Laplacian for a minimized function

General Usage
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FYI, it may relate to the nature of R(μ,E). Namely, R(μ,E) could take a minimum in a certain region (not at a single point) as a function of (μ[1],μ[2]) for a fixed E, say E=E0. See this topic as an example.
The important features are

  1. When R(μ,E0) has a minimizer region, the integrand of R(μ,E0) is well-defined in there.
  2. When R(μ,E0) has a minimizer point, the integrand of R(μ,E0) shows a logarithmic singularity at that point.

although, in both cases, R(μ,E0) itself is well-defined and takes a finite value.

My observation tells that DOS1(E) doesn’t work when a minimizer point exists. For instance,

  1. At E0 = [0.0,0.0], which provides a minimizer point,
    (DOS1(E0),DOS2(E0))=(2.8271597168564594e-16, 0.1418357989357821).

  2. At E0 = [-3.0,3.0], which gives a minimizer region,
    (DOS1(E0),DOS2(E0))=(0.0, -0.00018425790497728805).

For a given E0, you can see whether R(μ,E0) has a minimizer point or a minimizer region by calculating Amoeba of the polynomial, p(z,w) = t*T+td*Td+γ*Γ-E0[1]-im*E0[2], where z = exp(μ[1]+im*θ[1]) and w = exp(μ[2]+im*θ[2]).