As is known, a nonnegative-definite Hamiltonian H that has a tridiagonal matrix representation in a basis set allows defining forward (and backward) shift operators that can be used to determine the matrix representation of the supersymmetric partner Hamiltonian H (+) in the same basis. We show that if the Hamiltonian is also shape-invariant, then the matrix elements of the Hamiltonian are related such that the energy spectrum is known in terms of these elements. It is also possible to determine the matrix elements of the hierarchy of supersymmetric partner Hamiltonians. Moreover, we derive the coherent states associated with this type of Hamiltonian and illustrate our results with examples from well-studied shape-invariant Hamiltonians that also have a tridiagonal matrix representation.

中文翻译：

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形状不变的三对角哈密顿量的性质

众所周知，在基组中具有三对角矩阵表示的非负定哈密顿量 H 允许定义向前（和向后）移位算子，这些算子可用于确定超对称伙伴哈密顿量 H (+) 的矩阵表示。基础。我们表明，如果哈密顿量也是形状不变的，那么哈密顿量的矩阵元素是相关的，因此根据这些元素，能谱是已知的。也可以确定超对称伙伴哈密顿量的层次结构的矩阵元素。此外，我们推导出与这种类型的哈密顿量相关的相干状态，并用来自经过充分研究的形状不变哈密顿量的例子来说明我们的结果，这些哈密顿量也具有三对角矩阵表示。