Numerical Bootstrap in Quantum Mechanics

Bhattacharya, Jyotirmoy; Das, Diptarka; Das, Sayan Kumar; Jha, Ankit Kumar; Kundu, Moulindu

doi:10.48550/arxiv.2108.11416

Cited by 5 publications

(21 citation statements)

References 15 publications

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“…Comparing to the previous literatures in e.g. [2,4,5,6,7,8], we use a twoindexed operators with both x and p. This turns out to improve greatly the efficiency of the bootstrap procedure. We can achieve quite high numerical precisions with only very low level K ≤ 4, since the size of bootstrap matrix grows more quickly as K 2 , comparing to the linear growth in K in the previous literatures.…”

Section: Discussionmentioning

confidence: 99%

“…We underlie the digits in the bootstrap method which agree with the truncation method. literature [2,5,6]. The Hamiltonian is Ĥ = p2 + x2 + gx 4 .…”

Section: The Quartic Anharmonic Oscillatormentioning

confidence: 99%

“…To compare our method with that of [2], we consider the same choice of parameters = 1, g = 1. The case of a negative quadratic term gives a double-well potential and is considered in [5,6]. The quantization of this model also has a long history, see e.g.…”

Section: The Quartic Anharmonic Oscillatormentioning

confidence: 99%

“…Recently, a novel bootstrap method for numerical calculations in matrix models [1] and matrix quantum mechanics [2] is proposed, and has been studied in the literature [3,4,5,6,7,8]. This is related to and inspired by early works on matrix models, see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…In the Calabi-Yau models, it is natural to treat the momentum and position operators equally, and the bootstrap operator is a linear combination of operators in a two-dimensional space indexed by both momentum and position, instead of the one-dimensional space in the literature [2,4,5,6,7,8]. This turns out to also work for the more conventional quantum mechanical models with the standard nonrelativistic kinetic term, and improves the bootstrap efficiency.…”

Section: Introductionmentioning

confidence: 99%

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Bootstrapping Calabi-Yau Quantum Mechanics

Du,

Huang,

Zeng

2021

Preprint

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Recently, a novel bootstrap method for numerical calculations in matrix models and quantum mechanical systems is proposed. We apply the method to certain quantum mechanical systems derived from some well-known local toric Calabi-Yau geometries, where the exact quantization conditions have been conjecturally related to topological string theory. We find that the bootstrap method provides a promising alternative for the precision numerical calculations of the energy eigenvalues. An improvement in our approach is to use a larger set of two-dimensional operators instead of one-dimensional ones. We also apply our improved bootstrap methods to some non-relativistic models in the recent literature and demonstrate better numerical accuracies.

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Section: Discussionmentioning

confidence: 99%

“…We underlie the digits in the bootstrap method which agree with the truncation method. literature [2,5,6]. The Hamiltonian is Ĥ = p2 + x2 + gx 4 .…”

Section: The Quartic Anharmonic Oscillatormentioning

confidence: 99%

Section: The Quartic Anharmonic Oscillatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bootstrapping Calabi-Yau Quantum Mechanics

Du,

Huang,

Zeng

2021

Preprint

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Bootstrapping Calabi–Yau quantum mechanics

Du¹,

Huang²,

Zeng³

2022

Commun. Theor. Phys.

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Bootstrapping Bloch bands

Tchoumakov¹,

Florens²

2021

J. Phys. A: Math. Theor.

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Bootstrap methods, initially developed for solving statistical and quantum ﬁeld theories, have recently been shown to capture the discrete spectrum of quantum mechanical problems, such as the single particle Schrödinger equation with an anharmonic potential. The core of bootstrap methods builds on exact recursion relations of arbitrary moments of some quantum operator and the use of an adequate set of positivity criteria. We extend this methodology to models with continuous Bloch band spectra, by considering a single quantum particle in a periodic cosine potential. We ﬁnd that the band structure can be obtained accurately provided the bootstrap uses moments involving both position and momentum variables. We also introduce several new techniques that can apply generally to other bootstrap studies. First, we devise a trick to reduce by one unit the dimensionality of the search space for the variables parametrizing the bootstrap. Second, we employ statistical techniques to reconstruct the distribution probability allowing to compute observables that are analytic functions of the canonical variables. This method is used to extract the Bloch momentum, a quantity that is not readily available from the bootstrap recursion itself.

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Numerical Bootstrap in Quantum Mechanics

Cited by 5 publications

References 15 publications

Bootstrapping Calabi-Yau Quantum Mechanics

Bootstrapping Calabi-Yau Quantum Mechanics

Bootstrapping Calabi–Yau quantum mechanics

Bootstrapping Bloch bands

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