On representations of Lie algebras for quantized Hamiltonians (Q1372960)

The authors consider Lie algebras \(L^s\) generated by three operators \(K_0\), \(K_\pm\) with the multiplication rules \([K_+,K_-] =s \cdot K_0\), \([K_0,K_\pm] =\pm K_\pm\), where \(s\in\mathbb{R}\), \(K_0\) is a Hermitian diagonal operator and the Hermitian adjoint of \(K_-\) is equal to \(K_+\). The operators are used as a Hamiltonian model of coupled quantized harmonic oscillators \(H=K_0 +\lambda \cdot(K_+ +K_-)\), where \(\lambda\) is the coupling parameter. To solve the corresponding Schrödinger wave equation of this model a faithful matrix representation of the algebra \(L^s\) is needed. The authors prove that \(L^s\) has nontrivial representations only for \(s\geq 0\). If \(s=0\), then the representation is not faithful. If \(s>0\), then \(L^s\) has nontrivial faithful representations.