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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2412.04022 |
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| _version_ | 1866913603799482368 |
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| author | Abraham, Stalin Bhagwat, Ameeya A. |
| author_facet | Abraham, Stalin Bhagwat, Ameeya A. |
| contents | $2\times2$ matrix polynomials of the form $P_{n}(z)= Σ^{n}_{j=0}\,σ_{j}\,z^{j}$, 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 $P_{n}(z)$ 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 $h(x,y)\geq 0$, 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 $k\in \mathbb{R}$, are shown to lie on an ellipse, hyperbola, two lines passing through the origin, or a straight line, depending on the nature of PT-symmetry of the matrix polynomial. The PT-symmetric behaviour of $P_{n}(z)$ at the zeros of the matrix polynomial is also studied. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04022 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | PT-Symmetry in $2\times 2$ Matrix Polynomials Formed by Pauli Matrices Abraham, Stalin Bhagwat, Ameeya A. Mathematical Physics $2\times2$ matrix polynomials of the form $P_{n}(z)= Σ^{n}_{j=0}\,σ_{j}\,z^{j}$, 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 $P_{n}(z)$ 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 $h(x,y)\geq 0$, 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 $k\in \mathbb{R}$, are shown to lie on an ellipse, hyperbola, two lines passing through the origin, or a straight line, depending on the nature of PT-symmetry of the matrix polynomial. The PT-symmetric behaviour of $P_{n}(z)$ at the zeros of the matrix polynomial is also studied. |
| title | PT-Symmetry in $2\times 2$ Matrix Polynomials Formed by Pauli Matrices |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2412.04022 |