On the completeness of generalized eigenfunctions of the Hamiltonian of reggeon field theory in Bargmann space

TL;DR

This paper investigates the spectral properties of a non-Hermitian Hamiltonian in Bargmann space relevant to reggeon field theory, establishing boundary conditions and proving the completeness of its generalized eigenfunctions.

Contribution

It determines boundary conditions for the Hamiltonian's eigenvalue problem and proves the completeness of its generalized eigenfunctions, advancing understanding of its spectral structure.

Findings

Boundary conditions for the eigenvalue problem are identified.

Completeness of generalized eigenfunctions is proven.

An explicit integral form of the inverse operator is provided.

Abstract

The Hamiltonian reggeon acting in Bargmann space is non-Hermitian with respect to the standard scalar product associated to Bargmann space. Hence the question arises, whether the eigenfunctions by the finite norm condition form a complete basis. The main new results of this article are the determination of the boundary conditions for the eigenvalue problem associated to this Hamiltonian and the proof of the completeness of the generalized eigen-functions of this operator. This article also provides an explicit integral form of the inverse of reggeon's operator on the negative imaginary axis and we give some spectral properties of our operator.