Green's function for a linear PDE initial value problem

Question

For $x\in\mathbb{R}^{n}$ and $t\in[0,\infty)$, consider the linear PDE initial value problem $$\dfrac{\partial u}{\partial t} = \left(a \Delta - \dfrac{b}{|x|}\right)u, \quad u(x,0) = u_0(x)\quad\text{(given)},$$ where $\Delta$ is the standard Laplacian, $\vert\cdot\vert$ denotes the Euclidean 2-norm, and the constants $a,b>0$. I want to write its solution as $$u(x,t) = \int_{\mathbb{R}^n} K(x,y;t)u_0(y)\:{\rm{d}}y.$$

Is there a known explicit (in terms of $x,y,t,n$) expression for this kernel/Green's function $K$?

It feels like something well-known but I am having difficulty finding a reference stating/deriving such $K$. The solution may be related to time-dependent Schrödinger equation with Coulomb potential but I do not know enough background to be sure.

Answer 1

The time-dependent Schrödinger equation with Coulomb potential (hydrogen atom) has the form $$ i\frac{\partial u}{\partial t} = \left( -a^{\prime } \Delta -\frac{b^{\prime } }{|x|} \right) u $$ with $a^{\prime } ,b^{\prime} >0$, so that problem is related to yours by continuing to imaginary $t$. That problem has been treated for $n=2$ and $n=3$ in terms of the Fourier-transformed Green's function $$ K(x,y;E)=\int_{0}^{\infty } dt\, e^{iEt} K(x,y;t) $$ in I.H.Duru and H.Kleinert, Fortschritte der Physik 30 (1982) 401, available on the webpage of one of the authors. It becomes messy - for $n=2$, an integral representation of $K(x,y;E)$ is given in Eq. (35), and for $n=3$, it is given in Eq. (109). Not sure how useful this is for your purposes - they do present the extraction of (well-known) central properties of the hydrogen atom, such as wave functions, from these integral representations.