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A Semidefinite Programming algorithm for the Quantum Mechanical Bootstrap

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arxiv 2209.14332 v1 pith:KZ6BIUSI submitted 2022-09-28 hep-th quant-ph

David Berenstein , George Hulsey This is my paper · ORCID
classification hep-thquant-ph
keywords constraintsbootstrapalgorithmapproachenergyoperatorspositivityproblem
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We present a semidefinite program (SDP) algorithm to find eigenvalues of Schr\"{o}dinger operators within the bootstrap approach to quantum mechanics. The bootstrap approach involves two ingredients: a nonlinear set of constraints on the variables (expectation values of operators in an energy eigenstate), plus positivity constraints (unitarity) that need to be satisfied. By fixing the energy we linearize all the constraints and show that the feasability problem can be presented as an optimization problem for the variables that are not fixed by the constraints and one additional slack variable that measures the failure of positivity. To illustrate the method we are able to obtain high-precision, sharp bounds on eigenenergies for arbitrary confining polynomial potentials in 1-D.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Stationarity is not enough: tightness of the quantum mechanical bootstrap and the copositive cone

    cs.SC 2026-08 accept novelty 8.0 of 10

    The stationary quantum bootstrap is provably not tight in two dimensions, while the eigenstate bootstrap appears to remain tight in the tested settings.

  2. Bootstrapping periodic quantum systems

    hep-th 2025-07 conditional novelty 7.0 of 10

    A bootstrap method that includes the translation operator and uses reality conditions computes accurate Bloch-band dispersion relations for the cosine potential without positivity constraints.

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