Vol. 23 No. 4 (2024): Mapana Journal of Sciences
Research Articles

PT-Symmetry in 2 x 2 Matrix Polynomials Formed by Pauli Matrices

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  • Stalin Abraham,
  • Ameeya A. Bhagwat
Stalin Abraham
School of Physical Sciences, UM-DAE CEBS, University of Mumbai, Kalina Campus, Mumbai, Maharashtra, India - 400 098
Ameeya A. Bhagwat
School of Physical Sciences, UM-DAE-CEBS, Universit of Mumbai, Kalina Campus, Maharashtra, India - 400 098
DOI: https://doi.org/10.12723/mjs.71.2

Published 2024-12-23

Keywords

  • PT-Symmetric Operators,
  • Pseudo Hermitian Operators,
  • Matrix Polynomials

Copyright (c) 2024

Creative Commons License

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Abstract

2 × 2 matrix polynomials of the form Pn(z) = Σnj=0 σjzj, for the cases n = 1,2,3 are constructed, and the nature of PT-symmetry is examined across different points z = (x, y) in the complex plane. The PT-symmetric properties
of Pn(z) can be characterized by two functions, denoted by s(x, y) and h(x, y). If the trace of the matrix polynomial is
real, then the points at which it can exhibit PT-symmetry are defined by the family of curves s(x, y) = 0. Additionally,
at points where the function h(x, y) ≥ 0, the matrix polynomial exhibits unbroken PT-symmetry; otherwise, it exhibits
broken PT-symmetry. The intersection points of the curves s(x, y) = 0 and h(x, y) = k, for a given k ∈ R, are shown
to lie on an ellipse, hyperbola, two lines passing through the origin, or a straight line, depending on the nature of PTsymmetry of the matrix polynomial. The PT-symmetric behaviour of Pn(z) at the zeros of the matrix polynomial is also
studied.

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