Bootstrapping More QM Systems

Abstract

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.

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• Semidefinite Programs at Finite Fermion Density
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• Non-Hermitian Generalization of Rayleigh-Schrödinger Perturbation Theory
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• A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems
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• Bootstrapping entanglement in quantum spin systems
• Bootstrap Method in Theoretical Physics