Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a One-Dimensional Potential Exhibiting Non-Exponential Decay at All Times

The departure at large times from exponential decay in the case of resonance wavefunctions is mathematically demonstrated. Then, exact, analytical solutions to the time-dependent Schr?dinger equation in one dimension are developed for a time-independent potential consisting of an infinite wall and a repulsive delta function. The exact solutions are obtained by means of a superposition of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions for which the resulting integrals can be evaluated exactly. Square-integrability and the boundary conditions are satisfied. The simplest of the obtained solutions is presented and the probability for the particle to be found inside the potential well as a function of time is calculated. The system exhibits non-exponential decay for all times;the probability decreases at large times as . Other exact solutions found exhibit power law behavior at large times. The results are generalized to all normalizable solutions to this problem. Additionally, numerical solutions are obtained using the staggered leap-frog algorithm for select potentials exhibiting the prevalence of non-exponential decay at short times.

Equilibrium dividend strategies in the dual model with a random time horizon

This paper investigates the dividend problem with non-exponential discounting in a dual model.We assume that the dividends can only be paid at a bounded rate and that the surplus process is killed by an exponential random variable.Since the non-exponential discount function leads to a time inconsistent control problem,we study the equilibrium HJB-equation and give the associated verification theorem.For the case of a mixture of exponential discount functions and exponential gains,we obtain the explicit equilibrium dividend strategy and the corresponding equilibrium value function.Besides,numerical examples are shown to illustrate our results.

Effects of a Periodic Decay Rate on the Statistics of Radioactive Decay: New Methods to Search for Violations of the Law of Radioactive Change

It is a long-held tenet of nuclear physics, from the early work of Rutherford and Soddy up to present times that the disintegration of each species of radioactive nuclide occurs randomly at a constant rate unaffected by interactions with the external environment. During the past 15 years or so, reports have been published of some 10 or more unstable nuclides with non-exponential, periodic decay rates claimed to be of geophysical, astrophysical, or cosmological origin. Deviations from standard exponential decay are weak, and the claims are controversial. This paper examines the effects of a periodic decay rate on the statistical distributions of 1) nuclear activity measurements and 2) nuclear lifetime measurements. It is demonstrated that the modifications to these distributions are approximately 100 times more sensitive to non-standard radioactive decay than measurements of the decay curve, power spectrum, or autocorrelation function for corresponding system parameters.

Energy Decay for a Type of Plate Equation with Degenerate Energy Damping and Source Term

In this paper,we mainly investigate the initial boundary value problem for a plate equation with non-local degenerate energy damping term and source term.By using potential well and Nakao's inequality,we establish the global existence and the energy decay rate when the initial data is starting in the stable set.Finally,we derive some further estimate on the stability.