Discrete Kernel Preserving Model for 3D Electron–Optical Phonon Scattering Under Arbitrary Band Structures

  • Wenqi YaoEmail author
  • Tiao Lu


In Li et al. (J Sci Comput 62:317–335, 2015), we thoroughly investigated the structure of the kernel space of the discrete one-dimensional (1D) non-polar optical phonon (NPOP)-electron scattering matrix, and proposed a strategy to setup grid points so that the uniqueness of the discrete scattering kernel is preserved. In this paper, we extend the above work to the three dimensional (3D) case, and also investigate the polar optical phonon (POP)-electron case. In numerical discretization, it is important to get a discretization scattering matrix that keeps as many properties of the continuous scattering operator as possible. We prove that for the 3D NPOP-electron scattering, (i) the dimension of the kernel space of the discrete scattering matrix is one and (ii) the equilibrium distribution is constant with respect to the angular coordinates as long as the mesh over the energy interval obeys the rule proposed in Li et al. (2015). For the POP-electron scattering, (i) is also kept under the same condition, but to keep (ii) becomes a challenging task, since generally a simple uniform mesh in the azimuth and polar angular coordinates will not preserve (ii). Based on high degree of symmetry of the Platonic solids and regular pyramids, we propose two conditions and prove they are sufficient to guarantee (ii). Numerical experiments strongly support our theoretical findings.


Three dimensional electron–phonon scattering Polar phonons Kernel space Equilibrium distribution The Platonic solids and regular pyramids 

Mathematics Subject Classification

45J05 82B05 34K28 



The work was supported in part by NSFC (11801183, 91630130, 11671038).


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Authors and Affiliations

  1. 1.School of MathematicsSouth China University of TechnologyGuangzhouChina
  2. 2.CAPT, HEDPS, LMAM, IFSA Collaborative Innovation Center of MoE, and School of Mathematical SciencesPeking UniversityBeijingChina

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