Painleve II, anharmonic oscillators & degenerate orthogonal polynomials

In this thesis we study a surprising connection relating the second Painlevé transcendent, anharmonic oscillators and degenerate orthogonal polynomials. This connection arose from the investigations into the similarity of two sets of points in the complex plane. On one side is the set of zeroes of the Vorob’evYablonskii polynomials, that is, the poles of rational solutions of the second Painlevé transcendent where u = u(t) is a complex function of the complex variable t. On the other side is the values of of the parameter t for which the spectrum of a certain quartic anharmonic oscillator in the complex domain has eigenvalues of algebraic multiplicity at least 2 (under suitable boundary conditions). The similarity between these two sets of points when α = n and J = n + 1, with n ∈ N, was first observed by Shapiro and Tater and we give an explanation for this phenomenon. The study of this problem naturally lead us to the notion of certain non-hermitian orthogonal polynomial polynomials satisfying an excess of orthogonality conditions with a certain exponential weight. These degenerate orthogonal polynomials are in one-to-one correspondence to the solutions of the Stieltjes-Fekete equilibrium problem. This generalises the famous result of Stieltjes, which relates the zeroes of the classical orthogonal polynomials to the configuration of points on the line that minimize a suitable potential with logarithmic interactions under an external field. We study the case when the derivative of the external field is an arbitrary rational complex function. When the differential of the external field is of degree 3 on the Riemann sphere our result reproduces Stieltjes original findings and, for more than a century after the original result, provides a direct generalisation for higher degree