Let \(M_{\kappa}\) be a simply connected Riemann surface with constant scalar curvature \(\kappa\) equipped with the volume measure \(d\mu_{\kappa}=(1+\kappa z\overline{z})^{-2}d\lambda\) associated with the Hermitian metric \(ds_{\kappa}^{2}=(1+\kappa z\overline{z})^{-2}dz\otimes d\overline{z}\) on \(M_{\kappa}\). (Here \(d\lambda\) denotes the Lebesgue measure.) For \(\nu>0\), define the (magnetic Schrödinger) Laplacian
\[ \Delta_{\kappa}^{\nu}=(d+i\nu\theta_{\kappa})^{\star}(d+i\nu\theta_{\kappa}) \]
where \(\theta_{\kappa}\) is the one-form \((1+\kappa z\overline{z})^{-1}(\overline{z}dz-zd\overline{z})\). It is known that \(\Delta_{\kappa}^{\nu}\) is a selfadjoint elliptic differential operator acting on the Hilbert space \(L^{2}(M_{\kappa},d\mu_{\kappa})\).
In the paper under review, the author describes, using the so-called factorization method, the discrete spectrum of \(\Delta_{\kappa}^{\nu}\). As a byproduct, some orthogonal polynomials depending on the geometry of the surface \(M_{\kappa}\) are obtained.