On tau functions for orthogonal polynomials and matrix models
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arXiv
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| Format: | Preprint |
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2010
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| _version_ | 1866929509924601856 |
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| author | Blower, Gordon |
| author_facet | Blower, Gordon |
| contents | Let v be a real polynomial of even degree, and let ρbe the equilibrium probability measure for v with support S; so that v(x)\geq 2\int \log |x-y| ρ(dy)+C_v for some constant C_v with support S. Then S is the union of finitely many bounded intervals with endpoints delta_j, and ρis given by an algebrais weight w(x) on S. The system of orthogonal polynomials for w gives rise to the Magnus--Schlesinger differential equations. This paper identifies the tau function of this system with the Hankel determinant det[\in x^{j+k}ρ(dx)] of ρ. The solutions of the Magnus--Schlesinger equations are realised by a linear system, which is used to compute the tau function in terms of a Gelfand--Levitan equaiton. The tau function is associated with a potential q and a scattering problem for the Schrodinger operator with potential q. For some algebro-geometric potentials, the paper solves the scattering problem in terms of linear systems. The theory extends naturally to elliptic curves and resolves the case where S has exactly two intervals. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1008_2352 |
| institution | arXiv |
| publishDate | 2010 |
| record_format | arxiv |
| spellingShingle | On tau functions for orthogonal polynomials and matrix models Blower, Gordon Classical Analysis and ODEs 60B20, 37K15 Let v be a real polynomial of even degree, and let ρbe the equilibrium probability measure for v with support S; so that v(x)\geq 2\int \log |x-y| ρ(dy)+C_v for some constant C_v with support S. Then S is the union of finitely many bounded intervals with endpoints delta_j, and ρis given by an algebrais weight w(x) on S. The system of orthogonal polynomials for w gives rise to the Magnus--Schlesinger differential equations. This paper identifies the tau function of this system with the Hankel determinant det[\in x^{j+k}ρ(dx)] of ρ. The solutions of the Magnus--Schlesinger equations are realised by a linear system, which is used to compute the tau function in terms of a Gelfand--Levitan equaiton. The tau function is associated with a potential q and a scattering problem for the Schrodinger operator with potential q. For some algebro-geometric potentials, the paper solves the scattering problem in terms of linear systems. The theory extends naturally to elliptic curves and resolves the case where S has exactly two intervals. |
| title | On tau functions for orthogonal polynomials and matrix models |
| topic | Classical Analysis and ODEs 60B20, 37K15 |
| url | https://arxiv.org/abs/1008.2352 |