Generalized Freud’s equation and level densities with polynomial potential

[摘要] Orthogonal polynomials with weight exp[$−NV (x)$] are studied where $V (x) = sum_{k=1}^{d} a_{2k} x^{2k}$ is a polynomial of order 2𝑑. The generalized Freud’s equations for 𝑑 = 3, 4 and 5 are derived and using this $R_{𝜇} = h_{𝜇} / h{𝜇−1}$ is obtained, where $h_{𝜇}$ is the normalization constant for the corresponding orthogonal polynomials. Moments of the density functions, expressed in terms of $R_{𝜇}$ , are obtained using Freud’s equation and using this, explicit results of level densities as $N → ∞$ are derived using the method of resolvents. The results are compared with those using Dyson–Mehta method.

[学科分类] 物理（综合）

[关键词] Orthogonal polynomial;Freud’s equation;Dyson–Mehta method;methods of resolvents;level density.