My question is a mathematical one based on this physics paper. Suppose that $\lambda_i $ is an eigenvalue of a reduced density matrix. Up to a normalization factor, the distribution of eigenvalues is given by the following formula. See the first page of the paper to find the formula.

$$ P(\lambda) = \sum_i \delta(\lambda - \lambda_i) $$

The paper also mentions the following, where $\rho$ is the reduced density matrix:

$$ R_n = Trace(\rho^n) = c_n L_{\text{eff}}^{-\frac{c}{6}(n - 1/n)} \equiv c_n e^{-b(n - 1/n)} $$

where $b = \frac{c}{6} \ln (L_{\text{eff}})$

The claim made on the second page of the paper is the following:

$$ \lambda P(\lambda) = \lim_{\epsilon \to 0}\mathbb{Im}(f(\lambda - i\epsilon)) \text{ where } f(z) = \frac{1}{\pi} \sum_{n=1}^\infty R_n z^{-n} = \frac{1}{\pi}\int d\lambda \frac{\lambda P(\lambda)}{z - \lambda} $$ I am trying to prove this claim. The comment below the above claim is

Here, $f(λ − i\epsilon)$ has an imaginary part when $\epsilon → 0$ only on the support of $P(λ)$, due to the pole in the r.h.s. of [the above equation]

It seems like the formula mentioned in the claim is based on complex analysis, since the above text from the paper mentions poles. I first thought about using the residue theorem on the last integral, but then it occurred to me that the integral might not be over a contour. Can this integral be carried out by using the Residue Theorem? If so, what would be the contour? If not, is there a formula from complex analysis that relates an impulse train to a sum over $R_n$? If so, what is it?

My question is a mathematical one based on this physics paper. Suppose that $\lambda_i $ is an eigenvalue of a reduced density matrix. Up to a normalization factor, the distribution of eigenvalues is given by the following formula. See the first page of the paper to find the formula.

$$ P(\lambda) = \sum_i \delta(\lambda - \lambda_i). $$

The paper also mentions the following, where $\rho$ is the reduced density matrix:

$$ R_n = Trace(\rho^n) = c_n L_{\text{eff}}^{-\frac{c}{6}(n - 1/n)} \equiv c_n e^{-b(n - 1/n)} $$

where $$b = \frac{c}{6} \ln (L_{\text{eff}}).\tag{3}$$

The claim made on the second page of the paper is the following:

$$ \lambda P(\lambda) = \lim_{\epsilon \to 0}\mathbb{Im}(f(\lambda - i\epsilon)) $$ where $$ f(z) = \frac{1}{\pi} \sum_{n=1}^\infty R_n z^{-n} = \frac{1}{\pi}\int d\lambda \frac{\lambda P(\lambda)}{z - \lambda}.\tag{4} $$ I am trying to prove this claim. The comment below the above claim is

Here, $f(λ − i\epsilon)$ has an imaginary part when $\epsilon → 0$ only on the support of $P(λ)$, due to the pole in the r.h.s. of [the above equation]

It seems like the formula mentioned in the claim is based on complex analysis, since the above text from the paper mentions poles. I first thought about using the residue theorem on the last integral, but then it occurred to me that the integral might not be over a contour. Can this integral be carried out by using the Residue Theorem? If so, what would be the contour? If not, is there a formula from complex analysis that relates an impulse train to a sum over $R_n$? If so, what is it?

My question is a mathematical one based on this physics paper. Suppose that $\lambda_i $ is an eigenvalue of a reduced density matrix. Up to a normalization factor, the distribution of eigenvalues is given by the following formula. See the first page of the paper to find the formula.

$$ P(\lambda) = \sum_i \delta(\lambda - \lambda_i) $$

The paper also mentions the following, where $\rho$ is the reduced density matrix:

$$ R_n = Trace(\rho^n) = c_n L_{\text{eff}}^{-\frac{c}{6}(n - 1/n)} \equiv c_n e^{-b(n - 1/n)} $$

where $b = \frac{c}{6} \ln (L_{\text{eff}})$

The claim made on the second page of the paper is the following:

$$ \lambda P(\lambda) = \lim_{\epsilon \to 0}\mathbb{Im}(f(\lambda - i\epsilon)) \text{ where } f(z) = \frac{1}{\pi} \sum_{n=1}^\infty R_n z^{-n} = \frac{1}{\pi}\int d\lambda \frac{\lambda P(\lambda)}{z - \lambda} $$ I am trying to prove this claim. The comment below the above claim is

Here, $f(λ − i\epsilon)$ has an imaginary part when $\epsilon → 0$ only on the support of $P(λ)$, due to the pole in the r.h.s. of [the above equation]

It seems like the formula mentioned in the claim is based on complex analysis, since the above text from the paper mentions poles. I first thought about using the residue theorem on the last integral, but then it occurred to me that the integral might not be over a contour. Can this integral be carried out by using the Residue Theorem? If so, what would be the contour? If not, is there a formula from complex analysis that related an impulse train to a sum over $R_n$? If so, what is it?

My question is a mathematical one based on this physics paper. Suppose that $\lambda_i $ is an eigenvalue of a reduced density matrix. Up to a normalization factor, the distribution of eigenvalues is given by the following formula. See the first page of the paper to find the formula.

$$ P(\lambda) = \sum_i \delta(\lambda - \lambda_i) $$

The paper also mentions the following, where $\rho$ is the reduced density matrix:

$$ R_n = Trace(\rho^n) = c_n L_{\text{eff}}^{-\frac{c}{6}(n - 1/n)} \equiv c_n e^{-b(n - 1/n)} $$

where $b = \frac{c}{6} \ln (L_{\text{eff}})$

The claim made on the second page of the paper is the following:

$$ \lambda P(\lambda) = \lim_{\epsilon \to 0}\mathbb{Im}(f(\lambda - i\epsilon)) \text{ where } f(z) = \frac{1}{\pi} \sum_{n=1}^\infty R_n z^{-n} = \frac{1}{\pi}\int d\lambda \frac{\lambda P(\lambda)}{z - \lambda} $$ I am trying to prove this claim. The comment below the above claim is

Here, $f(λ − i\epsilon)$ has an imaginary part when $\epsilon → 0$ only on the support of $P(λ)$, due to the pole in the r.h.s. of [the above equation]

It seems like the formula mentioned in the claim is based on complex analysis, since the above text from the paper mentions poles. I first thought about using the residue theorem on the last integral, but then it occurred to me that the integral might not be over a contour. Can this integral be carried out by using the Residue Theorem? If so, what would be the contour? If not, is there a formula from complex analysis that relates an impulse train to a sum over $R_n$? If so, what is it?