On spectral asymptotic of quasi-exactly solvable quartic potential (Q2243938)

The paper is focused on the quasi-exactly solvable quartic oscillator \[ L_J(w)=w''-(x^4/4-ax^2/2-Jx)w=\lambda w,\tag{1} \] with the boundary conditions \(w(t)\to 0\) and \(w(t e^{2\pi i/3})\to 0\) as \(t\to+\infty\), where \(a\in \mathbb{C}\) and \(J\) are parameters. The authors study the asymptotic of the solvable part of spectrum for \((1)\) in two different regimes. A numerical study of the spectral monodromy for oscillator \((1)\) is also presented.