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Register a new userSolving the linear semiclassical Schrodinger equation on the real line
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Added by
Katharina Schratz
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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The numerical solution of a linear Schrodinger equation in the semiclassical regime is very well understood in a torus $mathbb{T}^d$. A raft of modern computational methods are precise and affordable, while conserving energy and resolving high oscillations very well. This, however, is far from the case with regard to its solution in $mathbb{R}^d$, a setting more suitable for many applications. In this paper we extend the theory of splitting methods to this end. The main idea is to derive the solution using a spectral method from a combination of solutions of the free Schrodinger equation and of linear scalar ordinary differential equations, in a symmetric Zassenhaus splitting method. This necessitates detailed analysis of certain orthonormal spectral bases on the real line and their evolution under the free Schrodinger operator.
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In this paper we compare three different orthogonal systems in $mathrm{L}_2(mathbb{R})$ which can be used in the construction of a spectral method for solving the semi-classically scaled time dependent Schrodinger equation on the real line, specifically, stretched Fourier functions, Hermite functions and Malmquist--Takenaka functions. All three have banded skew-Hermitian differentiation matrices, which greatly simplifies their implementation in a spectral method, while ensuring that the numerical solution is unitary -- this is essential in order to respect the Born interpretation in quantum mechanics and, as a byproduct, ensures numerical stability with respect to the $mathrm{L}_2(mathbb{R})$ norm. We derive asymptotic approximations of the coefficients for a wave packet in each of these bases, which are extremely accurate in the high frequency regime. We show that the Malmquist--Takenaka basis is superior, in a practical sense, to the more commonly used Hermite functions and stretched Fourier expansions for approximating wave packets
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The logarithmic nonlinearity has been used in many partial differential equations (PDEs) for modeling problems in various applications.Due to the singularity of the logarithmic function, it introducestremendous difficulties in establishing mathematical theories, as well asin designing and analyzing numerical methods for PDEs with such nonlinearity. Here we take the logarithmic Schrodinger equation (LogSE)as a prototype model. Instead of regularizing $f(rho)=ln rho$ in theLogSE directly and globally as being done in the literature, we propose a local energy regularization (LER) for the LogSE byfirst regularizing $F(rho)=rholn rho -rho$ locally near $rho=0^+$ with a polynomial approximation in the energy functional of the LogSE and then obtaining an energy regularized logarithmic Schrodinger equation (ERLogSE) via energy variation. Linear convergence is established between the solutions of ERLogSE and LogSE in terms of a small regularization parameter $0<epll1$. Moreover, the conserved energy of the ERLogSE converges to that of LogSE quadratically, which significantly improvesthe linear convergence rate of the regularization method in the literature. Error estimates are alsopresented for solving the ERLogSE by using Lie-Trotter splittingintegrators. Numerical results are reported to confirm our errorestimates of the LER and of the time-splitting integrators for theERLogSE. Finally our results suggest that the LER performs better than regularizing the logarithmic nonlinearity in the LogSE directly.
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