# How does the Laplace operator act on a functional determinant?

Consider the functional $$n \times n$$ matrix $$D_n = \begin{pmatrix} f_{1,1} & f_{1,2} & \ldots & f_{1,n}\\ f_{2,1} & f_{2,2} & \ldots & f_{2,n}\\ \vdots & \vdots & \dots & \vdots \\ f_{n,1} & f_{n,2} & \ldots & f_{n,n} \end{pmatrix},$$ where $$f_{i,j}$$ are differentiable functions of the variables $$x_1, x_2, \ldots, x_n$$.

The partial derivatives $$\partial_i = \frac{\partial}{\partial x_i}$$ act so nicely on the determinant of the matrix $$D_n$$ $$\partial_i \left( \begin{vmatrix} f_{1,1} & f_{1,2} & \ldots & f_{1,n}\\ f_{2,1} & f_{2,2} & \ldots & f_{2,n}\\ \vdots & \vdots & \dots & \vdots \\ f_{n,1} & f_{n,2} & \ldots & f_{n,n} \end{vmatrix} \right) = \begin{vmatrix} \partial_i(f_{1,1}) & \partial_i(f_{1,2}) & \ldots & \partial_i(f_{1,n}) \\ f_{2,1} & f_{2,2} & \ldots & f_{2,n}\\ \vdots & \vdots & \dots & \vdots \\ f_{n,1} & f_{n,2} & \ldots & f_{n,n} \end{vmatrix} + \cdots + \begin{vmatrix} f_{1,1} & f_{1,2} & \ldots & f_{1,n}\\ f_{2,1} & f_{2,2} & \ldots & f_{2,n}\\ \vdots & \vdots & \dots & \vdots \\ \partial_i(f_{n,1}) & \partial_i(f_{n,2}) & \ldots & \partial_i(f_{n,n}) \end{vmatrix}.$$ I am interested in the action of the Laplace operator $$\Delta_n = \frac{\partial^2}{\partial x_1^2} + \frac{\partial^2}{\partial x_2^2} + \cdots + \frac{\partial^2}{\partial x_n^2},$$ on the determinant of the matrix $$D_n$$.

Of course, since $$\Delta_n$$ is not merely derivation, its action will not be as straightforward. However, the following examples for $$n=2,3$$ gives hope that the deviation of its action from derivations is not very significant: direct computation can show that

$$\Delta_2 \left( \begin{vmatrix} f_{1,1} & f_{1,2} \\ f_{2,1} & f_{2,2} \end{vmatrix} \right)= \begin{vmatrix} \Delta_2(f_{1,1}) & \Delta_2(f_{1,2}) \\ f_{2,1} & f_{2,2} \end{vmatrix} +\begin{vmatrix} f_{1,1} & f_{1,2} \\ \Delta_2(f_{2,1}) & \Delta_2(f_{2,2}) \end{vmatrix} +2 \begin{vmatrix} \partial_1( f_{1,1}) & \partial_1(f_{1,2}) \\ \partial_1(f_{2,1}) & \partial_1(f_{2,2}) \end{vmatrix}+2 \begin{vmatrix} \partial_2( f_{1,1}) & \partial_2(f_{1,2}) \\ \partial_2(f_{2,1}) & \partial_2(f_{2,2}) \end{vmatrix}$$

and $$\begin{gather*} \Delta_3 \left( \begin{vmatrix} f_{1,1} & f_{1,2} & f_{1,3} \\ f_{2,1} & f_{2,2} & f_{2,3}\\ f_{3,1} & f_{3,2} & f_{3,3}\\ \end{vmatrix} \right)=\begin{vmatrix} \Delta_3(f_{1,1}) & \Delta_3(f_{1,2}) & \Delta_3(f_{1,3}) \\ f_{2,1} & f_{2,2} & f_{2,3}\\ f_{3,1} & f_{3,2} & f_{3,3}\\ \end{vmatrix} +\begin{vmatrix} f_{1,1} & f_{1,2} & f_{1,3} \\ \Delta_3(f_{2,1}) & \Delta_3(f_{2,2}) & \Delta_3(f_{2,3})\\ f_{3,1} & f_{3,2} & f_{3,3}\\ \end{vmatrix} +\\ \\ +\begin{vmatrix} f_{1,1} & f_{1,2} & f_{1,3} \\ f_{2,1} & f_{2,2} & f_{2,3}\\ \Delta_3(f_{3,1}) & \Delta_3(f_{3,2}) & \Delta_3(f_{3,3})\\ \end{vmatrix} +\color{red}{2\,\begin{vmatrix} \partial_1(f_{1,1}) & \partial_1(f_{1,2}) & \partial_1(f_{1,3}) \\ \partial_1(f_{2,1}) & \partial_1(f_{2,2}) & \partial_1(f_{2,3})\\ f_{3,1} & f_{3,2} & f_{3,3}\\ \end{vmatrix}+ 2\,\begin{vmatrix} \partial_2(f_{1,1}) & \partial_2(f_{1,2}) & \partial_2(f_{1,3}) \\ \partial_2(f_{2,1}) & \partial_2(f_{2,2}) & \partial_2(f_{2,3})\\ f_{3,1} & f_{3,2} & f_{3,3}\\ \end{vmatrix}+\\ \\ + 2\,\begin{vmatrix} \partial_3(f_{1,1}) & \partial_3(f_{1,2}) & \partial_3(f_{1,3}) \\ \partial_3(f_{2,1}) & \partial_3(f_{2,2}) & \partial_3(f_{2,3})\\ f_{3,1} & f_{3,2} & f_{3,3}\\ \end{vmatrix}+ 2\,\begin{vmatrix} f_{1,1} & f_{1,2} & f_{1,3} \\ \partial_1(f_{2,1} & \partial_1(f_{2,2}) & \partial_1(f_{2,3})\\ \partial_1(f_{3,1}) & \partial_1(f_{3,2}) & \partial_1(f_{3,3})\\ \end{vmatrix} + 2\,\begin{vmatrix} f_{1,1} & f_{1,2} & f_{1,3} \\ \partial_2(f_{2,1} & \partial_2(f_{2,2}) & \partial_2(f_{2,3})\\ \partial_2(f_{3,1}) & \partial_2(f_{3,2}) & \partial_2(f_{3,3})\\ \end{vmatrix} +\\ \\ + 2\,\begin{vmatrix} f_{1,1} & f_{1,2} & f_{1,3} \\ \partial_3(f_{2,1} & \partial_3(f_{2,2}) & \partial_3(f_{2,3})\\ \partial_3(f_{3,1}) & \partial_3(f_{3,2}) & \partial_3(f_{3,3})\\ \end{vmatrix} + 2\,\begin{vmatrix} \partial_1(f_{1,1}) & \partial_1(f_{1,2}) & \partial_1(f_{1,3}) \\ f_{2,1} & f_{2,2} & f_{2,3}\\ \partial_1(f_{3,1}) & \partial_1(f_{3,2}) & \partial_1(f_{3,3})\\ \end{vmatrix}+ 2\,\begin{vmatrix} \partial_2(f_{1,1}) & \partial_2(f_{1,2}) & \partial_2(f_{1,3}) \\ f_{2,1} & f_{2,2} & f_{2,3}\\ \partial_2(f_{3,1}) & \partial_2(f_{3,2}) & \partial_2(f_{3,3})\\ \end{vmatrix} +\\ \\ + 2\,\begin{vmatrix} \partial_3(f_{1,1}) & \partial_3(f_{1,2}) & \partial_3(f_{1,3}) \\ f_{2,1} & f_{2,2} & f_{2,3}\\ \partial_3(f_{3,1}) & \partial_1(f_{3,2}) & \partial_1(f_{3,3}) \end{vmatrix}}. \end{gather*}$$

Unfortunately, with this approach, the long red tail of the formula will only continue to grow.

Question. Since the Laplacian operator is well-studied, is there a known good formula for $$\Delta_n (|D_n|)$$ when $$n > 3?$$

The determinant is the sum over all products with signed index permutation $$\sigma\in \mathit S_n$$ $$\det f = \sum_{\sigma\in \mathit S_n} \ \text{sig}(\sigma) f_{1,\sigma(1)}\dots f_{n,\sigma(n)}$$
$$\partial_{kk} \det f = \sum_{\sigma\in \mathit S_n} \ \text{sig}(\sigma) \ \sum_l \frac{\partial_{kk} f_{l,\sigma(l)}}{ f_{l,\sigma(l)}} \ f_{1,\sigma(1)} \dots f_{n,\sigma(n))}$$ $$+ \sum_{\sigma\in \mathit S_n} \ \text{sig}(\sigma) \ \sum_{l\ne m} \frac{\partial_{k} f_{l,\sigma(l)}}{f_{l,\sigma(l)}} \ \frac{\partial_{k} f_{m,\sigma(m)})}{ f_{m,\sigma(m)}} \ f_{1,\sigma(1)} \dots f_{n,\sigma(n))}$$