Suppose you have a particle in the box $[0,L]^d$, with an attractive Dirac delta potential $-\delta_{\vec w}(x)$ at $\vec w$. How do you solve the Schroedinger equation for this system?
In the case $d=1$, you know your solutions are linear combinations of $e^{ikx}$ and $e^{-ikx}$. If you assume that you have two different linear combinations on $[0,w]$ and $[w,L]$, you have four coefficients you need to find. You can enforce the vanishing at the endpoints of the interval, continuity at $w$, and the discontinuity in the first derivative that you get from integrating across $w$, in order to get enough conditions on the coefficients to be able to find the quantization condition for $k$, and then solve for the coefficients.
However, for $d>1$, this doesn't work -- the delta potential doesn't divide the box into disjoint pieces. What do you do?
