Exactly Solvable Anharmonic Oscillator, Degenerate Orthogonal Polynomials and Painlevé II

Abstract

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\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\in \\mathbb {C}$$\\end{document} for which the spectrum of the quartic anharmonic oscillator in the complex plane d2ydx2-x4+tx2+2Jxy=Λy,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{\ extrm{d}^{2} y}{\ extrm{d} x^{2}} - \\left( x^4 + tx^2 + 2Jx \\right) y = \\Lambda y, \\end{aligned}$$\\end{document}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.