On the nodal patterns of the 2D isotropic quantum harmonic oscillator

Pierre Bérard (1) , Bernard Helffer (2, 3)

1 IF - Institut Fourier
2 LMJL - Laboratoire de Mathématiques Jean Leray
3 LM-Orsay - Laboratoire de Mathématiques d'Orsay

For the spherical Laplacian on the sphere and for the Dirichlet Laplacian in the square, Antonie Stern claimed in her PhD thesis (1924) the existence of an infinite sequence of eigenvalues whose corresponding eigenspaces contain an eigenfunction with exactly two nodal domains. These results were given complete proofs respectively by Hans Lewy in 1977, and the authors in 2014 (see also Gauthier-Shalom–Przybytkowski (2006)). In this paper, we obtain similar results for the two dimensional isotropic quantum harmonic oscillator. In the opposite direction, we construct an infinite sequence of regular eigenfunctions with " many " nodal domains. We finally provide bounds for the length of the nodal set of an eigenfunction with energy $\lambda$ in the classically permitted region $\{ |x| < \sqrt{\lambda} \}$.

Mots clés: Quantum harmonic oscillator; Nodal lines; Nodal domains; Courant theorem

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Théorie spectrale [math.SP]; Géométrie différentielle [math.DG]

Pierre Bérard, Bernard Helffer. On the nodal patterns of the 2D isotropic quantum harmonic oscillator. 2015. ⟨hal-01160620v1⟩