scholarly journals Discrete Transparent Boundary Conditions for General Schrödinger-type Equations

VLSI Design ◽  
1999 ◽  
Vol 9 (4) ◽  
pp. 325-338 ◽  
Author(s):  
Matthias Ehrhardt
Keyword(s):  
Boundary Conditions ◽  
Underwater Acoustics ◽  
Wide Angle ◽  
Transparent Boundary Conditions ◽  
Space Problem ◽  
Whole Space ◽  
Non Local ◽  
Memory Type ◽  
The Given ◽  
Stable Scheme

Transparent boundary conditions (TBCs) for general Schrödinger-type equations on a bounded domain can be derived explicitly under the assumption that the given potential V is constant on the exterior of that domain. In 1D these boundary conditions are non-local in time (of memory type).Existing discretizations of these TBCs have accuracy problems and render the overall Crank–Nicolson finite difference method only conditionally stable. In this paper a novel discrete TBC is derived directly from the discrete whole-space problem that yields an unconditionally stable scheme. Numerical examples illustrate the superiority of the discrete TBC over other existing consistent discretizations of the differential TBCs.As an application of these boundary conditions to wave propagation problems in underwater acoustics results for the so-called standard and wide angle “parabolic” equation (SPE, WAPE) models are presented.

scholarly journals Numerically Absorbing Boundary Conditions for Quantum Evolution Equations

VLSI Design ◽  
1998 ◽  
Vol 6 (1-4) ◽  
pp. 313-319 ◽  
Author(s):  
Anton Arnold
Keyword(s):  
Boundary Conditions ◽  
Ad Hoc ◽  
Evolution Equations ◽  
Absorbing Boundary Conditions ◽  
Quantum Evolution ◽  
Transparent Boundary Conditions ◽  
Absorbing Boundary ◽  
Quantum Devices ◽  
Non Local ◽  
The Given

Transparent boundary conditions for the transient Schrödinger equation on a domain Ω can be derived explicitly under the assumption that the given potential V is constant outside of this domain. In 1D these boundary conditions are non-local in time (of memory type). For the Crank-Nicolson finite difference scheme, discrete transparent boundary conditions are derived, and the resulting scheme is proved to be unconditionally stable. A numerical example illustrates the superiority of discrete transparent boundary conditions over existing ad-hoc discretizations of the differential transparent boundary conditions. As an application of these boundary conditions to the modeling of quantum devices, a transient 1D scattering model for mixed quantum states is presented.

Transparent boundary conditions for discretized non-Hermitian eigenvalue problems

2021 ◽  
pp. 2150138
Author(s):  
Jonathan Heinz ◽  
Miroslav Kolesik
Keyword(s):  
Quantum Mechanics ◽  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Eigenvalue Problems ◽  
Complex Scaling ◽  
Transparent Boundary Conditions ◽  
Drastic Reduction ◽  
Optical Cavities ◽  
Energy Dependent ◽  
Lossy Waveguides

A method is presented for transparent, energy-dependent boundary conditions for open, non-Hermitian systems, and is illustrated on an example of Stark resonances in a single-particle quantum system. The approach provides an alternative to external complex scaling, and is applicable when asymptotic solutions can be characterized at large distances from the origin. Its main benefit consists in a drastic reduction of the dimesnionality of the underlying eigenvalue problem. Besides application to quantum mechanics, the method can be used in other contexts such as in systems involving unstable optical cavities and lossy waveguides.

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