A classical problem in quantum mechanics involving the Dirac Delta function is given by $$ y''+(\delta(x)-\lambda^2)y=0 $$ Then, to find ''bound states'', you solve on the right and find the converging solution as $x\rightarrow \infty$, then solve on the left similarly. Assume continuity of the solution $y$. The jump condition on $y'$ is found by integrating from $-\epsilon$ to $\epsilon$ and take $\epsilon\rightarrow 0$. The only value of $\lambda$ giving a bound state is then found to be $1/2$.
My issue with this is that to make this solution mathematically correct, one has to make sense of the multiplication of a continuous non-differentiable function by the delta Dirac function. In a previous question of mine, Alan U. Kennington suggested to use Radon measures. However, there must be something simpler that makes sense of the problem above by mathematically making sense of multiplying a continuous function by a Dirac delta function.
The reason I am asking this question is because I am facing a third order equation with coefficients involving the Dirac delta function. I am able to find a solution but I would like to make my computations more mathematically sound.
I also posted this question on MathOverFlow and Paul Garrett gave the answer I was looking for here.