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

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.