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arXiv · 1408.1610

The $BC_{1}$ Elliptic model: algebraic forms, hidden algebra $sl(2)$, polynomial eigenfunctions

Alexander V. Turbiner
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Abstract

The potential of the $BC_1$ quantum elliptic model is a superposition of two Weierstrass functions with doubling of both periods (two coupling constants). The $BC_1$ elliptic model degenerates to $A_1$ elliptic model characterized by the Lam\'e Hamiltonian. It is shown that in the space of $BC_1$ elliptic invariant, the potential becomes a rational function, while the flat space metric becomes a polynomial. The model possesses the hidden $sl(2)$ algebra for arbitrary coupling constants: it is equivalent to $sl(2)$-quantum top in three different magnetic fields. It is shown that there exist three one-parametric families of coupling constants for which a finite number of polynomial eigenfunctions (up to a factor) occur.

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BibTeXRIS

Alexander V. Turbiner. 2014-08-07. The $BC_{1}$ Elliptic model: algebraic forms, hidden algebra $sl(2)$, polynomial eigenfunctions. https://doi.org/10.1088/1751-8113/48/9/095205

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