dorsal/arxiv
View SchemaThe One Dimensional Approachissimo Quantum Harmonic Oscillator: The Hilbert-Polya Hamiltonian for the Primes and the Zeros of the Riemann Function
| Authors | N. Garcia |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0611134 |
| URL | https://arxiv.org/abs/quant-ph/0611134 |
Abstract
I have made an ample study of one dimensional quantum oscillators, ranging from logarithmic to exponential potentials. I have found that the eigenvalues of the hamiltonian of the oscillator with the limiting (approachissimo) harmonic potential (~ p(x)2) maps the zeros of the Riemann function height up in the Riemann line. This is the potential created by the field of J(x) that is the Riemann generator of the prime number counting function, p(x), that in turn can be defined by an integral transformation of the Riemann zeta function. This plays the role of the spring strength of the quantum limiting harmonic oscillator. The number theory meaning of this result is that the roots height up of the zeta function are the eigenvalues of a Hamiltonian whose potential is the number of primes squared up to a given x. Therefore this may prove the never published Hilbert-Polya conjecture. The conjecture is true but does not imply the truth of the Riemann hypothesis. We can have complex conjugated zeros off the Riemman line and map them with another hermitic operator and a general expression is given for that. The zeros off the line affect the fluctuation of the eigenvalues but not their mean values.
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"abstract": "I have made an ample study of one dimensional quantum oscillators, ranging\nfrom logarithmic to exponential potentials. I have found that the eigenvalues\nof the hamiltonian of the oscillator with the limiting (approachissimo)\nharmonic potential (~ p(x)2) maps the zeros of the Riemann function height up\nin the Riemann line. This is the potential created by the field of J(x) that is\nthe Riemann generator of the prime number counting function, p(x), that in turn\ncan be defined by an integral transformation of the Riemann zeta function. This\nplays the role of the spring strength of the quantum limiting harmonic\noscillator. The number theory meaning of this result is that the roots height\nup of the zeta function are the eigenvalues of a Hamiltonian whose potential is\nthe number of primes squared up to a given x. Therefore this may prove the\nnever published Hilbert-Polya conjecture. The conjecture is true but does not\nimply the truth of the Riemann hypothesis. We can have complex conjugated zeros\noff the Riemman line and map them with another hermitic operator and a general\nexpression is given for that. The zeros off the line affect the fluctuation of\nthe eigenvalues but not their mean values.",
"arxiv_id": "quant-ph/0611134",
"authors": [
"N. Garcia"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP",
"math.NT"
],
"title": "The One Dimensional Approachissimo Quantum Harmonic Oscillator: The Hilbert-Polya Hamiltonian for the Primes and the Zeros of the Riemann Function",
"url": "https://arxiv.org/abs/quant-ph/0611134"
},
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