Analytic Bootstrap Solution for Quantum Mechanics on a Circle
Abstract
We solve the Schrödinger equation for a particle in the circle with a quantum mechanical bootstrap method. The main idea is that in physical situations one needs solutions of the recursion relations for the Fourier coefficients of the measure that vanish exponentially fast. We show that this results in a (generalized) continued fraction representation of the solution. The solution is an analytical function of the couplings and the energy. We also show how one this method directly computes the density of states and checking that the answer is physical is reduced to checking an inequality for a single variable. We also show how to reconstruct the wave function and how to extract the quasimomentum of the particle as a function of the energy. We illustrate our solution in the simplest case of the Mathieu equation, where we show that the method can easily resolve exponentially small bands in the band spectrum.
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BibTeX
@misc{berenstein2026analytic,
title = {{Analytic Bootstrap Solution for Quantum Mechanics on a Circle}},
author = {David Berenstein and Upamanyu Moitra},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36031},
eprint = {2610.36031},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36031},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Analytic Bootstrap Solution for Quantum Mechanics on a Circle
AU - David Berenstein
AU - Upamanyu Moitra
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36031
UR - https://arxiv.org/abs/2610.36031
AB - We solve the Schrödinger equation for a particle in the circle with a quantum mechanical bootstrap method. The main idea is that in physical situations one needs solutions of the recursion relations for the Fourier coefficients of the measure that vanish exponentially fast. We show that this results in a (generalized) continued fraction representation of the solution. The solution is an analytical function of the couplings and the energy. We also show how one this method directly computes the density of states and checking that the answer is physical is reduced to checking an inequality for a single variable. We also show how to reconstruct the wave function and how to extract the quasimomentum of the particle as a function of the energy. We illustrate our solution in the simplest case of the Mathieu equation, where we show that the method can easily resolve exponentially small bands in the band spectrum.
ER -