Riemannζfunction from wave-packet dynamics

Abstract

We show that the time evolution of a thermal phase state of an anharmonic oscillator with logarithmic energy spectrum is intimately connected to the generalized Riemann ζ function ζ (s,a). Indeed, the autocorrelation function at a time t is determined by ζ (σ + iτ,a), where σ is governed by the temperature of the thermal phase state and τ is proportional to t. We use the JWKB method to solve the inverse spectral problem for a general logarithmic energy spectrum; that is, we determine a family of potentials giving rise to such a spectrum. For large distances, all potentials display a universal behavior; they take the shape of a logarithm. However, their form close to the origin depends on the value of the Hurwitz parameter a in ζ (s,a). In particular, we establish a connection between the value of the potential energy at its minimum, the Hurwitz parameter and the Maslov index of JWKB. We compare and contrast exact and approximate eigenvalues of purely logarithmic potentials. Moreover, we use a numerical method to find a potential which leads to exact logarithmic eigenvalues. We discuss possible realizations of Riemann ζ wave-packet dynamics using cold atoms in appropriately tailored light fields.

Dirichlet series as interfering probability amplitudes for quantum measurements

“…Hence, the phase factor n exp[ i ln ( 1)] τ − + is reminiscent of the unitary time evolution of an energy eigenstate of a quantum system with a logarithmic energy spectrum. Here τ plays the role of a dimensionless time and the terms ( ) In [14] we have constructed a non-linear oscillator whose energy eigenstates n | 〉 have the eigenvalues…”

“…where the coefficients c ( , ) nj σ τ depend on the real part σ as well as the imaginary part τ of the argument s. The function  can of course be realized by (14), that is the overlap between the phase-conjugated initial state σ τ and the sum D ( , ) σ τ in the previous sections.…”

Entanglement and analytical continuation: an intimate relation told by the Riemann zeta function

“…Here, the fact that the interaction has entangled the two systems is of utmost importance. Indeed, a single quantum system does not [11] allow us to obtain the behavior of ζ along the critical line σ = 1/2. Conservation of probability restricts us to a domain of the complex plane to the right of σ = 1 where ζ is defined by the Dirichlet sum (1).…”

“…In [11] we have followed this line of thought and have represented the Riemann zeta function by wave packet dynamics. In particular, we have analyzed the motion of a particle in a potential whose energy eigenvalues E n satisfy the law…”

“…In this appendix we derive a Jaynes-Cummings-like Hamiltonian as an approximation for the Riemann-Siegel HamiltonianĤ RS defined by (11). Although this approximate Hamiltonian does not quite display the logarithmic dependence ofĤ RS , it nevertheless provides us with a surprisingly good approximation of the Riemann zeta function.…”