Physical and unphysical solutions of random-phase approximation equation

TL;DR

This paper reanalyzes the mathematical properties of solutions to the RPA equation, focusing on both physical and unphysical solutions, and explores the role of Jordan blocks in understanding these solutions.

Contribution

It provides a detailed mathematical analysis of RPA solutions, including the role of Jordan blocks and duality of eigenvectors, enhancing understanding of physical and unphysical solutions.

Findings

Identification of duality in eigenvectors and basis vectors.

Analysis of Jordan blocks in the stability matrix.

Clarification of properties of physical and unphysical solutions.

Abstract

Properties of solutions of the RPA equation is reanalyzed mathematically, which is defined as a generalized eigenvalue problem of the stability matrix $\mathsf{S}$ with the norm matrix $\mathsf{N}=\mathrm{diag.}(1,-1)$. As well as physical solutions, unphysical solutions are examined in detail, with taking the possibility of Jordan blocks of the matrix $\mathsf{N\,S}$ into consideration. Two types of duality of eigenvectors and basis vectors of the Jordan blocks are pointed out and explored, which disclose many basic properties of the RPA solutions.