A lifting theorem for planar mixed automorphic functions

We address the concrete spectral analysis of an invariant magnetic Schrödinger operator, which acts on one-dimensional \(L^2\) $L^2$ -mixed automorphic functions associated with a given equivariant pair \((\rho , \tau )\) $(\rho ,\tau )$ and a discrete subgroup of the semi-direct group \(\textsf{U}(1) \ltimes \mathbb {C}\) $\mathsf{U}(1)⋉\mathbb{C}$ . To achieve this, we employ a lifting theorem to the classical automorphic functions associated with a specific pseudo-character. In addition, we offer a partial characterization of the equivariant pairs relative to our setting and discuss possible generalization to higher dimensions.