A functional model for the Lie algebra $\operatorname{ISO}(1,1)$ of linear non-self-adjoint operators

A functional model is constructed for the Lie algebra $\operatorname{ISO}(1,1)$ of linear non-self-adjoint operators subject to the commutation relations $[A_1,A_2]=0$, $[A_1,A_3]=iA_2$, $[A_2,A_3]=iA_1$. The construction is based on a non-Abelian generalization of the Lax–Phillips scattering scheme on the group of transformations of the pseudo-Euclidean plane preserving the quadratic form $x^2-y^2$.