Eigenvalue Spectrum of this Operator

[Joschua_S]

Hello

I have this Hamiltonian:

$\mathcal{H} = \alpha S_{+} + \alpha^{*}S_{-} + \beta S_{z}$

with $\alpha, \beta \in \mathbb{C}$. The Operators $S_{\pm}$ are ladder-operators on the spin space that has the dimension $2s+1$ and $S_{z}$ is the z-operator on spin space.

Do you know how to get (if possible with algebraic argumentation) the eigenvalue spectrum $\sigma( \mathcal{H} )$?

This Hamiltonian describes anisotropy of g-factor.

Thanks
Greetings

 

[fresh_42]

The representations of $\mathfrak{sl}(2)\cong \mathfrak{su}(2)$ are all known. So you must simply solve a linear equation system. Here is the corresponding theorem (see Examples):
https://www.physicsforums.com/insights/lie-algebras-a-walkthrough-the-representations/