Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials
We investigate the asymptotic zero distribution of Heine-Stieltjes polynomials - polynomial solutions of a second order differential equations with complex polynomial coefficients. In the case when all zeros of the leading coefficients are all real, zeros of the Heine-Stieltjes polynomials were inte...
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| Format: | info:eu-repo/semantics/article |
| Language: | English |
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2012
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| Subjects: | |
| Online Access: | http://hdl.handle.net/10835/1628 |
| _version_ | 1789406537716858880 |
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| author | Martínez-Finkelshtein, Andrei Rakhmanov, Evgenii A. |
| author_facet | Martínez-Finkelshtein, Andrei Rakhmanov, Evgenii A. |
| author_sort | Martínez-Finkelshtein, Andrei |
| collection | DSpace |
| description | We investigate the asymptotic zero distribution of Heine-Stieltjes polynomials - polynomial solutions of a second order differential equations with complex polynomial coefficients. In the case when all zeros of the leading coefficients are all real, zeros of the Heine-Stieltjes polynomials were interpreted by Stieltjes as discrete distributions minimizing an energy functional. In a general complex situation one deals instead with a critical point of the energy. We introduce the notion of discrete and continuous critical measures (saddle points of the weighted logarithmic energy on the plane), and prove that a weak-* limit of a sequence of discrete critical measures is a continuous critical measure. Thus, the limit zero distributions of the Heine-Stieltjes polynomials are given by continuous critical measures. We give a detailed description of such measures, showing their connections with quadratic differentials. In doing that, we obtain some results on the global structure of rational quadratic differentials on the Riemann sphere that have an independent interest. |
| format | info:eu-repo/semantics/article |
| id | oai:repositorio.ual.es:10835-1628 |
| institution | Universidad de Cuenca |
| language | English |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oai:repositorio.ual.es:10835-16282023-04-12T19:38:17Z Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials Martínez-Finkelshtein, Andrei Rakhmanov, Evgenii A. Distribución asintótica a cero Polinomios de Heine-Stieltjes Ecuaciones diferenciales Coeficientes polinomiales complejos Asymptotic zero distribution Heine-Stieltjes polynomials Differential equations Complex polynomial coefficients We investigate the asymptotic zero distribution of Heine-Stieltjes polynomials - polynomial solutions of a second order differential equations with complex polynomial coefficients. In the case when all zeros of the leading coefficients are all real, zeros of the Heine-Stieltjes polynomials were interpreted by Stieltjes as discrete distributions minimizing an energy functional. In a general complex situation one deals instead with a critical point of the energy. We introduce the notion of discrete and continuous critical measures (saddle points of the weighted logarithmic energy on the plane), and prove that a weak-* limit of a sequence of discrete critical measures is a continuous critical measure. Thus, the limit zero distributions of the Heine-Stieltjes polynomials are given by continuous critical measures. We give a detailed description of such measures, showing their connections with quadratic differentials. In doing that, we obtain some results on the global structure of rational quadratic differentials on the Riemann sphere that have an independent interest. 2012-08-01T08:01:46Z 2012-08-01T08:01:46Z 2011 info:eu-repo/semantics/article http://hdl.handle.net/10835/1628 en info:eu-repo/semantics/openAccess Communications in Mathematical Physics Volume 302, Number 1 (2011) |
| spellingShingle | Distribución asintótica a cero Polinomios de Heine-Stieltjes Ecuaciones diferenciales Coeficientes polinomiales complejos Asymptotic zero distribution Heine-Stieltjes polynomials Differential equations Complex polynomial coefficients Martínez-Finkelshtein, Andrei Rakhmanov, Evgenii A. Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials |
| title | Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials |
| title_full | Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials |
| title_fullStr | Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials |
| title_full_unstemmed | Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials |
| title_short | Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials |
| title_sort | critical measures, quadratic differentials, and weak limits of zeros of stieltjes polynomials |
| topic | Distribución asintótica a cero Polinomios de Heine-Stieltjes Ecuaciones diferenciales Coeficientes polinomiales complejos Asymptotic zero distribution Heine-Stieltjes polynomials Differential equations Complex polynomial coefficients |
| url | http://hdl.handle.net/10835/1628 |
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