# Hamiltonians defined by biorthogonal sets

## TL;DR

This paper explores the use of biorthogonal and $\

## Contribution

It introduces the concept of $\

## Key findings

- Conditions for defining non self-adjoint Hamiltonians with real spectra.
- Extension from Riesz bases to biorthogonal sets in quantum models.
- Framework for Hamiltonians using $\

## Abstract

In some recent papers, the studies on biorthogonal Riesz bases has found a renewed motivation because of their connection with pseudo-hermitian Quantum Mechanics, which deals with physical systems described by Hamiltonians which are not self-adjoint but still may have real point spectra. Also, their eigenvectors may form Riesz, not necessarily orthonormal, bases for the Hilbert space in which the model is defined. Those Riesz bases allow a decomposition of the Hamiltonian, as already discussed is some previous papers. However, in many physical models, one has to deal not with o.n. bases or with Riesz bases, but just with biorthogonal sets. Here, we consider the more general concept of $\mathcal{G}$-quasi basis and we show a series of conditions under which a definition of non self-adjoint Hamiltonian with purely point real spectra is still possible.