Geodesic
Conditions for the self-adjointness of a~quasi-elliptic operator
Matematičeskie zametki, Tome 20 (1976) no. 5, pp. 709-716

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We prove the following theorem for the operator $L=\sum_{k=1}^n(-1)^{m_k}D_k^{2m_k}+q$ considered in $L_2(R^n)$ (the $m_k$ are natural numbers): If $q(x)\ge-C\max\limits_k|x_k|^{\frac1{1-1/2m_k}}$ ($C>0$) for sufficiently large $|x|$, then L is a self-adjoint operator. 
@article{MZM_1976_20_5_a9,
     author = {M. G. Gimadislamov},
     title = {Conditions for the self-adjointness of a~quasi-elliptic operator},
     journal = {Matemati\v{c}eskie zametki},
     pages = {709--716},
     publisher = {mathdoc},
     volume = {20},
     number = {5},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1976_20_5_a9/}
}
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M. G. Gimadislamov. Conditions for the self-adjointness of a~quasi-elliptic operator. Matematičeskie zametki, Tome 20 (1976) no. 5, pp. 709-716. http://geodesic.mathdoc.fr/item/MZM_1976_20_5_a9/