Title: A composite operator and it’s applications to Biorthogonal Quantum Mechanics

Abstract: In some physics settings, including Quantum Mechanics there is an underlying operator that defines a differential equation modeling a system. For physical reasons, it is assumed to be Hermitian. However, this requirement has been loosened to allow for non-Hermitian operators, or in finite dimension the matrix representation. This is a result of wanting to model a system that can lose information to the environment. To this end we investigate the role of a generalization of the quadratic composite operator construction first introduced by A. Cerjan, T. Lorin, and F. Vides. We will explore how a specific composite operator connects with non-Hermitian problems in the biorthogonal framework of quantum mechanics with an emphasis on approximate joint eigenvectors and how they evolve over time.