Diagonal of Green's Function

I am looking to numerically calulate the diagonal of Green's function. I am interested in Green's functions of elliptic PDEs and in those that arise from stochastic processes (discrete and continuous). I am aware of the connection - that is why I am interested in whatever I can find. Most literature I found is concerned with showing the diagonal is not regular in some sense and with bounds on it.

Let $G$ be the Green's function of $(-\Delta + \kappa)^2$ ($\kappa$ a constant killing rate) on $\Omega \subseteq \mathbb{R}^d, d \leq 3$. This operator needs two boundary conditions on $\Omega$. Take these conditions to be any of the standard three homogeneous boundary conditions taken with each application of the operator.

Is there a PDE that $G(x,x)$ satisfies? The probabilistic interpretation of the of Green's function is the expected number of visits to the start point (at least for random walks on graphs). I expect the diagonal to be harmonic if $k=0$. Is that true? Reasonable?

Here's what I've tried: For $d=1$, taking $y(x) = x$ and differentiating $\frac{d^2 G(x,y(x))}{dx^2}$, I get a term $\frac{\partial^2 G}{\partial x \partial y}|_{y=x}$ which I don't know what to do with. Using Mercer's theorem (assuming eigenfunctions are real, at least for now):

$$G(x,y) = \sum_n \lambda_{n} \Psi_{n}(x) \Psi_n(y)$$

I get

$$\frac{d^2 G(x,x)}{dx^2} = \sum_n \lambda_{n} \frac{d^2 \Psi_n^2(x)}{dx^2}\\ % % =2\sum_n \lambda_{n} \bigg (\Psi_n''(x)\Psi_n(x) + [\Psi_n'(x)]^2 \bigg )\\ % % =2\frac{\partial ^2 G(x,y(x))}{\partial x^2} + \sum_n \lambda_{n} [\Psi_n'(x)]^2\\$$

which is (not surprisingly) pretty much the same problem. I considered taking an eigenvalue expansion of $(-\Delta + \kappa)^{-1}$ but I'm hoping for a better approach.

References I know: The authors of Diagonals of Green's functions and applications consider Green's functions for Sturm-Liouville operators that look similar to the operator I described above. They consider only one dimension - time. I don't know if this can be genralized to more than one spatial dimension. They do present a differential equation for the diagonal of the corresponding Green's function. However, their operator has non-homogeneous Dirichlet BC. They say their derivation was originally proposed in Semilinear Elliptic Equations Involving Critical Sobolev Exponents but I couldn't find any reference to diagonals in the latter.

• Consider an (elliptic) differential operator $D(\partial)$ of degree $k$ with constant coefficients on $\mathbb{R}^n$ (e.g., a power of the Laplacian). Its Green function $G(x,y)=g(x-y)$ is the Fourier transform of $1/D(ip)$, interpreted as a distribution. Roughly speaking, for large $|p|$, it will behave like $|p|^{-k}$, which translates to $g(x) \sim |x|^{n-k}$ for small $|x|$, which is generic behavior even on bounded domains and with non-constant coefficients. In other words, unless $k>n$, you should expect $G(x,x)=g(0)=\infty$. – Igor Khavkine May 24 '16 at 1:26
• @IgorKhavkine thanks, you are absolutely right. I should have phrased my question better, since I am specifically interested in cases where $k > n$. I will change my question accordingly. – Yair Daon May 24 '16 at 3:33