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# Parity-decomposition and Moment Analysis for Stationary Wigner Equation with Inflow Boundary Conditions

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## Abstract

We study the stationary Wigner equation on a bounded, one-dimensional spatial domain with inflow boundary conditions by using the parity decomposition in (Barletti and weifel, Trans. Theory Stat. Phys., 507--520, 2001). The decomposition reduces the half-range, two-point boundary value problem into two decoupled initial value problems of the even part and the odd part. Without using a cutoff approximation around zero velocity, we prove that the initial value problem for the even part is well-posed. For the odd part, we prove the uniqueness of the solution in the odd \$L^2\$-space by analyzing the moment system. An example is provided to show that how to use the analysis to obtain the solution of the stationary Wigner equation with inflow boundary conditions.

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... The wellposedness of the stationary Wigner inflow boundary value problem (SWIBVP), i.e., (1)-(3), is still an open problem, where many efforts have been made to solve it [14][15][16][17]. In [16], the wellposedness of the SWIBVP with periodic potential has been proved. ...
... In [17], the authors have discussed the wellposedness in general cases by using the paritydecomposition and the moment method, of which the wellposedness is obtained partially. Besides Frensley's method [13], plenty of works have been done in solving the Wigner equation by using the finite difference method [18][19][20][21], spectral methods [22][23][24][25], moment methods [26,27], the Monte-Carlo method [28,29], etc. Due to lack of a unified theory of wellposedness of the SWIBVP, numerical results solved with different numerical methods are controversial. ...
... , then the solution of the discrete velocity SWIBVP, i.e., (17) and (18), is also symmetric about x = L 2 , i.e., ...
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The stationary Wigner inflow boundary value problem (SWIBVP) is modeled as an optimization problem by using the idea of shooting method in this paper. To remove the singularity at \(v=0\), we consider a regularized SWIBVP, where a regularization constraint is considered along with the original SWIBVP, and a modified optimization problem is established for it. A shooting algorithm is proposed to solve the two optimization problems, involving the limited-memory BFGS (L-BFGS) algorithm as the optimization solver. Numerical results show that solving the optimization problems with respect to the SWIBVP with the shooting algorithm is as effective as solving the SWIBVP with Frensley’s numerical method (Frensley in Phys Rev B 36:1570–1580, 1987). Furthermore, the modified optimization problem gets rid of the singularity at \(v=0\), and preserves symmetry of the Wigner function, which implies the optimization modeling with respect to the regularized SWIBVP is successful.
... long-standing open problem even in a single spatial dimension [1,14,15]. ...
... 1. Setting k = 0 in Eq. (8) gives the following integrability constraint In this degenerate case, the left hand side in Eq. (8) vanishes and we do not get a differential equation. This special case is called an algebraic constraint in [1,15] and is needed to avoid poles on the right hand side of Eq. (17). The integrability constraint has also a physical interpretation in the particle picture: The total potential inscattering rate at k = 0 must vanish in the steady state. ...
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We discuss boundary value problems for the characteristic stationary von Neumann equation (stationary sigma equation) and the stationary Wigner equation in a single spatial dimension. The two equations are related by a Fourier transform in the non-spatial coordinate. In general, a solution to the characteristic equation does not produce a corresponding Wigner solution as the Fourier transform will not exist. Solution of the stationary Wigner equation on a shifted k-grid gives unphysical results. Results showing a negative differential resistance in IV-curves of resonant tunneling diodes using Frensley’s method are a numerical artefact from using upwinding on a coarse grid. We introduce the integro-differential sigma equation which avoids distributional parts at k=0 in the Wigner transform. The Wigner equation for k=0 represents an algebraic constraint needed to avoid poles in the solution at k=0. We impose the inverse Fourier transform of the integrability constraint in the integro-differential sigma equation. After a cutoff, we find that this gives fully homogeneous boundary conditions in the non-spatial coordinate which is overdetermined. Employing an absorbing potential layer double homogeneous boundary conditions are naturally fulfilled. Simulation results for resonant tunneling diodes from solving the constrained sigma equation in the least squares sense with an absorbing potential reproduce results from the quantum transmitting boundary with high accuracy. We discuss the zero bias case where also good agreement is found. In conclusion, we argue that properly formulated open boundary conditions have to be imposed on non-spatial boundaries in the sigma equation both in the stationary and the transient case. When solving the Wigner equation, an absorbing potential layer has to be employed.
... The well-posedness of the stationary Wigner transport equation with inflow boundary conditions have attracted the attention of many mathematicians, but it is still an open problem and is only partially solved in [1,2,7,8]. One big issue is that if L 2 (R) is a suitable solution space for (1.10). ...
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In this work, the abstract Cauchy problem for an initial value system with singular integral is considered. The system is of closed form of two evolution equations for a real-valued function and a function-valued function. By proposing an appropriate Banach space, we prove the existence and uniqueness of classical solutions to the evolution system under assumptions on the boundedness and smoothness of data. Furthermore, we connect by an isomorphism the solution of the evolution system and a class of integral-differential equations with singular convolution kernels and extend our results to the corresponding problem. It is revealed that our findings also improve the understanding of the open problem on the well-posedness of the stationary Wigner equation with inflow boundary conditions.
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