Spectral properties of the twisted bi-Laplacian (Q1047905)

The twisted Laplacian \(L\) is the Hermite operator \[ H= -\Delta+ \frac{1}{4} (x^{2} + y^{2}) \] perturbed by the partial differential operator \(-iN\), where \[ N= x \frac{\partial}{\partial y}- y \frac{\partial}{\partial x}. \] The authors show that \(L\) is globally hypoelliptic in the Schwartz space \(S(\mathbb R^2)\) in the sense that \(u\in S'(\mathbb R^2)\) and \(Lu\in S(\mathbb R^2)\) imply \(u\in S(\mathbb R^2)\); \(L\) is essential selfadjoint; the spectrum of its closure is formed by the odd natural numbers each of which being an eigenvalue with infinite multiplicity.