Let $\Delta =\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\frac{\partial^2}{\partial x_3^2}$ be the Laplacian on $\mathbb{R}^3$. Consider a self adjoint operator $H$ on complex valued functions on $\mathbb{R}^3$ $$H\psi=\Delta\psi(x) +i\sum_{p=1}^3A_p(x)\frac{\partial \psi(x)}{\partial x_p} +B(x)\psi(x),$$ where $A_i,B$ are smooth functions.
I am looking for a precise result of the following approximate form: (1) if $A_i$ and $B$ are 'small' then the discrete spectrum of $H$ is non-positive. (2) If $A_i,B$ are 'large' then the discrete spectrum of $H$ contains necessarily a positive element.
