论文标题
离散近似dirac运算符和norm Resolvent收敛
Discrete approximations to Dirac operators and norm resolvent convergence
论文作者
论文摘要
我们考虑在$ \ mathbf {r}^d $，$ d \ in \ {1,2,3 \} $上定义的连续狄拉克运算符，以及它们的各种离散版本。前向后和对称有限差异都用作部分衍生物的近似值。我们还允许在离散设置在网格上评估离散设置的有界，霍尔德连续和自动配合矩阵值。我们的主要目标是调查提出的离散模型是否以规范的分辨率融合了其连续的对应物，因为网格尺寸倾向于零，并且最多将离散空间的自然嵌入到连续的空间中。在维度一个方面，我们表明前向差异会导致标准分辨率收敛，而在第二和第三方面，它们却没有。当使用对称差异时，相同的负结果在所有维度中都保持。另一方面，在所有这些情况下，强有力的分解融合均持有。然而，非常明显的是，对离散模型的一种相当简单但非标准的修改，涉及质量术语，可确保规范总结融合一般。
We consider continuous Dirac operators defined on $\mathbf{R}^d$, $d\in\{1,2,3\}$, together with various discrete versions of them. Both forward-backward and symmetric finite differences are used as approximations to partial derivatives. We also allow a bounded, Hölder continuous, and self-adjoint matrix-valued potential, which in the discrete setting is evaluated on the mesh. Our main goal is to investigate whether the proposed discrete models converge in norm resolvent sense to their continuous counterparts, as the mesh size tends to zero and up to a natural embedding of the discrete space into the continuous one. In dimension one we show that forward-backward differences lead to norm resolvent convergence, while in dimension two and three they do not. The same negative result holds in all dimensions when symmetric differences are used. On the other hand, strong resolvent convergence holds in all these cases. Nevertheless, and quite remarkably, a rather simple but non-standard modification to the discrete models, involving the mass term, ensures norm resolvent convergence in general.
