Isospectrality of spherical MHD dynamo operators: pseudo-Hermiticity and a no-go theorem

TL;DR

This paper investigates the spectral properties of spherical MHD dynamo operators, demonstrating their pseudo-Hermiticity and establishing a no-go theorem that prevents constructing isospectral classes via first-order differential intertwining operators.

Contribution

It shows that the spherical alpha^2-dynamo operator is pseudo-Hermitian and proves a no-go theorem ruling out isospectral classes constructed with first-order intertwining operators.

Findings

The operator is formally pseudo-Hermitian and J-symmetric.

An intrinsic structural inconsistency prevents the use of first-order differential intertwining operators.

A no-go theorem is established for constructing isospectral dynamo operator classes.

Abstract

The isospectrality problem is studied for the operator of the spherical hydromagnetic alpha^2-dynamo. It is shown that this operator is formally pseudo-Hermitian (J-symmetric) and lives in a Krein space. Based on the J-symmetry, an operator intertwining Ansatz with first-order differential intertwining operators is tested for its compatibility with the structure of the alpha^2-dynamo operator matrix. An intrinsic structural inconsistency is obtained in the set of associated matrix Riccati equations. This inconsistency is interpreted as a no-go theorem which forbids the construction of isospectral alpha^2-dynamo operator classes with the help of first-order differential intertwining operators.