I am considering the following ODE
\begin{equation}
\begin{split}
&\frac{d^2}{dy^2}u + \frac{\alpha}{(1+y^2)^{\frac{r}{2}}}u = \delta(y)\\
&\lim\_{|y|\to \infty}u(y) = 0.
\end{split}
\end{equation}
Here $\alpha$ is a constant and $r > 0$ and $\delta(y)$ is the Dirac delta function. Is it possible to solve this ODE? I expect the solution to be singular in $\alpha$ as $\alpha$ approaches zero. If we can't get the explicit expression of the solution, can we analyze its singularity in $\alpha$ asymptotically as $\alpha \to 0$? For example, when $r = 0$, the solution behaves like $\frac{e^{-|y|\sqrt{-\alpha}}}{\sqrt{-\alpha}}$ which is singular in $\alpha$.



How to solve the following ODE with a parameter?
This is essentially an elliptic second-order ODE in $(0,\infty)$:
$$ (1 + y^2)^{r/2} u''(y) = (-\alpha) u(y) , $$
with boundary conditions $u'(0) = -1$ and $u(\infty) = 0$. This kind of problems are well-studied, the corresponding theory is known as *Krein's spectral theory of strings*.


The value $h(\alpha)$ of $u$ at $0$ for the given parameter $\alpha$ is the corresponding *spectral function*, and $h(-\alpha)$ is a Stieltjes function of $\alpha$ (other names: Nevanlinna–Pick function, Herglotz function). The relation between the coefficient (here: $(1+y^2)^{-r/2}$, roughly known as the *string*) and the spectral function $h$ is not explicit, but some comparison results are known.


I do not have time now to search for the relevant results, but I would start by looking at the following paper:


* S. Kotani and S. Watanabe, *Krein's spectral theory of strings and generalized diffusion processes*, in: Functional Analysis in Markov processes (Katata/Kyoto, 1981), Lecture Notes in Math. 923, Springer, Berlin, 1982, 235–259, [DOI:10.1007/BFb0093046](https://doi.org/10.1007/BFb0093046).


Another excellent reference with a chapter on Krein's theory is


* R. Schilling, R. Song, Z. Vondraček, *Bernstein Functions: Theory and Applications*. De Gruyter, Studies in Math. 37, Berlin, 2012, [DOI:10.1515/9783110269338](https://doi.org/10.1515/9783110269338).




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*Edit: some additional details.*


Strictly speaking, our case corresponds to the *string* $m(dy) = (1 + y^2)^{-r/2} dy$, which is commonly identified with its distribution function $$m(y) = m([0, y)) = \int\_0^y (1 + s^2)^{-r/2} ds.$$


The spectral function can be defined in a number of equivalent ways: one of them inolves the integral of $(\phi\_N)^{-2}$, where $\phi\_N$ is the solution with "Neumann" initial condition $\phi\_N(0) = 1$, $\phi\_N'(0) = 0$. The one I am most familiar with defines it to be the reciprocal of $-\phi'(0)$, where $\phi$ is the solution with boundary conditions $\phi(0) = 1$ and $\phi(\infty) = 0$. This is clearly equivalent to the definition as $-u'(0)$ with $u$ as in the statement of the problem.


I failed once to find a simple reference for the above calculation in analytical terms, so together with Jacek Mucha we included a very brief discussion in the appendix to our paper (see Section A.3 therein):


* J. Mucha, M. Kwaśnicki, *Extension technique for complete Bernstein functions of the Laplace operator*, J. Evol. Equ. 18(3) (2018): 1341–1379 [DOI:10.1007/s00028-018-0444-4](https://doi.org/10.1007/s00028-018-0444-4).


