Generalized Riesz systems and orthonormal sequences in Krein spaces

TL;DR

This paper explores special biorthogonal vector sets in Hilbert and Krein spaces, their connection with quasi bases, and their application to quantum systems with non-self-adjoint Hamiltonians.

Contribution

It introduces new classes of biorthogonal sets in Krein spaces and examines their relevance to quantum mechanics with non-self-adjoint operators.

Findings

Identification of biorthogonal sets related to $ ext{G}$-quasi bases

Analysis of their properties in Krein spaces

Application to quantum systems with non-self-adjoint Hamiltonians

Abstract

We analyse special classes of biorthogonal sets of vectors in Hilbert and in Krein spaces, and their relations with $\mathcal{G}$- quasi bases. We also discuss their relevance in some concrete quantum mechanical system driven by manifestly non self-adjoint Hamiltonians.