03362},
  eprint = {2610.03362},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03362},
}
RIS
TY  - UNPB
TI  - Strict non-tightness of the level-3 quantum bootstrap for a natural three-dimensional Hamiltonian, at an excited level
AU  - Daniel Keren
AU  - Ye Zhou
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.03362
UR  - https://arxiv.org/abs/2610.03362
AB  - For the coupled three-dimensional double well H=1/2|p|^2+2∑_i(q_i^2−1)^2−∑_i<jq_i^2q_j^2 we exhibit, at the first excited energy level (and also at the ground state), a linear functional on polynomials of degree ≤ 6 that satisfies every constraint of the standard level-3 eigenstate bootstrap, has a \emph{strictly} positive definite moment matrix, and nevertheless assigns a negative value to a nonnegative polynomial. It is therefore an interior point of the level-3 relaxation that is realized by no quantum state. The construction reflects a fact that holds at every level: the eigenstate rows never reach the top-degree momentum moments. At level 3, where strictly feasible points exist, this means that the corresponding block of the moment matrix carries no information about which energies admit one. Every algebraic ingredient is an exact rational certificate, and the only numerical ingredients (spectral enclosures of an eigenfunction) are carried out in rigorous ball arithmetic.
ER  -