Geodesic
An Invariant Symmetric Non-Selfadjoint Differential Operator
Journal of Lie theory, Tome 12 (2002) no. 1, pp. 245-257 Cet article a éte moissonné depuis la source Heldermann Verlag

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Let $D$ be a symmetric left invariant differential operator on a unimodular Lie group $G$ of type $I$. Then we show that $D$ is essentially self-adjoint if and only if for almost all $\pi \in \widehat{G}$, with respect to the Plancherel measure, the operator $\pi(D)$ is essentially self-adjoint. This, in particular, allows one to exhibit a left invariant symmetric differential operator on the Heisenberg group, which is not essentially self-adjoint. 
@article{JLT_2002_12_1_JLT_2002_12_1_a11,
     author = {E. G. F. Thomas },
     title = {An {Invariant} {Symmetric} {Non-Selfadjoint} {Differential} {Operator}},
     journal = {Journal of Lie theory},
     pages = {245--257},
     year = {2002},
     volume = {12},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JLT_2002_12_1_JLT_2002_12_1_a11/}
}
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E. G. F. Thomas . An Invariant Symmetric Non-Selfadjoint Differential Operator. Journal of Lie theory, Tome 12 (2002) no. 1, pp. 245-257. http://geodesic.mathdoc.fr/item/JLT_2002_12_1_JLT_2002_12_1_a11/