I am a theoretical physicist dealing with Matrix Models. In this context, one often encounters singular integral equations whose solution gives the eigenvalue density in the so-called large $N$ limit. A very generic version has been written down in Eqn (2.47) of https://arxiv.org/pdf/hep-th/0410165.pdf. Essentially, one ends up with an equation of the form: \begin{equation} \Pr \int_{a}^b \frac{\rho(\lambda')}{\lambda-\lambda'} d\lambda'=W(\lambda) \end{equation} where $W(\lambda)$ is some known function and $\Pr$ is the Cauchy principal value. One way to solve for $\rho(\lambda)$ is to define a resolvent which at genus zero ($\omega_0(\lambda)$) has a closed form expression given by Eqn (2.60) (for a proof of this result, one can check pp 64, the passage appearing below Eqn (3.2.34) of https://arxiv.org/pdf/1510.04430.pdf). The discontinuity in $\omega_0(\lambda)$ across the $x$-axis gives us $\rho(\lambda)$ as given in Eqn (2.58). Another class of such singular integral that arises (refer to Eqn (3.20) of https://arxiv.org/pdf/hep-th/0509002.pdf) is given by \begin{equation} \Pr \int_a^b d\lambda' \rho(\lambda') \coth \frac{t}{2}(\lambda-\lambda') = G(\lambda) \end{equation} where again $\Pr$ refers to the Cauchy principal value while $G(\lambda)$ is a known function. In order to solve this, one can use the transformation $h=e^{t\lambda}$. Using this along with the normalization condition $\int_a^b \rho(\lambda) d\lambda =1$ coverts this to the form of the Cauchy kernel. Then one uses the standard techniques for the Cauchy kernel to get the eigenvalue density which can eventually be expressed in term of the original varibale $\lambda$.

I am looking at a certain matrix model which curiously enough gives a sort of "hybrid", namely, a singular integral equation of the form \begin{equation} \Pr \int_a^b d\lambda' \rho(\lambda') \left[\frac{1}{\lambda -\lambda'}+\frac{t}{2}\coth \frac{t}{2}(\lambda-\lambda') \right]=F(\lambda) \end{equation} where $F(\lambda)$ is a known function. Clearly a simple transformation of the form $h=e^{th}$ does not help here much. My questions are the following:

  1. Is there an analog of the resolvent that one can define for kernels like the one above that I encountered?
  2. What will be the analogous conditions to determine the limits $a$ and $b$ (equivalent constraints of Eqn (3.2.39) in https://arxiv.org/pdf/1510.04430.pdf)?

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