The effect of a $\delta$ distribution potential on a quantum mechanical particle in a box

We study the effect of a $\delta$ distribution potential placed at $x_0\geq 0$ and multiplied by a parameter $\alpha$ on a quantum mechanical particle in an infinite square well over the segment $\left[-\,\frac{L}{2},\frac{L}{2}\right]$. We obtain the limit of the eigenfunctions of the time independent Schr\"{o}dinger equation as $\alpha\nearrow+\infty$ and as $\alpha\searrow-\infty$. We see how each solution of the Schr\"{o}dinger equation corresponding to $\alpha=0$ changes as $\alpha$ runs through the real line. When $x_0$ is a rational multiple of $L$, there exist solutions of the Schr\"{o}dinger equation which vanish at $x_0$ and are unaffected by the value of $\alpha$. We show that each one of these has an energy that coincides with the energy of a certain limiting eigenfunction obtained by taking $|\alpha|\to\infty$. The expectation value of the position of a particle with wave function equal to the limiting eigenfunction is $x_0$.