The Berry–Keating operator on L^2({\mathbb R}_\gt,{\rm d}x) and on compact quantum graphs with general self-adjoint realizations

Endres, Sebastian; Steiner, Frank

doi:10.1088/1751-8113/43/9/095204

Cited by 29 publications

(31 citation statements)

References 47 publications

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“…H is an essentially self-adjoint operator acting on the Hilbert space L 2 (0, ∞) of square integrable functions in the half line IR+ = (0, ∞) [23,24,30]. The eigenfunctions, with eigenvalue E, are given by…”

Section: The Quantum Xp Modelmentioning

confidence: 99%

The Riemann Zeros as Spectrum and the Riemann Hypothesis

Sierra

2019

Symmetry

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We present a spectral realization of the Riemann zeros based on the propagation of a massless Dirac fermion in a region of Rindler spacetime and under the action of delta function potentials localized on the square free integers. The corresponding Hamiltonian admits a self-adjoint extension that is tuned to the phase of the zeta function, on the critical line, in order to obtain the Riemann zeros as bound states. The model suggests a proof of the Riemann hypothesis in the limit where the potentials vanish. Finally, we propose an interferometer that may yield an experimental observation of the Riemann zeros. ContentsI. Introduction 2 II. The semiclassical XP Berry, Keating and Connes model 2 III. The quantum XP model 4 IV. The Landau model and XP 5 V. The XP model revisited 7 VI. The space-time geometry of the modified XP models 10 VII. Diracization of H = X(P + 2 p /P ) 12 VIII. ξ-functions: Pólya's is massive and Riemann's is massless 14 IX. The massive Dirac Model in Rindler coordinates 16 X. The massless Dirac equation with delta function potentials 18 XI. Heuristic approach to the spectrum 21 XII. The Riemann zeros as spectrum and The Riemann hypothesis 23 1. Normalizable eigenstates 23 2. The Magnus expansion 24 3. Perron formula 25 XIII. The Riemann interferometer 28 XIV. Dirac models for a class of modified ζ and L functions 29 XV. Conclusions 30 Acknowledgements 31 References 31 arXiv:1601.01797v4 [math-ph]

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Section: The Quantum Xp Modelmentioning

confidence: 99%

The Riemann Zeros as Spectrum and the Riemann Hypothesis

Sierra

2019

Symmetry

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“…Much of the success for the analysis of these problems on graphs can be attributed to the existence of well behaved spectral functions. Consequently spectral determinants and trace formulae in graphs have become important problems widely studied in the literature [2,3,4,5,7,9,8,10,11,13,15,18,19,23,26,30,33]. Here we consider the Schödinger operator −△ + V (x) on a metric graph where each bond of the B bonds is associated with an interval [0, L b ], b = 1, .…”

Section: Introductionmentioning

confidence: 99%

Spectral determinants and zeta functions of Schrödinger operators on metric graphs

Harrison¹,

Kirsten²,

Texier³

2012

J. Phys. A: Math. Theor.

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Abstract. A derivation of the spectral determinant of the Schrödinger operator on a metric graph is presented where the local matching conditions at the vertices are of the general form classified according to the scheme of Kostrykin and Schrader. To formulate the spectral determinant we first derive the spectral zeta function of the Schrödinger operator using an appropriate secular equation. The result obtained for the spectral determinant is along the lines of the recent conjecture [32].

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“…For the Hamiltonian operator as given by Eq. (14), the Hilbert space is H = L 2 (1, ∞) [26][27][28]. We then impose on Eq.…”

Section: Riemann Zeta Schrödinger Equationmentioning

confidence: 99%

A Schrödinger Equation for Solving the Riemann Hypothesis

Moxley¹

2017

Preprint

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The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann zeta function analogies have led to the Bender-Brody-Müller (BBM) conjecture, which involves a non-Hermitian Hamiltonian that maps to the zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a unique Hermitian Hamiltonian that maps to the zeros of the analytic continuation of the Riemann zeta function, and provide an analytical expression for the eigenvalues of the results. The holomorphicity of the resulting eigenvalues is examined. Moreover, a second quantization of the resulting Schrödinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the eigenvalues it is shown that the real part of every nontrivial zero of the Riemann zeta function converges at σ = 1/2.

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The Berry–Keating operator on L^2({\mathbb R}_\gt,{\rm d}x) and on compact quantum graphs with general self-adjoint realizations

Cited by 29 publications

References 47 publications

The Riemann Zeros as Spectrum and the Riemann Hypothesis

The Riemann Zeros as Spectrum and the Riemann Hypothesis

Spectral determinants and zeta functions of Schrödinger operators on metric graphs

A Schrödinger Equation for Solving the Riemann Hypothesis

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The Berry–Keating operator on L^2({\mathbb R}_\gt,{\rm d}x) and on compact quantum graphs with general self-adjoint realizations

Cited by 29 publications

References 47 publications

The Riemann Zeros as Spectrum and the Riemann Hypothesis

The Riemann Zeros as Spectrum and the Riemann Hypothesis

Spectral determinants and zeta functions of Schrödinger operators on metric graphs

A Schr&ouml;dinger Equation for Solving the Riemann Hypothesis

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Product

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A Schrödinger Equation for Solving the Riemann Hypothesis