Hamiltonian for the Hilbert-Pólya Conjecture

• URL: http://arxiv.org/abs/2309.00405v6
• Date: Fri, 21 Jun 2024 07:39:44 GMT
• Title: Hamiltonian for the Hilbert-Pólya Conjecture
• Authors: Enderalp Yakaboylu,
• Abstract summary: We introduce a Hamiltonian to address the Hilbert-P'olya conjecture. Our result can be extended to a broader class of functions whose zeros lie on the critical line.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We introduce a Hamiltonian to address the Hilbert-P\'olya conjecture. The eigenfunctions of the introduced Hamiltonian, subject to the Dirichlet boundary conditions on the positive half-line, vanish at the origin by the nontrivial zeros of the Riemann zeta function. Consequently, the eigenvalues are determined by these nontrivial Riemann zeros. If the Riemann hypothesis (RH) is true, the eigenvalues become real and represent the imaginary parts of the nontrivial zeros. Conversely, if the Hamiltonian is self-adjoint, or more generally, admits only real eigenvalues, then the RH follows. In our attempt to demonstrate the latter, we establish the existence of a similarity transformation of the introduced Hamiltonian that is self-adjoint on the domain specified by an appropriate boundary condition, which is satisfied by the eigenfunctions through the vanishing of the Riemann zeta function. Our result can be extended to a broader class of functions whose zeros lie on the critical line.

Related papers

• Reality of the Eigenvalues of the Hilbert-Pólya Hamiltonian [0.0]
  We propose a Hamiltonian for the Hilbert-P'olya Conjecture. We show that the eigenfunctions of the transformed operator are square-integrable, and crucially, that the eigenvalues are real.
  arXiv  Detail & Related papers  (2024-08-27T15:16:00Z)
• The one-dimensional Coulomb Hamiltonian: Properties of its Birman-Schwinger operator [0.0]
  We study the Birman-Schwinger operator for a self-adjoint realisation of the one-dimensional Hamiltonian with the Coulomb potential. In both cases, the Birman-Schwinger operator is Hilbert-Schmidt, even though it is not trace class.
  arXiv  Detail & Related papers  (2024-05-14T13:59:10Z)
• Non-Abelian observable-geometric phases and the Riemann zeros [1.3597551064547502]
  We introduce the notion of non-Abelian observable-geometric phases. Since the observable-geometric phases are connected with the geometry of the observable space, this sheds some light on the investigation of the Heisenberg equation.
  arXiv  Detail & Related papers  (2024-03-28T03:23:46Z)
• A non-hermitean momentum operator for the particle in a box [49.1574468325115]
  We show how to construct the corresponding hermitean Hamiltonian for the infinite as well as concrete example. The resulting Hilbert space can be decomposed into a physical and unphysical subspace.
  arXiv  Detail & Related papers  (2024-03-20T12:51:58Z)
• Curvature-Independent Last-Iterate Convergence for Games on Riemannian Manifolds [77.4346324549323]
  We show that a step size agnostic to the curvature of the manifold achieves a curvature-independent and linear last-iterate convergence rate. To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence has not been considered before.
  arXiv  Detail & Related papers  (2023-06-29T01:20:44Z)
• Rigorous derivation of the Efimov effect in a simple model [68.8204255655161]
  We consider a system of three identical bosons in $mathbbR3$ with two-body zero-range interactions and a three-body hard-core repulsion of a given radius $a>0$.
  arXiv  Detail & Related papers  (2023-06-21T10:11:28Z)
• Supersymmetric quantum mechanics and the Riemann hypothesis [0.0]
  We show that the trivial and nontrivial zeros of the Riemann zeta function naturally correspond to the vanishing ground state energies in this model. The model provides a natural form of supersymmetry.
  arXiv  Detail & Related papers  (2022-11-08T17:13:46Z)
• Formally Self-Adjoint Hamiltonian for the Hilbert-P\'olya Conjecture [0.0]
  We consider a two-dimensional Hamiltonian which couples the Berry-Keating Hamiltonian to the number operator on the half-line via a unitary transformation. We demonstrate that the unitary operator confines the eigenfunction of the Hamiltonian to one dimension as the squeezing parameter tends towards infinity.
  arXiv  Detail & Related papers  (2022-11-03T15:32:32Z)
• Topological Quantum Gravity of the Ricci Flow [62.997667081978825]
  We present a family of topological quantum gravity theories associated with the geometric theory of the Ricci flow. First, we use BRST quantization to construct a "primitive" topological Lifshitz-type theory for only the spatial metric. We extend the primitive theory by gauging foliation-preserving spacetime symmetries.
  arXiv  Detail & Related papers  (2020-10-29T06:15:30Z)
• Models of zero-range interaction for the bosonic trimer at unitarity [91.3755431537592]
  We present the construction of quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range. For a large part of the presentation, infinite scattering length will be considered.
  arXiv  Detail & Related papers  (2020-06-03T17:54:43Z)
• Quantum Geometric Confinement and Dynamical Transmission in Grushin Cylinder [68.8204255655161]
  We classify the self-adjoint realisations of the Laplace-Beltrami operator minimally defined on an infinite cylinder. We retrieve those distinguished extensions previously identified in the recent literature, namely the most confining and the most transmitting.
  arXiv  Detail & Related papers  (2020-03-16T11:37:23Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.