Generalized Riesz systems and orthonormal sequences in Krein spaces

TL;DR

This paper explores bi-orthogonal vector sets in Hilbert and Krein spaces, introduces the concept of first/second type sequences, and discusses their relevance to quantum systems with non-self-adjoint Hamiltonians.

Contribution

It introduces the notions of first/second type sequences in Krein spaces and analyzes their relation to generalized Riesz systems, with applications to quantum mechanics.

Findings

Introduction of first/second type sequences in Krein spaces

Analysis of bi-orthogonal sets and their properties

Relevance to quantum systems with non-self-adjoint Hamiltonians

Abstract

We analyze special classes of bi-orthogonal sets of vectors in Hilbert and in Krein spaces, and their relations with generalized Riesz systems. In this way, the notion of the first/second type sequences is introduced and studied. We also discuss their relevance in some concrete quantum mechanical system driven by manifestly non self-adjoint Hamiltonians.