A hybrid method for computing the Schrödinger equations with periodic potential with band-crossings in the momentum space. (English) Zbl 1475.65143
Summary: We propose a hybrid method which combines the Bloch decomposition-based time splitting (BDTS) method and the Gaussian beam method to simulate the Schrödinger equation with periodic potentials in the case of band-crossings. With the help of the Bloch transformation, we develop a Bloch decomposition-based Gaussian beam (BDGB) approximation in the momentum space to solve the Schrödinger equation. Around the band-crossing a BDTS method is used to capture the inter-band transitions, and away from the crossing, a BDGB method is applied in order to improve the efficiency. Numerical results show that this method can capture the inter-band transitions accurately with a computational cost much lower than the direct solver. We also compare the Schrödinger equation with its Dirac approximation, and numerically show that, as the rescaled Planck number \(\varepsilon \rightarrow 0\), the Schrödinger equation converges to the Dirac equations when the external potential is zero or small, but for general external potentials there is an \(\mathcal{O}(1)\) difference between the solutions of the Schrödinger equation and its Dirac approximation.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 74Q10 | Homogenization and oscillations in dynamical problems of solid mechanics |
| 35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |
Keywords:
Schrödinger equation; band-crossing; Dirac point; Bloch decomposition; time-splitting spectral method; Gaussian beam methodReferences:
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