Abstract
AbstractUsing 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 $$t\in \mathbb {C}$$
t
∈
C
for which the spectrum of the quartic anharmonic oscillator in the complex plane $$\begin{aligned} \frac{\textrm{d}^{2} y}{\textrm{d} x^{2}} - \left( x^4 + tx^2 + 2Jx \right) y = \Lambda y, \end{aligned}$$
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.
Funder
H2020 Marie Sklodowska-Curie Actions
Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada
Scuola Internazionale Superiore di Studi Avanzati - SISSA
Publisher
Springer Science and Business Media LLC
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