How to solve the following ODE with a parameter?

Question

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 $\alpha$.

Answer 1

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.

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.

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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.