On spectral asymptotic of quasi-exactly solvable quartic potential

Abstract

Motivated by the earlier results of Masoero and De Benedetti (Nonlinearity 23:2501, 2010) and Shapiro et al. (Commun Math Phys 311(2):277–300, 2012), we discuss below the asymptotic of the solvable part of the spectrum for the quasi-exactly solvable quartic oscillator. In particular, we formulate a conjecture on the coincidence of the asymptotic shape of the level crossings of the latter oscillator with the asymptotic shape of zeros of the Yablonskii–Vorob’ev polynomials. Further we present a numerical study of the spectral monodromy for the oscillator in question.

Exactly solvable anharmonic oscillator, degenerate orthogonal polynomials and Painlevé II

“…Figure 1: Scaled roots of the Vorob'ev-Yablonsky polynomials Y n (n 2/3 s) in red, and roots of the discriminant D n (n 2/3 s) in black, for n = 30. This particular scaling was conjectured by Shapiro and Tater in [ST22].…”

“…the zeroes of the Vorob'ev-Yablonskii polynomials). This similarity is the object of a conjecture that B. Shapiro and M. Tater floated several years ago, but only recently formalized in [ST22] (Conjecture 2 ibidem).…”

“…Following the definition in [BB98] a quantum mechanical potential is Quasi Exactly Solvable (QES) if a finite portion of the energy spectrum and associated eigenfunctions can be found exactly and in closed form, while the potential is Exactly Solvable (ES) if all the spectrum and associated eigenfunctions can be found exactly and in closed form. In particular it was shown in [ST22] that the eigenvalue problem (1.1) with only two out of the three boundary conditions at infinity is Quasi Exactly Solvable. Here we show that the quartic anharmonic oscillator in (1.1), with the boundary conditions (1.2) is Exactly Solvable and admits a solution if and only if J = n + 1 ∈ N is a positive integer.…”

“…( The roots of D n (t) = 0 are precisely the values of t such that the matrix M n (t) has repeated eigenvalues (algebraic multiplicity greater than 1). The Shapiro-Tater conjecture [ST22] relates the zeros of D n (t) to the poles of rational solutions of PII. The Painlevé II equation has rational solutions u(t) if and only if α = n ∈ Z, and we denote them by u n (t).…”

Critical graph of a polynomial quadratic differential related to a Schrödinger equation with quartic potential

“…where b is real, m is integer and we choose = 1; see [3], [10], [11]. This problem is a quasi-exactly solvable : for each b, there are m eigenvalues λ m,k , with elementary eigenfunctions…”