Related papers: Hamiltonian with Energy Levels Corresponding to Ri…

A Hamiltonian operator $\hat H$ is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. The classical…

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…

We construct a formally self-adjoint Hamiltonian whose eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. We consider a two-dimensional Hamiltonian which couples the Berry-Keating Hamiltonian to the number operator…

We construct a supersymmetric quantum mechanical model in which the energy eigenvalues of the Hamiltonians are the products of Riemann zeta functions. We show that the trivial and nontrivial zeros of the Riemann zeta function naturally…

Following the Hilbert-P\'olya approach to the Riemann Hypothesis, we present an exact spectral realization of the nontrivial zeros of the Riemann zeta function $\zeta(z)$ with a Mellin-Barnes integral that explicitly contains it. This…

The local Riemann hypothesis states that the zeros of the Mellin transform of a harmonic-oscillator eigenfunction (on a real or p-adic configuration space) have real part 1/2. For the real case, we show that the imaginary parts of these…

Towards the Hilbert-P\'olya conjecture, in this paper, we present a general construction of Hamiltonian $\hat{H}_{f},$ which leads a general family of Hurwitz zeta functions $(-1)^{z_{n}-1}L(f,z_{n},x+1)$ defined by Mellin transform becomes…

The concept of energy-dependent forces in quantum mechanics is re-analysed. We suggest a simplification of their study via the representation of each self-adjoint and energy-dependent Hamiltonian H=H(E) with real spectrum by an auxiliary…

We study the algebraic structure of the eigenvalues of a Hamiltonian that corresponds to a many-body fermionic system. As the Hamiltonian is quadratic in fermion creation and/or annihilation operators, the system is exactly integrable and…

Effective (i.e., subspace-constrained) Hamiltonians become, by construction, energy-dependent while all the energy-dependent forces prove non-linear because the energy itself is merely an eigenvalue of the Hamiltonian H. One of the most…

A general recipe to define, via Noether theorem, the Hamiltonian in any natural field theory is suggested. It is based on a Regge-Teitelboim-like approach applied to the variation of Noether conserved quantities. The Hamiltonian for General…

First, we recount a history of how certain methods using natural self-adjoint operators have, thus far, failed to prove the Riemann Hypothesis. In Section 2, we set the analytical context necessary to have genuine proofs in later sections,…

We present a brief review of the spectral approach to the Riemann hypothesis, according to which the imaginary part of the non trivial zeros of the zeta function are the eigenvalues of the Hamiltonian of a quantum mechanical system.

A proof of the Riemann's hypothesis (RH) about the non-trivial zeros of the Riemann zeta-function is presented. It is based on the construction of an infinite family of operators D^{(k,l)} in one dimension, and their respective…

The differential-equation eigenvalue problem associated with a recently-introduced Hamiltonian, whose eigenvalues correspond to the zeros of the Riemann zeta function, is analyzed using Fourier and WKB analysis. The Fourier analysis leads…

Berry and Keating conjectured that the classical Hamiltonian H = xp is related to the Riemann zeros. A regularization of this model yields semiclassical energies that behave, in average, as the non trivial zeros of the Riemann zeta…

The Hamiltonian $H={1\over2} p^2+{1\over2}m^2x^2+gx^2(ix)^\delta$ with $\delta,g\geq0$ is non-Hermitian, but the energy levels are real and positive as a consequence of ${\cal PT}$ symmetry. The quantum mechanical theory described by $H$ is…

We first explain the link between the Berry-Keating Hamiltonian and the spectral realization of zeros of the Riemann zeta function of the first author, and why there is no conflict at the semi-classical level between the "absorption"…

We have given some arguments that a two-dimensional Lorentz-invariant Hamiltonian may be relevant to the Riemann hypothesis concerning zero points of the Riemann zeta function. Some eigenfunction of the Hamiltonian corresponding to…

The Riemann hypothesis, which states that the non-trivial zeros of the Riemann zeta function all lie on a certain line in the complex plane, is one of the most important unresolved problems in mathematics. Inspired by the P\'olya-Hilbert…