Is there, similar to the Mehler kernel, a closed formula for the heat kernel of the heat equation associated to the Laplacian
$$ -\sum\_j \frac{d^2}{dx\_j^2} + 2\sqrt{-1} \sum\_j \lambda\_j \frac{d}{dx\_j} + \sum\_{ij} a\_{ij}x\_ix\_j$$
on $\mathbb{R}^n$? Here, the matrix $(a\_{ij})$ is supposed to be symmetric and positive definite, while the $\lambda\_j$ can be arbitrary.



Closed formula for heat kernel
Yes there is. Here is how you do it. First find an orthogonal change in variables 


$$ x\_j=\sum\_{jk} s\_{jk}y\_k $$


$(s\_{jk})$ orthogonal matrix, so that in the new coordinates we have


$$ \sum\_{i,j}a\_{ij} x\_ix\_j = \sum\_j \mu\_j^2 y\_j^2, $$


where $\mu\_j^2$ are the eigenvalues of the symmetric matrix $(a\_{ij})$. 


Note that $\newcommand{\ii}{\boldsymbol{i}}$ $\newcommand{\pa}{\partial}$


$$\frac{\pa}{\pa x\_k}=\sum\_j\frac{\pa y\_j}{\pa x\_k}\frac{\pa }{\pa y\_j} =\sum\_j t\_{jk}\frac{\pa}{\pa y\_j}, $$


where $(t\_{jk})$ is the inverse of the orthogonal matrix $(s\_{jk})$ so that $t\_{jk}=s\_{kj}$.


Then for some real numbers $r\_j$ 


$$ -\sum\_j\frac{\pa^2}{\pa x\_j^2}+2\ii\sum\_j\lambda\_j\frac{\pa }{\pa x\_j}+\sum\_{i,j}a\_{ij}x\_ix\_j $$


$$ =-\sum\_j\frac{\pa^2}{\pa y\_j^2} +2\ii\sum\_j r\_j\frac{\pa}{\pa y\_j} +\sum\_j \mu\_j^2 y\_j^2 $$


$$=\underbrace{\sum\_j \left(\ii\frac{\pa}{\pa y\_j}+r\_j\right)^2 +\sum\_j\mu\_j^2y\_j^2-\sum\_j r\_j^2}\_{=: L}. $$


Next set


$$ R^2 :=\sum\_j r\_j^2, \;\; w(t,y) := R^2t +\sum\_j \ii r\_j y\_j, $$


$$ L\_0 :=\sum\_j\left(-\frac{\pa^2}{\pa y\_j^2} +\mu\_j^2y\_j^2\right), $$


and observe that


$$\pa\_t +L =e^{w}(\pa\_t+L\_0) e^{-w}. $$


Thus, if $K$ is a fundamental solution of $\pa\_t+L\_0$, then


$$(\pa\_t +L) (e^{w} K) = e^{w} (\pa\_t+L\_0)K=e^{w} \delta\_0=\delta\_0 $$


so that $e^{w}K$ is a fundamental solution of $\pa\_t+L$. You can find $K$ using Mehler's formula.


