Eigenvalue density in Hermitian matrix models by the Lax pair method

In this paper, a new method is discussed to derive the eigenvalue density in a Hermitian matrix model with a general potential. The density is considered on one interval or multiple disjoint intervals. The method is based on Lax pair theory and the Cayley-Hamilton theorem by studying the orthogonal polynomials associated with the Hermitian matrix model. It is obtained that the restriction conditions for the parameters in the density are connected to the discrete Painleve I equation, and the results are related to the scalar Riemann-Hilbert problem. Some special density functions are also discussed in association with the known results in this subject.