On $L^2$-eigenfunctions of Twisted Laplacian on curved surfaces and suggested orthogonal polynomials

TL;DR

This paper uses the factorization method to fully characterize the $L^2$-eigenspaces of the twisted Laplacian on curved surfaces, deriving new orthogonal polynomials linked to surface geometry.

Contribution

It provides a unified approach to describe eigenspaces and introduces new orthogonal polynomials related to the geometry of constant curvature surfaces.

Findings

Complete description of $L^2$-eigenspaces for the twisted Laplacian

Derivation of new subclasses of orthogonal polynomials

Connection between polynomials and surface geometry

Abstract

We show in a unified manner that the factorization method describes completely the $L^2$-eigenspaces associated to the discrete part of the spectrum of the twisted Laplacian on constant curvature Riemann surfaces. Subclasses of two variable orthogonal polynomials are then derived and arise by successive derivations of elementary complex valued functions depending on the geometry of the surface.