Conditions for the self-adjointness of a quasi-elliptic operator

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.