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 author's webpage. 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.

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.