0
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0answers
58 views

Equivalence of ensembles and $\sum_{N=0}^{\infty} \int_{-Bn}^{\infty} 1_{\text{{N>K or u>A}}}(u)e^{-\beta |\lambda_m|(\frac{u}{2}+n)}du $

I am thinking about a step in the proof of the equivalence of ensembles in StatMech on page 25. step from 3.19 to 3.20. It seems to be argued there that for $A$ and $K$ large enough, the term ...
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0answers
203 views

Contour integral (inverse Laplace transform) with arctan

I have what I think is a relatively simple contour integral involving arctan, but it is giving me difficulty. I would really appreciate any help. The integral itself is, with $\tau$, $\lambda$, and ...
2
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0answers
250 views

How to perform this matrix integral?

Edit: some backgrouds added. In quiver matrix model which is reviewed DV or CKR, the path integral reduce to the matrix integral $$Z \sim \int \prod_{i=1}^r d\Phi_i \prod_{<a,b>} dQ_{ab} ...
1
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2answers
195 views

Positivity of the Coulomb energy in two dimensions

In dimensions $d\geq 3$ the Coulomb energy is always non-negative (since the Fourier transform of $\frac{1}{|\cdot|^{d-2}}$ is non-negative). What can one say about positivity properties of the ...
1
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1answer
102 views

Is $C_c(\mathbb{R}^2,\mathbb{R}^2)$ dense in the irrotational square integrable functions?

Let $L_D(\mathbb{R}^n)^n$ be the set of square integrable functions which are the weak derivative of a locally square integrable function. That is $$L_D(\mathbb{R}^n)^n=\{Du\colon u\in ...