zeta2d

Module: luescher_nd.zeta.zeta2d

Computation of 2D zeta function

class DispersionZeta2D(L, epsilon, nstep=None)

Two dimensional dispersion Zeta function for discretized finite volume.

property N

Returns L/epsillon as an int

Return type

int

__call__(x)

Computes the result of the three-d Zeta function.

__post_init__()

Initializes the elements of the sum for the zeta function denominator

class Zeta2D(N, spherical=False)

Two dimensional Zeta function for discretized finite volume.

This class implements $$ S_2^{A}(x; N) = \sum_{n_i \in M^A(N)} \frac{1}{\vec{n}^2 - x} - 2 \pi \log(N) + \delta^A $$ where \(N = \Lambda L / (2 \pi)\) is the cutoff of the zeta function, \(A\) means either spherical or cartesian $$ M^A(N) = \begin{cases} \left\{ (n_1, n_2) \in \mathbb Z^2 \middle\vert -N \leq n_i < N \right\} & A = \text{cartesian} \\ \left\{ (n_1, n_2) \in \mathbb Z^2 \middle\vert n_1^2 + n_2^2 < N \right\} & A = \text{spherical} \end{cases} $$ and \(\delta^{\bigcirc} = 0\) but \(\delta^{\square} = 4G - 2\pi\log(2) \approx -0.69\).

__call__(x)

Computes the result of the three-d Zeta function.

__post_init__()

Initializes the elements of the sum for the zeta function denominator