Zero distribution of complex orthogonal polynomials with respect to exponential weights

We study the limiting zero distribution of orthogonal polynomials with respect to some particular exponential weights e^-^n^V^(^z^) along contours in the complex plane. We are especially interested in the question under which circumstances the zeros of the orthogonal polynomials accumulate on a single analytic arc (one cut case), and in which cases they do not. In a family of cubic polynomial potentials V(z)=-iz^33+iKz, we determine the precise values of K for which we have the one cut case. We also prove the one cut case for a monomial quintic V(z)=-iz^55 on a contour that is symmetric in the imaginary axis.

[1]  Alexander Tovbis,et al.  Asymptotics of Orthogonal Polynomials with Complex Varying Quartic Weight: Global Structure, Critical Point Behavior and the First Painlevé Equation , 2011, 1108.0321.

[2]  A. Martínez-Finkelshtein,et al.  Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters , 2013 .

[3]  Arno B. J. Kuijlaars,et al.  S-curves in polynomial external fields , 2013, J. Approx. Theory.

[4]  Сергей Павлович Суетин,et al.  Вариация равновесной энергии и $S$-свойство стационарного компакта@@@Variation of the equilibrium energy and the $S$-property of stationary compact sets , 2011 .

[5]  G. Lagomasino,et al.  Recent advances in orthogonal polynomials, special functions, and their applications : 11th International Symposium, August 29-September 2, 2011, Universidad Carlos III de Madrid, Leganés, Spain , 2012 .

[6]  Daan Huybrechs,et al.  Complex Gaussian quadrature of oscillatory integrals , 2009, Numerische Mathematik.

[7]  E. Rakhmanov,et al.  EQUILIBRIUM DISTRIBUTIONS AND DEGREE OF RATIONAL APPROXIMATION OF ANALYTIC FUNCTIONS , 1989 .

[8]  Marco Bertola,et al.  Boutroux curves with external field: equilibrium measures without a variational problem , 2011 .

[9]  L. Alonso,et al.  Superpotentials, quantum parameter space and phase transitions in $ \mathcal{N} $ = 1 supersymmetric gauge theories , 2013, 1301.5982.

[10]  B. Konopelchenko,et al.  Spectral curves in gauge/string dualities: integrability, singular sectors and regularization , 2013, 1301.7082.

[11]  N. Ayırtman,et al.  Univalent Functions , 1965, Series and Products in the Development of Mathematics.

[12]  E. Rakhmanov,et al.  Orthogonal Polynomials and $S$-curves , 2011, 1112.5713.

[13]  E. Saff,et al.  Logarithmic Potentials with External Fields , 1997 .

[14]  Elena Medina,et al.  Determination of S-curves with applications to the theory of non-Hermitian orthogonal polynomials , 2013, 1305.3028.

[15]  G. Pólya,et al.  Problems and theorems in analysis , 1983 .

[16]  Dinesh Manocha,et al.  SOLVING SYSTEMS OF POLYNOMIAL EQUATIONS , 2002 .

[17]  Daan Huybrechs,et al.  Asymptotic zero distribution of complex orthogonal polynomials associated with Gaussian quadrature , 2010, J. Approx. Theory.

[18]  M. Y. Mo,et al.  Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights , 2006 .

[19]  E. Rakhmanov,et al.  Critical Measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes Polynomials , 2009, 0902.0193.