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 for which the spectrum of the quartic anharmonic oscillator in the complex plane with certain boundary conditions, has repeated eigenvalues. On the other side is the set of zeroes of the Vorob'ev-Yablonskii polynomials, i.e.