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 $1/2$.
In the literature, there are a lot of results concerning more complicated singular Schrödinger eigenvalue problems. More generally, I am looking at linear systems of ODEs with singular coefficients such as the coefficient in the example given above.
The problem given above can simply be thought of as the limit of smooth systems where you would replace the Dirac function by a parameter dependent function, which converges to the Dirac function when the parameter goes to zero. So, one would think that the eigenvector goes to the eigenvector given above as the parameter goes to zero.
My question is this: Are there results, known examples, or references in the literature dealing with singular systems which have solutions, eigenvectors, or behaviors, which strictly appear in the singular limit and which are not obtained by taking the limit of smooth systems. In other words, I would like to know if there is something more to those systems involving Dirac functions than being the limit of smooth systems.
