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arXiv · 1812.10749

Properties of shape-invariant tridiagonal Hamiltonians

Hashim A. Yamani·Zouhaïr Mouayn
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Abstract

It has been established that a positive semi-definite Hamiltonian,$H$, that has a tridiagonal matrix representation in a basis set, allows a definition of forward (and backward) shift operators that can be used to define the matrix representation of the supersymmetric partner Hamiltonian $H^{\left( +\right) \text{\ }}$ in the same basis. \ We show that if, additionally, the Hamiltonian has a shape invariant property, the matrix elements of the Hamiltonian are related in a such a way that the energy spectrum is known in terms of these elements. It is also possible to determine the matrix elements of the hierarchy of super-symmetric partner Hamiltonians. Additionally, we derive the coherent states associated with this type of Hamiltonians and illustrate our results with examples from well-studied shape-invariant Hamiltonians that also has tridiagonal matrix representation.

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Hashim A. Yamani, Zouhaïr Mouayn. 2018-12-27. Properties of shape-invariant tridiagonal Hamiltonians. https://arxiv.org/abs/1812.10749

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