The boundary condition for delta function

Question

Beginning with the Schrödinger equation for $N$ particles in one dimension interacting via a $\delta$-function potential

$$\left(-\sum_{i=1}^{N}\frac{\partial^2}{\partial x_i^2}+2c\sum_{<i,j>}\delta(x_i-x_j)\right)\psi=E\psi$$

Why the boundary condition equivalent to the $\delta$ function potential is

$$\left(\frac{\partial}{\partial x_j}-\frac{\partial}{\partial x_k}\right)\psi |_{x_j=x_{k+}}-\left(\frac{\partial}{\partial x_j}-\frac{\partial}{\partial x_k}\right)\psi |_{x_j=x_{k-}}=2c\psi |_{x_j=x_k}.$$

Answer 1

Try integrating the original differential equation over an interval $[x_k-\epsilon,x_k+\epsilon]$, then take the limit for $\epsilon\rightarrow0$.

The integral over the right-hand-side vanishes (if $\psi$ is continuous in $x_k$), the integral containing the delta function leads to the rhs of your resulting boundary condition and the integral over the second derivative leads to the first derivative terms approaching $x_k$ from above and from below (there is a discontinuity in the derivative of $\psi$ as a result of the delta function).