Sesquilinear forms as eigenvectors in quasi *-algebras, with an application to ladder elements

TL;DR

This paper introduces a class of sesquilinear forms called eigenstates in Banach quasi *-algebras, explores their properties, and applies them to ladder elements with implications for orthogonality in GNS-representations.

Contribution

It defines eigenstates of elements in Banach quasi *-algebras and applies this concept to ladder elements, revealing new properties and orthogonality relations.

Findings

Eigenstates exhibit specific orthogonality properties.

Application to ladder elements demonstrates physical relevance.

GNS-representation elucidates form properties.

Abstract

We consider a particular class of sesquilinear forms on a {Banach quasi *-algebra} $(\A[\|.\|],\Ao[\|.\|_0])$ which we call {\em eigenstates of an element} $a\in\A$, and we deduce some of their properties. We further apply our definition to a family of ladder elements, i.e. elements of $\A$ obeying certain commutation relations physically motivated, and we discuss several results, including orthogonality and biorthogonality of the forms, via GNS-representation.