Semiclassical Dynamics with Exponentially Small Error Estimates

We construct approximate solutions to the time--dependent Schr\"odinger equation $i \hbar (\partial \psi)/(\partial t) = - (\hbar^2)/2 \Delta \psi + V \psi$ for small values of $\hbar$. If $V$ satisfies appropriate analyticity and growth hypotheses and $|t|\le T$, these solutions agree with exact solutions up to errors whose norms are bounded by $C \exp{-\gamma/\hbar}$, for some $C$ and $\gamma>0$. Under more restrictive hypotheses, we prove that for sufficiently small $T', |t|\le T' |\log(\hbar)|$ implies the norms of the errors are bounded by $C' \exp{-\gamma'/\hbar^{\sigma}}$, for some $C', \gamma'>0$, and $\sigma>0$

Dynamics in the Sherrington-Kirkpatrick Ising spin glass at and above Tg

A detailed numerical study is made of relaxation at equilibrium in the Sherrington-Kirkpatrick Ising spin glass model, at and above the critical temperature Tg. The data show a long time stretched exponential relaxation q(t) ~ exp[-(t/tau(T))^beta(T)] with an exponent beta(T) tending to ~ 1/3 at Tg. The results are compared to those which were observed by Ogielski in the 3d ISG model, and are discussed in terms of a phase space percolation transition scenario.Comment: 6 pages, 7 figure

A Time-Dependent Born-Oppenheimer Approximation with Exponentially Small Error Estimates

We present the construction of an exponentially accurate time-dependent Born-Oppenheimer approximation for molecular quantum mechanics. We study molecular systems whose electron masses are held fixed and whose nuclear masses are proportional to $\epsilon^{-4}$, where $\epsilon$ is a small expansion parameter. By optimal truncation of an asymptotic expansion, we construct approximate solutions to the time-dependent Schr\"odinger equation that agree with exact normalized solutions up to errors whose norms are bounded by \ds C \exp(-\gamma/\epsilon^2), for some C and $\gamma>0$

Exponentially Accurate Semiclassical Dynamics: Propagation, Localization, Ehrenfest Times, Scattering and More General States

We prove six theorems concerning exponentially accurate semiclassical quantum mechanics. Two of these theorems are known results, but have new proofs. Under appropriate hypotheses, they conclude that the exact and approximate dynamics of an initially localized wave packet agree up to exponentially small errors in $\hbar$ for finite times and for Ehrenfest times. Two other theorems state that for such times the wave packets are localized near a classical orbit up to exponentially small errors. The fifth theorem deals with infinite times and states an exponentially accurate scattering result. The sixth theorem provides extensions of the other five by allowing more general initial conditions

Some remarks on barycentric-sum problems over cyclic groups

We derive some new results on the k-th barycentric Olson constants of abelian groups (mainly cyclic). This quantity, for a finite abelian (additive) group (G,+), is defined as the smallest integer l such that each subset A of G with at least l elements contains a subset with k elements {g_1, ..., g_k} satisfying g_1 + ... + g_k = k g_j for some 1 <= j <= k.Comment: to appear in European Journal of Combinatoric

Multi-overlap simulations of spin glasses

We present results of recent high-statistics Monte Carlo simulations of the Edwards-Anderson Ising spin-glass model in three and four dimensions. The study is based on a non-Boltzmann sampling technique, the multi-overlap algorithm which is specifically tailored for sampling rare-event states. We thus concentrate on those properties which are difficult to obtain with standard canonical Boltzmann sampling such as the free-energy barriers F^q_B in the probability density P_J(q) of the Parisi overlap parameter q and the behaviour of the tails of the disorder averaged density P(q) = [P_J(q)]_av.Comment: 14 pages, Latex, 18 Postscript figures, to be published in NIC Series - Publication Series of the John von Neumann Institute for Computing (NIC

Age differences in fMRI adaptation for sound identity and location

We explored age differences in auditory perception by measuring fMRI adaptation of brain activity to repetitions of sound identity (what) and location (where), using meaningful environmental sounds. In one condition, both sound identity and location were repeated allowing us to assess non-specific adaptation. In other conditions, only one feature was repeated (identity or location) to assess domain-specific adaptation. Both young and older adults showed comparable non-specific adaptation (identity and location) in bilateral temporal lobes, medial parietal cortex, and subcortical regions. However, older adults showed reduced domain-specific adaptation to location repetitions in a distributed set of regions, including frontal and parietal areas, and to identity repetition in anterior temporal cortex. We also re-analyzed data from a previously published 1-back fMRI study, in which participants responded to infrequent repetition of the identity or location of meaningful sounds. This analysis revealed age differences in domain-specific adaptation in a set of brain regions that overlapped substantially with those identified in the adaptation experiment. This converging evidence of reductions in the degree of auditory fMRI adaptation in older adults suggests that the processing of specific auditory “what” and “where” information is altered with age, which may influence cognitive functions that depend on this processing