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| 물리:스펙트럴_차원_spectral_dimension [2025/12/12 16:14] – minwoo | 물리:스펙트럴_차원_spectral_dimension [2025/12/17 16:31] (current) – minwoo | ||
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| - | 이때, 라플라시안 행렬 $L$의 각 성분은 $L_{ij}=D_{ii} - A_{ij}= D_0\delta_{ij}-A_{ij}$로 주어지므로, | + | 이때, 라플라시안 행렬 $L$의 각 성분은 $L_{ij}=D_{ii} - A_{ij}= D_0\delta_{ij}-A_{ij}$로 주어지므로, |
| $$ | $$ | ||
| \sum_j L_{ij} v^k_j = \lambda_\mathbf{k} v_i\\ | \sum_j L_{ij} v^k_j = \lambda_\mathbf{k} v_i\\ | ||
| - | \to \sum_j (D_0\delta_{ij}-A_{ij}) e^{i \mathbf{k} \cdot \mathbf{r}_j} = \lambda_\mathbf{k} e^{i \mathbf{k} \cdot \mathbf{r}_j} \\ | + | \to \sum_j (D_0\delta_{ij}-A_{ij}) e^{i \mathbf{k} \cdot \mathbf{r}_j} = \lambda_\mathbf{k} e^{i \mathbf{k} \cdot \mathbf{r}_i} \\ |
| =D_0 e^{i \mathbf{k} \cdot \mathbf{r}_i} -\sum_j J(|\mathbf{r}_i -\mathbf{r}_j|)e^{i \mathbf{k} \cdot \mathbf{r}_j}. | =D_0 e^{i \mathbf{k} \cdot \mathbf{r}_i} -\sum_j J(|\mathbf{r}_i -\mathbf{r}_j|)e^{i \mathbf{k} \cdot \mathbf{r}_j}. | ||
| $$ | $$ | ||