Group-Theoretical Origin of the Sectoral-Tesseral-Zonal Trichotomy in Spherical Harmonics
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912785317756928 |
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| author | Bakr, Mustafa Amari, Smain |
| author_facet | Bakr, Mustafa Amari, Smain |
| contents | The spherical harmonics $Y_\ell^m$ fall into three families -- sectoral ($\ell = |m|$), tesseral ($\ell > |m| > 0$), and zonal ($m = 0$) -- which exhibit fundamentally different behaviour under analytic continuation to non-integer parameters. We demonstrate that this trichotomy has a natural explanation in the representation theory of SO(3). Sectoral harmonics correspond to highest-weight vectors annihilated by the raising operator $L_+$; this annihilation condition reduces to a first-order differential equation admitting solutions for any real $m > 0$, independent of representation-theoretic constraints. Tesseral harmonics arise from the full ladder algebra acting on highest-weight states; for non-integer $m$, this construction yields tesseral modes at $ν= m + k$ for positive integer $k$, with the hypergeometric series terminating when $ν- m$ is a non-negative integer. Zonal harmonics with $m = 0$ require integer $ν$ on the full sphere, but TE-polarised zonal modes survive in wedge geometries because their electric field components automatically satisfy the conducting boundary conditions. Numerical simulations of electromagnetic cavities with conducting wedges confirm these predictions quantitatively: both sectoral modes ($ν= m$) and tesseral modes ($ν= m + k$) are observed with sub-percent frequency agreement, validating the extended framework for non-integer azimuthal index. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20119 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Group-Theoretical Origin of the Sectoral-Tesseral-Zonal Trichotomy in Spherical Harmonics Bakr, Mustafa Amari, Smain Mathematical Physics The spherical harmonics $Y_\ell^m$ fall into three families -- sectoral ($\ell = |m|$), tesseral ($\ell > |m| > 0$), and zonal ($m = 0$) -- which exhibit fundamentally different behaviour under analytic continuation to non-integer parameters. We demonstrate that this trichotomy has a natural explanation in the representation theory of SO(3). Sectoral harmonics correspond to highest-weight vectors annihilated by the raising operator $L_+$; this annihilation condition reduces to a first-order differential equation admitting solutions for any real $m > 0$, independent of representation-theoretic constraints. Tesseral harmonics arise from the full ladder algebra acting on highest-weight states; for non-integer $m$, this construction yields tesseral modes at $ν= m + k$ for positive integer $k$, with the hypergeometric series terminating when $ν- m$ is a non-negative integer. Zonal harmonics with $m = 0$ require integer $ν$ on the full sphere, but TE-polarised zonal modes survive in wedge geometries because their electric field components automatically satisfy the conducting boundary conditions. Numerical simulations of electromagnetic cavities with conducting wedges confirm these predictions quantitatively: both sectoral modes ($ν= m$) and tesseral modes ($ν= m + k$) are observed with sub-percent frequency agreement, validating the extended framework for non-integer azimuthal index. |
| title | Group-Theoretical Origin of the Sectoral-Tesseral-Zonal Trichotomy in Spherical Harmonics |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2512.20119 |