Inverse Power Of Time Research Articles

Maximal speed of quantum propagation

For Schroedinger equations with both time-independent and time-dependent Kato potentials, we give a simple proof of the maximal speed bound. The latter states that the probability to find the quantum system outside the ball of radius proportional to the time lapsed decays as an inverse power of time. We give an explicit expression for the constant of proportionality in terms of the maximal energy available to the initial condition. For the time-independent part of the interaction, we require neither decay at infinity nor smoothness.

Observational signature of the logarithmic terms in the soft-graviton theorem

We show that the recently discovered logarithmic terms in the soft graviton theorem induce a late time component in the gravitational wave-form that falls off as inverse power of time, producing a tail term to the linear memory effect.

Instability of supersymmetric microstate geometries

We investigate the classical stability of supersymmetric, asymptotically flat, microstate geometries with five non-compact dimensions. Such geometries admit an "evanescent ergosurface": a timelike hypersurface of infinite redshift. On such a surface, there are null geodesics with zero energy relative to infinity. These geodesics are stably trapped in the potential well near the ergosurface. We present a heuristic argument indicating that this feature is likely to lead to a nonlinear instability of these solutions. We argue that the precursor of such an instability can be seen in the behaviour of linear perturbations: nonlinear stability would require that all linear perturbations decay sufficiently rapidly but the stable trapping implies that some linear perturbation decay very slowly. We study this in detail for the most symmetric microstate geometries. By constructing quasinormal modes of these geometries we show that generic linear perturbations decay slower than any inverse power of time.

Numerical solution of the 2 + 1 Teukolsky equation on a hyperboloidal and horizon penetrating foliation of Kerr and application to late-time decays

In this work, we present a formulation of the Teukolsky equation for generic spin perturbations on the hyperboloidal and horizon penetrating foliation of Kerr recently proposed by Rácz and Tóth. An additional, spin-dependent rescaling of the field variable can be used to achieve stable, long-term and accurate time-domain evolutions of generic spin perturbations. As an application (and a severe numerical test), we investigate the late-time decays of electromagnetic and gravitational perturbations at the horizon and future null infinity by means of 2 + 1 evolutions. As initial data we consider four combinations of (non-)stationary and (non-)compact-support initial data with a pure spin-weighted spherical harmonic profile. We present an extensive study of late-time decays of axisymmetric perturbations. We verify the power-law decay rates predicted analytically, together with a certain ‘splitting’ behaviour of the power-law exponent. We also present results for non-axisymmetric perturbations. In particular, our approach allows us to study the behaviour of the late-time decays of gravitational fields for nearly extremal and extremal black holes. For rapid rotation we observe a very prolonged, weakly damped, quasi-normal-mode phase. For extremal rotation, the field at future null infinity shows an oscillatory behaviour decaying as the inverse power of time, while at the horizon it is amplified by several orders of magnitude over long timescales. This behaviour can be understood in terms of the superradiance cavity argument.

Unified analytical description of the time evolution of decay for initial states formed by wave-packet scattering and by initial decaying states in quantum systems

We consider a unified analytical approach for scattering and decay processes in one-dimensional resonant tunneling systems using the formalism of resonant states to address the issue of the differences and similarities in the time evolution of decay between the decay of an arbitrary state prepared initially within a system and the formation and subsequent decay of a quasistationary state in the scattering of a Gaussian wavepacket on that system. We find three distinctive regimes. A first regime, which refers only to the quasistationary state, that is characterized by a buildup time of the probability density at a given position within the internal region of the potential. Here we find that the buildup time has a dependence on position. A second regime, dominated by the exponentially decaying terms, where the decay of the quasistationary state proceeds in an almost identical fashion as for the initially prepared decaying state. And finally, a third regime that involves the transition to nonexponential decay at long times and its ulterior behavior as an inverse power of time. Here we find that the time scale of the transition occurs at different times, which implies a dependence on the parameters of the initial state.

Late-time particle creation from gravitational collapse to an extremal Reissner-Nordström black hole

We investigate the late time behavior of particle creation from an extremal Reissner-Nordstrom (RN) black hole formed by gravitational collapse. We calculate explicitly the particle flux associated with a massless scalar field at late times after the collapse. Our result shows that the expected number of particles in any wave packet spontaneously created from the ``in'' vacuum state approaches zero faster than any inverse power of time. This result confirms the traditional belief that extremal black holes do not emit particles. We also calculate the expectation value of the stress energy tensor in a 1+1 RN black hole and show that it also drops to zero at late times. Some comments on previous work by other authors are provided.

Analytic controllability of the wave equation over a cylinder

We analyze the controllability of the wave equation on a cylinder when the control acts on the boundary, that does not satisfy the classical geometric control condition. We obtain precise estimates on the analyticity of reachable functions. As the control time increases, the degree of analyticity that is required for a function to be reachable decreases as an inverse power of time. We conclude that any analytic function can be reached if that control time is large enough. In the C ∞ class, a precise description of all reachable functions is given.

Ideal and resistive magnetohydrodynamic modes

This paper gives a further discussion of the analytical properties of both discrete and continuous Alfven wave spectra in an incompressible as well as compressible plasma. Although the continuous MHD modes produced by a well-behaved initial perturbation decay according to a power law, some singular solutions exist and are found to behave differently. In particular, it is possible to exhibit the existence of a new continuous mode which decays exponentially, and not as an inverse power of time, and this exponential damping is not the consequence of a continuous variation of the magnetic field. Even the set of discrete magnetohydrodynamic modes is shown to be empty unless certain conditions are satisfied. Next, we consider resistive modes and give explicit solutions for them which are valid in the neighborhood of the Alfven resonance layer and discuss their implications for plasma heating schemes. Finally, we study discrete and continuous Alfven wave spectra in a compressible plasma and discuss how they behave differently from those in an incompressible plasma. In particular, we show that though compressibility of the plasma is responsible for the slow mode continuum, strong compressibility will eliminate it. The discrete modes in a compressible plasma undergo an exponential damping even in an ideal plasma if the compressibility is strong.

The Alfvén wave equation for magnetospheric plasmas

It is shown that besides the continuous spectrum which damps away as inverse power of time, the coupled Alfvén wave equation, which gives coupling between a shear Alfvén wave and a surface wave, can also admit a well behaved harmonic solution in the closed form for a set of initial conditions. This solution, though valid for finite time intervals, points out that the Alfvén surface waves can have a band of frequency (instead of a monochromatic frequency for a nonsheared magnetic field) within which the local field line resonance frequency can lie, and thus can excite magnetic pulsations with latitude‐dependent frequency. By considering magnetic fields not only varying in magnitude but also in direction, it is shown that the time interval for the validity of the harmonic solution depend upon the angle between the magnetic field directions on either side of the magnetopause. For small values of the angle the time interval can become appreciably large.

Raman correlation function for generalized M-diffusion model

A generalized version of Gordon's M -diffusion model for reorientation of molecules in dense phase was proposed by Duttagupta and Sood. They calculated the IR correlation function and compared it with a few experimental data. We calculate the Raman correlation function for this model and compare it with experimental data for OCS, N 2 O in CCl 4 , and CH 3 I. We find that at short times, the agreement with the experimental data is much better than that for Gordon's M -diffusion model. We have also shown that, at long times the correlation function decays as an inverse power of time. However, the experimental correlation function at long times decays much faster.

Harmonic and transient fields on the surface of a two‐layer medium

The article deals with the behavior of the quasi‐static electromagnetic (EM) fields created by currents in a two‐layer medium when the source is a vertical axis circular loop. This analysis is of use for determining the depth of investigation of induction methods in mining prospecting. The relationship of the active and reactive components of the field with the geoelectrical parameters is different, depending on the range of frequencies. Unlike the case of a confined body, the low‐frequency part of the spectrum is presented as a series made up of integral and fractional powers of frequency ω and also logarithmic terms of ω. The late stage of a transient process is described with a sum of terms, proportional to the inverse power of time t. These representations are useful for determining the host rock effect against which the signal from an ore body of finite extent must be detected.

Order parameter relaxation in spin glasses

Relaxation in spin glasses is studied using a Langevin equation constructed from the free energy function given by Thouless, Anderson and Palmer. Below the freezing temperature the order parameter relaxes as an inverse power of time.

Dipole Resonances in a Homogeneous Plasma in a Magnetic Field

Radar observations recently made from a satellite orbiting above the ionosphere provide evidence for resonances of a plasma in a magnetic field which may be excited and detected by a dipole. The plasma may be said to be resonant, for a particular mode and frequency, if the group velocity is zero. These resonances are studied theoretically on the assumption that the dipole is of infinitesimal extent and that the plasma is excited by a charge or current impulse. The former assumption restricts the validity of the results to the asymptotic response of the plasma. The latter assumption is not a restriction. There are two electromagnetic resonances at the positive-frequency roots of ω2 ± Ωω − π2 = 0, where Ω is the electron gyro-frequency and π the electron plasma frequency, which occur at k = 0. There is also a resonance at ω = Ω, k = ∞, but this cannot be treated by the infinitesimal-dipole approximation and is deferred for separate study. ``Electrostatic'' resonances are treated in the quasi-electrostatic approximation. The resonant frequencies are π, (π2 + Ω2)½, and nΩ, n = 2, 3, …, and k = 0 at resonance. The infinitesimal-dipole model breaks down for the n = 2, 3, 4 cyclotron harmonics, but the infinitesimal line-dipole model and a line-charge model do not. The analysis shows that the oscillations decay asymptotically as an inverse power of time. The analysis also indicates that the response would be significantly stronger than is observed if measurements were made with a stationary dipole, indicating that the observed duration of the resonances is to be ascribed to the finite velocity of the satellite with respect to the exospheric plasma.