# Properties of shape-invariant tridiagonal Hamiltonians

## TL;DR

This paper explores the properties of shape-invariant, tridiagonal Hamiltonians, showing how their spectra and supersymmetric partners can be explicitly determined, and constructs associated coherent states with illustrative examples.

## Contribution

It demonstrates that shape invariance in tridiagonal Hamiltonians allows explicit spectral and partner Hamiltonian determination, extending the understanding of their algebraic structure.

## Key findings

- Explicit relations for energy spectra of shape-invariant tridiagonal Hamiltonians.
- Method to determine matrix elements of supersymmetric partner Hamiltonians.
- Construction of coherent states for these Hamiltonians.

## 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.