Numerical bootstrap in quantum mechanics

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

doi:10.1016/j.physletb.2021.136785

Jyotirmoy Bhattacharya, Diptarka Das + Show 3 more

Open Access

https://doi.org/10.1016/j.physletb.2021.136785

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Abstract

We study the effectiveness of the numerical bootstrap techniques recently developed in [1] for quantum mechanical systems. We find that for a double well potential the bootstrap method correctly captures non-perturbative aspects. Using this technique we then investigate quantum mechanical potentials related by supersymmetry and recover the expected spectra. Finally, we also study the singlet sector of O(N) vector model quantum mechanics, where we find that the bootstrap method yields results which in the large N agree with saddle point analysis.

Highlights

Bootstrap techniques have received a lot of attention in the context of conformal field theories - CFTs
In the bootstrap approach one first classifies the full set of ‘data’ necessary to specify a CFT completely and constrains this data based on physical inputs such as reflection positivity, crossing symmetry and causality
The basic idea is to first identify the complete set of the ‘data’ for the problem. This data a priori mainly consists of all the correlation function of the basic variables

Summary

Introduction

Bootstrap techniques have received a lot of attention in the context of conformal field theories - CFTs (see the reviews [2] and the references therein). We consider susy quantum mechanical problems [8] which can be reduced to a problem of two isospectral potentials (except the ground state) that follow from the same superpotential This susy system provides us with a unique opportunity to study and compare the convergence properties of the method, where it is known a priori that the spectra of two distinct potentials should converge to the same values of energy. The singlet sector of this model reduces to a quantum mechanical problem in collective coordinates, which can be subjected to the bootstrap This gives us the value of ground state energy in this system for all values of N and the quartic coupling λ. We shall choose different forms of the potential V and execute the bootstrap method to obtain some low-lying energy eigenstates

The harmonic oscillator as a limit of the anharmonic oscillator

The double-well potential

Susy quantum mechanics

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