Riemann $\zeta$ function from wave-packet dynamics

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 $\zeta$ function $\zeta (s,a)$. Indeed, the autocorrelation function at a time $t$ is determined by $\zeta (\sigma +i\tau ,a)$, where $\sigma$ is governed by the temperature of the thermal phase state and $\tau$ 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 $\zeta (s,a)$. 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 $\zeta$ wave-packet dynamics using cold atoms in appropriately tailored light fields.

Figure 1

Potential $V=V(x;V_0)$ for a nonrelativistic particle of mass $\mu$ giving rise to a logarithmic energy spectrum $E_n=\hslash \omega \ln [\gamma (n+a)]$ according to the RKR method. Here we display $V$ in units of $\hslash \omega$ in its dependence on the dimensionless position $\gamma \kappa x$ and the value $V_0/\hslash \omega$ of $V$ at $x=0$. The parameters $\gamma$ and $a$ of the spectrum determine via Eq. (40) the value $V_0$. At the origin the potential is approximately quadratic. For large values of $\gamma \kappa \left|x\right|$ it increases logarithmically. In the limit of $V_0\to -\infty$ we obtain a purely logarithmic potential with a singularity at $x=0$.

Figure 2

Comparison between the RKR potential $V(x;V_0)$ determined from Eq. (34) (depicted by the solid curve) and the approximations for small or large values of the dimensionless position $\gamma \kappa x$ given by Eqs. (35) or (37) (represented by dashed or dotted lines, respectively). The potential is scaled in units of $\hslash \omega$ and we have chosen a value of $V_0=-\hslash \omega \ln 2$.

Figure 3

Comparison between the numerically exact potential (solid line) leading to the energy spectrum $E_n=\hslash \omega \ln (n+1)$ and the RKR potential (dashed line). The two potentials are substantially different. However, for larger distances the deviation becomes less important, as shown by the inset. Here we have chosen the parameters $V_0=-\hslash \omega \ln 2$ and $a=\gamma =1$. Moreover, we have expressed the potentials in units of $\hslash \omega$. The dimensionless position variable is given by $\kappa x$.