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Bootstrapping More QM Systems
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We test the bootstrap approach for determining the spectrum of one dimensional Hamiltonians. In this paper we focus on problems that have a two parameter search space in the bootstrap approach: the double well and a periodic potential associated to the Mathieu equation. For the double well, we compare the energies with contributions from perturbative and non-perturbative results, finding good agreement. For the periodic potentials, we notice that the bootstrap approach gives the band structure of the periodic potential, but it has trouble finding the quasi-momentum of the system. To make further progress on the dispersion relation of the bands, new techniques are needed.
Forward citations
Cited by 2 Pith papers
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Stationarity is not enough: tightness of the quantum mechanical bootstrap and the copositive cone
The stationary quantum bootstrap is provably not tight in two dimensions, while the eigenstate bootstrap appears to remain tight in the tested settings.
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Bootstrapping periodic quantum systems
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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