zeta3d¶
Module: luescher_nd.zeta.zeta3d
Computation of 3D zeta function
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class
DispersionZeta3D(L, epsilon, nstep=None)[source]¶ Three dimensional Zeta function for discretized finite volume.
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property
N¶ Returns L/epsillon as an int
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int
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property
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class
Zeta3D(N, spherical=False)[source]¶ Three dimensional Zeta function for discretized finite volume.
This class implements $$ S_3^{A}(x; N) = \sum_{n_i \in M^A(N)} \frac{1}{\vec{n}^2 - x} - 2 \pi^2 N \mathcal L_A $$ where \(N = \Lambda L / (2 \pi)\) is the cutoff of the zeta function, \(A\) means either spherical or cartesian $$ M^A(N) = \begin{cases} \left\{ (n_1, n_2, n_3) \in \mathbb Z^3 \middle\vert -N \leq n_i < N \right\} & A = \text{cartesian} \\ \left\{ (n_1, n_2, n_3) \in \mathbb Z^3 \middle\vert n_1^2 + n_2^2 = n_3^2 < N \right\} & A = \text{spherical} \end{cases} $$ and \(\mathcal L^{\bigcirc} = \frac{2}{\pi}\) but \(\mathcal L^{\square} = 0.77755134963633393039\).