论文标题
LatticeSchrödinger操作员的连续限制
Continuum limit for lattice Schrödinger operators
论文作者
论文摘要
We study the behavior of solutions of the Helmholtz equation $(- Δ_{disc,h} - E)u_h = f_h$ on a periodic lattice as the mesh size $h$ tends to 0. Projecting to the eigenspace of a characteristic root $λ_h(ξ)$ and using a gauge transformation associated with the Dirac point, we show that the gauge transformed solution $u_h$ converges to that for the方程$（p（d_x） - e）v = g $对于$ {\ bf r}^d $上的连续模型，其中$λ_H（ξ）\ to p（ξ）$。对于六边形和相关晶格的情况，{在合适的能量区域}，它会收敛到dirac方程。对于平方晶格，三角晶格，{六角晶格（在另一个能量区域）}和平方晶格的细分，一个人可以添加标量的潜力，以及晶格schr {Ö} dinger方程的解决方案（ - schr {Ö} dinger方程$（p（d_x） + v（x）-e）u = f $。
We study the behavior of solutions of the Helmholtz equation $(- Δ_{disc,h} - E)u_h = f_h$ on a periodic lattice as the mesh size $h$ tends to 0. Projecting to the eigenspace of a characteristic root $λ_h(ξ)$ and using a gauge transformation associated with the Dirac point, we show that the gauge transformed solution $u_h$ converges to that for the equation $(P(D_x) - E)v = g$ for a continuous model on ${\bf R}^d$, where $λ_h(ξ) \to P(ξ)$. For the case of the hexagonal and related lattices, {in a suitable energy region}, it converges to that for the Dirac equation. For the case of the square lattice, triangular lattice, {hexagonal lattice (in another energy region)} and subdivision of a square lattice, one can add a scalar potential, and the solution of the lattice Schr{ö}dinger equation $( - Δ_{disc,h} +V_{disc,h} - E)u_h = f_h$ converges to that of the continuum Schr{ö}dinger equation $(P(D_x) + V(x) -E)u = f$.
