We propose an analytical approach for computing the eigenspectrum and corresponding eigenstates of a hyperbolic double well potential of arbitrary height or width, which goes beyond the usual techniques applied to quasi-exactly solvable models. We map the time-independent Schrödinger equation onto the Heun confluent differential equation, which is solved by using an infinite power series. The coefficients of this series are polynomials in the quantisation parameter, whose roots correspond to the system’s eigenenergies. This leads to a quantisation condition that allows us to determine a whole spectrum, instead of individual eigenenergies. This method is then employed to perform an in depth analysis of electronic wave-packet dynamics, with emphasis on intra-well tunneling and the interference-induced quantum bridges reported in a previous publication Chomet et al (2019 New J. Phys. 21 123004). Considering initial wave packets of different widths and peak locations, we compute autocorrelation functions and Wigner quasiprobability distributions. Our results exhibit an excellent agreement with numerical computations, and allow us to disentangle the different eigenfrequencies that govern the phase-space dynamics.
In order for an e-commerce platform to maximize its revenue, it must recommend customers items they are most likely to purchase. However, the company often has business constraints on these items, such as the number of each item in stock. In this work, our goal is to recommend items to users as they arrive on a webpage sequentially, in an online manner, in order to maximize reward for a company, but also satisfy budget constraints. We first approach the simpler online problem in which the customers arrive as a stationary Poisson process, and present an integrated algorithm that performs online optimization and online learning together. We then make the model more complicated but more realistic, treating the arrival processes as non-stationary Poisson processes. To deal with heterogeneous customer arrivals, we propose a time segmentation algorithm that converts a non-stationary problem into a series of stationary problems. Experiments conducted on large-scale synthetic data demonstrate the effectiveness and efficiency of our proposed approaches on solving constrained resource allocation problems.
We solve analytically the differential equations for a skier on a hemispherical hill and for a particle on a loop-the-loop track when the hill or track is endowed with a coefficient of kinetic friction μ. For each problem, we determine the exact “phase diagram” in the two-dimensional parameter plane.
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