The spectrum of the non-self-adjoint twisted Laplacian on \(\mathbb{R}^2\) (Q2238519)

As remarked by \textit{S. Thangavelu} [Math. Ann. 335, No. 4, 879--899 (2006; Zbl 1100.22008)], the spectrum and eigenvalues of the non-self-adjoint twisted Laplacian operator \[L_s = - \Delta + \frac{s^2}{4} |z|^2 + i s \sum_{j=1}^n \left(x_j \dfrac{\partial}{\partial y_j} - y_j \dfrac{\partial}{\partial x_j} \right)\] is well understood and studied when \(s\) is a real number, however, few or nothing is known about the spectrum when \(s\) is a complex number. More is known in case when \(s\) is purely imaginary, however, the eigenfunctions have not been characterised. The authors of the paper under review study this spectrum in the case of the twisted Laplacian operator on \(\mathbb{R}^2\) \[L_s = - \Delta_{\mathbb{R}^2} + \frac{s^2}{4} (x^2 + y^2) + i s \left(x \dfrac{\partial}{\partial y} - y \dfrac{\partial}{\partial x} \right)\] acting on \(L^2(\mathbb{R}^2)\) when \(s\) is a purely imaginary number. Concretely, the main result establishes that \[\sigma(L_{i \beta}) = \mathbb{R} + i \beta \mathbb{Z}\] for all \(\beta \in \mathbb{R}^*\).