On a conjecture of P. Jorgensen and W. Klink (Q1111082)

In the work by \textit{P. E. T. Jorgensen} and \textit{W. H. Klink} [Publ. Res. Inst. Math. Sci. 21, 969-991 (1985; Zbl 0601.58027)], the authors conjecture that if \(\lambda_ n(\alpha)\) is the series of the eigenvalues of the operator: \[ H_{\alpha}=-\Delta_ x+(\alpha +x_ 1x_ 2)^ 2\quad in\quad {\mathbb{R}}^ 2\quad (\alpha \in {\mathbb{R}}), \] then each of the \(\lambda_ n(\alpha)\) is continuous in \(\alpha\). It is relatively easy to prove this conjecture using the technics of \textit{B. Helffer} and \textit{J. Nourrigat} [Hypoellipticité maximale pour des opérateurs polynomes de champs de vecteurs (1985; Zbl 0568.35003)] combined with the more classical technics of \textit{T. Kato} [``Perturbation theory for linear operators (1984; Zbl 0531.47014)]. Using technics of \textit{B. Helffer} and \textit{J. Sjöstrand} [Commun. Partial Differ. Equations 9, 337-408 (1984; Zbl 0546.35053) and Pap. dedic. S. Mizohata Occas. 60th Birthday, 133-186 (1986; Zbl 0628.35024)], we study also the asymptotic behavior of \(\lambda_ n(\alpha)\) for \(| \alpha | \to \infty\).