Eigenfunctions of $L = \nabla^2 + \delta(x-x_o)$ I am looking for the eigenfunctions and eigenvalues of
$L = \nabla^2 + \delta(x-x_0)$, $\quad x \in \mathbb{R}^n, \quad n = 1, 2.$
The idea is to solve $(\nabla^2 + \delta(x-x_0))p(x) = -k^2p(x)$ $\quad(1).$
I have found nothing similar to this in the literature, any help is more than welcome.
 A: Since the equation is translation invariant, we may set $x_0=0$ with no loss of generality. When $n=1$, the eigenproblem is
$$\tag{1}
p''(x)+\delta(x)p(x)=-k^2p(x)
$$
Integrating (1) over a vanishing region around $x=0$ leads to the derivative discontinuity condition
$$\tag{2}
p'(0^+)-p'(0^-)+p(0)=0
$$
Continuity of $p$ at $x=0$ requires $p(0^+)=p(0^-)$. A direct approach would be to solve (1) in the regions $x<0$ and $x>0$ then patch the solutions together using (2) and the continuity condition. Instead, let us make an ansatz for the eigenfunctions of (1) by noting that for $x\neq0$ the solutions are sinusoids and that we can consider the even $P_k$ and odd $Q_k$ solutions separately
$$\tag{3}
P_k(x)=\sin(k|x|+\alpha_k)\\
Q_k(x)=\sin(kx+\beta_k)
$$
Substituting (3) into (1)  we find the constants
$$\tag{4}
\alpha_k=\tan^{-1}(-1/2k)\\
\beta_k=0
$$
When $n\geq 2$, it is not so simple. One needs to work with a regularized version of the delta (as done here). For example, with $n=2$ in polar $(r,\phi)$ co-ordinates, one choice is
$$\tag{5}
\delta_R(r)=\frac{H(R-r)}{\pi R^2}
$$
Then we would need to solve
$$\tag{6}
\frac{\partial^2 p}{\partial r^2}+\frac{1}{r}\frac{\partial p}{\partial r} + \frac{\partial^2 p}{\partial \phi^2} + \delta_R(r) p =-k^2 p
$$
Then, at the end of the calculation, we study the solutions as $R\to 0$. Since this is done in the linked paper, I will briefly describe a method:

*

*Separate variables $p(r,\phi)=f(r)g(\phi)$, substitute into (6) and get two ODEs.

*The ODE for $g(\phi)$ will be simple and have solutions $g(\phi)=e^{im\phi}$ with integer $m$.

*The ODE for $f(r)$ needs to be further split into regions $r<R$ and $r>R$. Call these solutions $f_-$ and $f_+$. They will be Bessel functions.

*Enforce continuity of $f$ and its derivative at $r=R$ so that $f_-(R)=f_+(R)$ and $\partial_r f_-(R)=\partial_r f_+(R)$.

