I am working on the operator $$ K_t:=-\sum_{i=1}^n\left(\frac{\partial}{\partial x^i}\right)-t\cdot\mathrm{tr}\sqrt{A^*A}+t^2\langle A^*Ax,x\rangle $$ acting on functions of $ \mathbb{R}^n $, where $ \det A\ne 0 $. I want to show that the kernel of $ K_t $ is one-dimensional and is generated by $ \exp\left(-\frac{t\lvert Ax \rvert^2}{2}\right) $.
I know that the harmonic oscillator $ H=-\nabla^2-t+t^2x^2 $ has one-dimensional kernel generated by $ \exp\left(-\frac{tx^2}{2}\right) $. But how I can apply this conclusion to the operator above?