1. A Transparent Boundary Approach for the Simulation of Resonances in Open Systems
- Author
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Moloney, Jerome V., Norwood, Robert A., Heinz, Jonathan Robert, Moloney, Jerome V., Norwood, Robert A., and Heinz, Jonathan Robert
- Abstract
Open systems, or systems that interact with their surroundings, can contain selfconsistent but unstable solutions that are referred to as resonances. Resonances are often sensitive to a narrow band of frequencies and can have long lifetimes, but they also couple to the environment and their calculation can be challenging compared to the simulation of the bound states of a closed system. In this dissertation, a new transparent boundary method is proposed to target the resonances of open systems. This method allows for the highly accurate calculation of resonances using smaller computational domains than are achievable with traditional boundary methods. To implement the proposed boundary approach for a given system, the domain is separated into two regions: one for the interior of the system and one for the environment. Expressions are first found for the resonances in the exterior region, which are then used to establish boundary conditions for a numerical treatment of the interior region. The use of the exterior expressions as a part of the interior treatment results in a nonlinear eigenvalue problem, which we resolve using a simple iteration procedure. The proposed method is first applied to the Stark system in quantum mechanics for the calculation of Stark resonances. These resonances are non-Hermitian quantum states that arise when a central potential is subjected to a linear field term, allowing the electron to escape and tunnel to infinity, and they can be used to calculate the nonlinear optical properties of atomic and molecular species. Two implementations are given for the treatment of the Stark system, using different assumptions and formulations to achieve highly accurate solutions for the Stark resonances of atomic hydrogen. The simple iteration procedure is shown to be sufficient to resolve the nonlinear nature of the eigenvalue problem for these scalar systems. The proposed method is also used to perform first-of-their-kind three-dimensional calculation
- Published
- 2023