Some studies on quantum equivalents of non-commutative operators via commutating eigenvalue relation: PT-symmetry

TL;DR

This paper explores quantum equivalents of non-commutative operators using a universal relation involving eigenvalues, applicable across various models and disciplines, including quantum mechanics and social sciences.

Contribution

It introduces a universal relation for non-commutative operators that reproduces eigenvalues and applies broadly across different fields and models.

Findings

The relation $B^{-1} AB$ reproduces eigenvalues of $A$.

Applicable to matrices and operators in physics and social sciences.

Includes models with logarithmic potential.

Abstract

We study quantum equivalents of non-commutative operators in quantum mechanics. Any matrix "$B$" satisfying the non-commuting relation $[A,B]\neq 0$ with "$A$", can be used via $B^{-1} AB$ to reproduce eigenvalues of "$A$". This universality relation is also equally valid for any matrix in any branch of physical or social science and also any operator involving co-ordinate$(x)$ or momentum$(p)$. Pictorially this is represented in fig. 1. Many interesting models including logarithmic potential have been considered.