By C.B. Wang

ISBN-10: 3642385648

ISBN-13: 9783642385643

ISBN-10: 3642385656

ISBN-13: 9783642385650

The eigenvalue densities in numerous matrix types in quantum chromodynamics (QCD) are eventually unified during this ebook through a unified version derived from the integrable structures. Many new density types and loose strength features are therefore solved and awarded. The part transition types together with severe phenomena with fractional power-law for the discontinuities of the unfastened energies within the matrix types are systematically labeled by way of a transparent and rigorous mathematical demonstration. The equipment the following will stimulate new study instructions equivalent to the real Seiberg-Witten differential in Seiberg-Witten conception for fixing the mass hole challenge in quantum Yang-Mills conception. 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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Commun. Math. Phys. 59, 35–51 (1978) 2. : Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955) 3. : Hamiltonian Methods in the Theory of Solitons. Springer, Berlin (1986) 4. : Discrete Painlevé equations and their appearance in quantum gravity. Commun. Math. Phys. 142, 313–344 (1991) 5. : Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Physica D 2, 407–448 (1981) 6. : Eigenvalue density in Hermitian matrix models by the Lax pair method.

The Lax pair obtained from the generalized Hermite polynomials can be applied to formulate the eigenvalue densities in the Hermitian matrix models with a general potential. The Lax pair method then solves the eigenvalue density problems on multiple disjoint intervals, which are associated with scalar Riemann-Hilbert problems for multi-cuts. The string equation can be applied to derive the nonlinear algebraic relations between the parameters in the density models by reformulating the potential function in terms of the trace function of the coefficient matrix obtained from the Lax pair and using the Cayley-Hamilton theorem in linear algebra.

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. 95) n (z ) dz = 12 V (z), as the level density [7], satisfies −∞ σn (z)dz = n and (P) −∞ σz−z that is consistent with the unified model discussed in Sect.

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