A lifting theorem for planar mixed automorphic functions and applications

TL;DR

This paper introduces a lifting theorem for planar mixed automorphic functions, enabling spectral analysis of magnetic Schrödinger operators, with potential extensions to higher dimensions and applications in mathematical physics.

Contribution

It presents a new lifting theorem connecting mixed automorphic functions to classical automorphic functions, facilitating spectral analysis of magnetic Schrödinger operators.

Findings

Spectral analysis of magnetic Schrödinger operators on mixed automorphic functions.

A partial characterization of equivariant pairs in the given setting.

Discussion on possible generalizations to higher dimensions.

Abstract

We deal with the concrete spectral analysis of an invariant magnetic Schr\"odinger operator acting on one dimensional $L^2$-mixed automorphic functions with respect to given equivariant pair $ (\rho,\tau) $ and given discrete subgroup of the semi-direct group $U(1)\ltimes\mathbb{C}$. This will be carried out by means of a lifting theorem to the classical automorphic functions associated with specific pseudo-character. We also provided a partial characterization of the equivariant pairs relative to our setting and discuss possible generalization to higher dimensions.