By C.B. Wang
The eigenvalue densities in numerous matrix types in quantum chromodynamics (QCD) are eventually unified during this booklet by way of a unified version derived from the integrable structures. Many new density types and loose power capabilities are for that reason solved and offered. The section transition types together with severe phenomena with fractional power-law for the discontinuities of the loose energies within the matrix types are systematically categorised through a transparent and rigorous mathematical demonstration. The equipment right here will stimulate new examine instructions reminiscent of the $64000 Seiberg-Witten differential in Seiberg-Witten concept for fixing the mass hole challenge in quantum Yang-Mills idea. The formulations and effects will gain researchers and scholars within the fields of part transitions, integrable structures, matrix versions and Seiberg-Witten theory.
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Additional resources for Application of Integrable Systems to Phase Transitions
1 Denote Φn (z) = e− 2 V (z) (pn (z), pn−1 (z))T . 24) 26 2 Densities in Hermitian Matrix Models for a matrix An (z). 24) are called the Lax pair for the string equation. This Lax pair structure was given in , as well as in  (Part 2, Chap. 1). The Lax pair method for the eigenvalue density starts from the construction of the matrix An . For m ≥ 1 and n ≥ 2m, consider ∂ pn = an,n−1 pn−1 + an,n−2 pn−2 + · · · + an,n−2m+1 pn−2m+1 , ∂z ∂ pn−1 = an−1,n−2 pn−2 + an−1,n−3 pn−3 + · · · + an−1,n−2m pn−2m , ∂z where an ,n −k hn −k = ∞ −∞ V (z)pn −k pn e−V (z) dz, for n = n or n−1, and k = 1, 2, .
93). By the discussions above, it can be seen that when l2 = 0, as , bs (s = 1, . . , l), and gj (j = 1, . . 91), then y(η) = ωl (η) − 12 W (η) satisfies the following relations: y(η) is analytic when η ∈ C Ω; y(η)|Ω + + y(η)|Ω − = −W (η); y(η) → 0, as η → ∞. These relations are important in complex analysis, called scalar Riemann-Hilbert problem. 94) then the density models and the corresponding scalar Riemann-Hilbert problems can be well solved. 94) is also equal to − det(J (l) − √ √ −1 (det J (l) )J (l) ) by considering (J (l) − det J (l) )(J (l) + det J (l) ) and calculating the determinants.
94) is also equal to − det(J (l) − √ √ −1 (det J (l) )J (l) ) by considering (J (l) − det J (l) )(J (l) + det J (l) ) and calculating the determinants. 95) n (z ) dz = 12 V (z), as the level density , satisfies −∞ σn (z)dz = n and (P) −∞ σz−z that is consistent with the unified model discussed in Sect. 1. When the density involves the parameter n, the string equation and the initial conditions when n is less than 2m need to be considered to calculate the functions un and vn . 96) √ √ for η ∈ [− 2, 2], which is the well known Wigner semicircle.