Riemann {zeta} function from wave-packet dynamics

Journal Article · Wed Sep 15 00:00:00 EDT 2010 · Physical Review. A

DOI:https://doi.org/10.1103/PHYSREVA.82.032119· OSTI ID:21448638

Mack, R; Schleich, W P ^[1]; Dahl, J P ^[1]; Moya-Cessa, H ^[1]; Strunz, W T ^[2]; Walser, R ^[1]

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1. Institut fuer Quantenphysik, Albert-Einstein-Allee 11, Universitaet Ulm, D-89069 Ulm (Germany)
2. Institut fuer Theoretische Physik, Technische Universitaet Dresden, D-01062 Dresden (Germany)

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.

• OSTI ID: 21448638
• Journal Information: Physical Review. A, Vol. 82, Issue 3; Other Information: DOI: 10.1103/PhysRevA.82.032119; (c) 2010 The American Physical Society; ISSN 1050-2947
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
74 ATOMIC AND MOLECULAR PHYSICS
ANHARMONIC OSCILLATORS
APPROXIMATIONS
ATOMS
COMPARATIVE EVALUATIONS
EIGENVALUES
ENERGY SPECTRA
FUNCTIONS
POTENTIAL ENERGY
POTENTIALS
TIME DEPENDENCE
VISIBLE RADIATION
WAVE PACKETS
CALCULATION METHODS
ELECTROMAGNETIC RADIATION
ENERGY
EVALUATION
RADIATIONS
SPECTRA