F4 Quantum Integrable, rational and trigonometric models: space-of-orbits view

8th International Symposium on Quantum Theory and Symmetries

Turbiner, A.V.^1 ; Vieyra, J. C. Lopez^1

Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, 04510 Mexico D.F, Mexico^1

关键词: Algebraic functions;    Characteristic vectors;    Configuration space;    Differential operators;    Functions with singularities;    Laplace-Beltrami operator;    Polynomial coefficients;    Quantum Hamiltonians;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/512/1/012014/pdf
DOI  :  10.1088/1742-6596/512/1/012014

来源: IOP

【 摘 要 】

Algebraic-rational nature of the four-dimensional, F4-invariant integrable quantum Hamiltonians, both rational and trigonometric, is revealed and reviewed. It was shown that being written in F4Weyl invariants, polynomial and exponential, respectively, both similarity-transformed Hamiltonians are in algebraic form, they are quite similar the second order differential operators with polynomial coefficients; the flat metric in the Laplace-Beltrami operator has polynomial (in invariants) matrix elements. Their potentials are calculated for the first time: they are meromorphic (rational) functions with singularities at the boundaries of the configuration space. Ground state eigenfunctions are algebraic functions in a form of polynomials in some degrees. Both Hamiltonians preserve the same infinite flag of polynomial spaces with characteristic vector (1, 2, 2, 3), it manifests exact solvability. A particular integral common for both models is derived. The first polynomial eigenfunctions are presented explicitly.