The Riemann Zeta Function and the Inverted Harmonic Oscillator

(from Annals of Physics, Volume 254, Number 1, February 1997)

R.K. Bhaduri, Avinash Khare, S.M. Reimann, and E.L. Tomusiak

Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1


The Riemann zeta function has phase jumps of pi every time it changes sign as the parameter t in the complex argument s = 1/2 + it is varied. We show analytically that as the real part of the argument is increased to rho > 1/2, the memory of the zeros fades only gradually through a Lorentzian smoothing of the density of the zeros. The corresponding trace formula, for rho >>1, is of the same form as that generated by a one-dimensional harmonic oscillator in one direction, along with an inverted oscillator in the transverse direction. It is pointed out that Lorentzian smoothing of the level density for more general dynamical systems may be done similarly. The Gutzwiller trace formula for the simple saddle plus oscillator model is obtained analytically, and is found to agree with the quantum result



The Riemann zeta function zeta(s) of the complex variable s = rho + it has an infinite number of zeros on the half-line rho = 1/2 [1]. Along this line, as a function of t, every time that zeta(s) changes sing a discontinuous jump by pi in the phase angle is introduced. Otherwise the phase angle is a smooth function of t. In a previous paper [2], it was noted that as rho is increased, the phase of the zeta function is smoothed gradually, and the smooth part is directly linked to the quantum scattering phase shift of a one-dimensional inverted harmonic oscillator. In this paper, we first examine the phase of zeta(s) and show that its derivative with respect to t for a fixed rho>1/2 is the Lorentz-smoothed oscillating part of the density of the zeros at rho = 1/2. The latter may also be expressed [3] as a Gutzwiller-like trace formula with primitive orbits whose periods are the logarithms of the prime numbers. By choosing rho>>1, the contributions of the larger primes are severely damped. In this limit of large damping, the residual oscillating part of the density of states appears to be of the form of a harmonic oscillator, but with a denominator corresponding to an unstable periodic orbit. In the present paper, we show that this instability in a periodic orbit may be brought about by a very simple model Hamiltonian. In particular, we examine the motion of a particle in a plane, bounded by a harmonic oscillator potential alnong one axis and by a parabolic saddle in the transverse direction. This dynamical model is a caricature of the Riemann zeta function for very large rho. Interestingly enough, the electrostatic potential at the bottleneck of a quantum point contact in a mesoscopic structure [4] has the same shape. In this constriction of the split-gate system, the electron transmission through the saddle may take place in quantised channels, corresponding to the bound states of the one-dimensional harmonic potential. This gives rise to quantised conductance steps that are seen experimentally. The relevance of this will be discussed later. The focus in this paper, however, is not the well-studied transmission problem through this potential, but the semiclassical density of states and its asymptotic connection to the Riemann zeta function. The classical motion perpendicular to the saddle has only one isolated (unstable) periodic orbit and its repetitions. The semiclassical trace formula using the Gutzwiller approach [5] and the Selberg zeta function [6] of this system are obtained analytically. The corresponding bound-state problem of the two- (and higher) dimensional harmonic oscillator has been recently solved by the same method [7]. In our example, the role of the inverted oscillator is to damp out the higher harmonics and improve the convergence of the Selberg zeta function. The quantum mechanical density of states is also obtained analytically. A detailed comparison of the semclassical and quantum results shows that the semiclassical trace formula accurately reproduces the quantum result.

Obviously, this toy model is not the dynamic Hamiltonian that describes the phase of zeta(s) even outside the strip rho>1. But asymptotically, for very large rho, when all but the lowest prime (=2) is dominatnt, the saddle-like planar potential appears to have relevance. In particular, the curvature of the inverted potential at the saddle is found to be directly proportional to the parameter rho of the zeta function. Recalling that the phase of zeta(s) on the rho = 1 line is described by the quantum scattering phase shift of a non-Euclidean surface of constant negative curvature [8], the current work poses the question of whether a single Hamiltonian may describe the phase of zeta(s) for the entire range rho >>1.



A. The Gutzwiller Trace Formula

B. The Quantum Density of States

C. Comparison of the Semiclassical and Quantum Formulae




[1] G.H. Hardy, Comptes Rendus 158 (1914), 280.

[2] R.K. Bhaduri, A. Khare, and J. Law, Physical Review E 52 (1995), 486

[3] M.V. Berry, Proceedings of the Royal Society of London, Series A 400 (1985), 229.

[4] M. Buttiker, Physical Review B 41 (1990), 7906; R. Taboryski. A. Kristensen, C.B. Sorensen, and P. Lindelof, Physical Review B 51 (1995), 2282.

[5] M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer-Verlag, New York, 1991.

[6] A. Voros, Journal of Physics A 21 (1988), 3462.

[7] M. Brack and S.R. Jain, Physical Review A 51 (1995), 3462.

[8] M.C.Gutzwiller, Physica D 7 (1983), 341.

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