Exactly Solvable Anharmonic Oscillator, Degenerate Orthogonal Polynomials and Painlevé II.

Using WKB analysis, the paper addresses a conjecture of Shapiro and Tater on the similarity between two sets of points in the complex plane; on one side is the set the values of t ∈ C for which the spectrum of the quartic anharmonic oscillator in the complex plane d 2 y d x 2 - x 4 + t x 2 + 2 J x y = Λ y , with certain boundary conditions, has repeated eigenvalues. On the other side is the set of zeroes of the Vorob'ev-Yablonskii polynomials, i.e. the poles of rational solutions of the second Painlevé equation. Along the way, we indicate a surprising and deep connection between the anharmonic oscillator problem and certain degenerate orthogonal (monic) polynomials.