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Riemann Hypothesis, Modified Morse Potential and Supersymmetric Quantum Mechanics
Mathematical Physics · Submitted 28 Feb 2020
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This allows us to formulate the Riemann hypothesis in terms of the location of the nodes of the ground state wave function in momentum space.
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
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The abstract, sentence by sentence
In this paper we discuss various potentials related to the Riemann zeta function and the Riemann Xi function. These potentials are modified versions of Morse potentials and can also be related to modified forms of the radial harmonic oscillator and modified Coulomb potential. We use supersymmetric quantum mechanics to construct their ground state wave functions and the Fourier transform of the ground state to exhibit the Riemann zeros. This allows us to formulate the Riemann hypothesis in terms of the location of the nodes of the ground state wave function in momentum space. We also discuss the relation these potentials to one and two matrix integrals and construct a few orthogonal polynomials associated with the matrix models. We relate the Schrodinger equation in momentum space to and finite difference equation in momentum space with an infinite number of terms. We computed the uncertainty relations associated with these potentials and ground states as well as the Shannon Information entropy and compare with the unmodified Morse and harmonic oscillator potentials. Finally we discuss the extension of these methods to other functions defined by a Dirichlet series such as the the Ramanujan zeta function.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper we discuss various potentials related to the Riemann zeta function and the Riemann Xi function. | contribution 0.75 | 0.65 | 0.04 | asserted 0.84 |
| s1 | These potentials are modified versions of Morse potentials and can also be related to modified forms of the radial harmonic oscillator and modified Coulomb potential. | contribution 0.64 | 0.37 | 0.04 | asserted 0.62 |
| s2 | We use supersymmetric quantum mechanics to construct their ground state wave functions and the Fourier transform of the ground state to exhibit the Riemann zeros. | method 0.95 | 0.94 | 0.13 | asserted 1.00 |
| s3 | This allows us to formulate the Riemann hypothesis in terms of the location of the nodes of the ground state wave function in momentum space. | interpretation 0.65 | 0.85 | 0.42 | asserted 0.87 |
| s4 | We also discuss the relation these potentials to one and two matrix integrals and construct a few orthogonal polynomials associated with the matrix models. | contribution 0.99 | 0.92 | 0.05 | asserted 1.00 |
| s5 | We relate the Schrodinger equation in momentum space to and finite difference equation in momentum space with an infinite number of terms. | method 0.52 | 0.87 | 0.05 | asserted 0.99 |
| s6 | We computed the uncertainty relations associated with these potentials and ground states as well as the Shannon Information entropy and compare with the unmodified Morse and harmonic oscillator potentials. | result 0.52 | 0.96 | 0.06 | asserted 1.00 |
| s7 | Finally we discuss the extension of these methods to other functions defined by a Dirichlet series such as the the Ramanujan zeta function. | future work 0.90 | 0.71 | 0.06 | asserted 0.83 |
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The paper
Original abstract
In this paper we discuss various potentials related to the Riemann zeta function and the Riemann Xi function. These potentials are modified versions of Morse potentials and can also be related to modified forms of the radial harmonic oscillator and modified Coulomb potential. We use supersymmetric quantum mechanics to construct their ground state wave functions and the Fourier transform of the ground state to exhibit the Riemann zeros. This allows us to formulate the Riemann hypothesis in terms of the location of the nodes of the ground state wave function in momentum space. We also discuss the relation these potentials to one and two matrix integrals and construct a few orthogonal polynomials associated with the matrix models. We relate the Schrodinger equation in momentum space to and finite difference equation in momentum space with an infinite number of terms. We computed the uncertainty relations associated with these potentials and ground states as well as the Shannon Information entropy and compare with the unmodified Morse and harmonic oscillator potentials. Finally we discuss the extension of these methods to other functions defined by a Dirichlet series such as the the Ramanujan zeta function.
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How the literature uses it
2 arXiv papers cite this in 2 sentences
Papers citing it
Cites it as background 2 papers
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In some recent studies, the relationship between the Riemann zeta function and various aspects of Physics has been explored [ref] .
A symmetry perspective of the Riemann zeros · Introduction · 0.99
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Recently there has been several investigations of the interesting relation of supersymmetric quantum mechanics to the Riemann hypothesis [ref] [ref] [ref] [ref] .
Quantum Computing and the Riemann Hypothesis · Relation to supersymmetric quantum mechanics · 0.99
What it cites · 10 sentences without a confident label
Uses the method or tool of 1 paper
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Using the definition for the Shannon information entropies in position and momentum basis [ref] [ref] [ref] : [equation] [equation] these satisfy the inequality: [equation] In table 3 we calculate then quantities for the six potentials.
Quantum information entropies of the eigenstates of the Morse potential · Shannon information inequality · 0.84
Compares with 2 papers
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Although the connection to supersymmetric quantum mechanics to the Riemann Zeta function and the Morse potential has been made in [ref] [ref] [ref] our approach is somewhat different in that we concentrate on the ground state in momentum space to form the hypothesis.
Supersymmetry and the Riemann zeros on the critical line · Fourier transform of the ground state · 0.80
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Although the connection to supersymmetric quantum mechanics to the Riemann Zeta function and the Morse potential has been made in [ref] [ref] [ref] our approach is somewhat different in that we concentrate on the ground state in momentum space to form the hypothesis.
Weyl Asymptotics for Perturbations of Morse Potential and Connections to the Riemann Zeta Function · Fourier transform of the ground state · 0.80
Cites as background 1 paper
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Another modern approach concerns the deformation of the $Φ$ function by a Gaussian term [ref] .
The De Bruijn-Newman constant is non-negative · Introduction · 0.92
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