So say I have a differential equation of the form: $$ \left(\alpha \frac{d^2}{dx^2}+fx^2 \right)y(x)=\lambda y(x) $$ Whose solutions are known (a Gaussian multiplying a Hermite polynomial.)
I am now curious how adding a delta function effects the solutions to this equation. So I am interested in figuring out the solutions to: $$ \left(\alpha \frac{d^2}{dx^2}-g \delta(x)+fx^2 \right)y(x)=\lambda y(x) $$
I can see that to the left and right of the origin, our original solution is still completely valid. At the origin however, we have an equation that approaches negative infinity from the delta potential. I can't really see if the solutions change at all from the added function, or if they do change if I could find the new solution using the original solution.