# Completeness Relation for Tricomi Confluent Hypergeometric Function

Consider the Kummer differential equation $$\frac{d}{dz}\left[z^be^{-z}\frac{dw}{dz}\right]=az^{b-1}e^{-z}w,\quad z\in\mathbb{R}.$$ It is an eigenvalue problem of Sturm-Liouville type with weight function $W(z)=z^{b-1}e^{-z}$. The two linearly independent solutions are the Kummer function $M(a,b,z)$ and the Tricomi function $U(a,b,z)$.

My question is what type of boundary conditions (BC) does one need to impose at $z=\pm\infty$ for the usual completeness relation to hold $$\int da\,w(a,b,z)w(a,b,z')W(z) =\delta(z-z').$$ Here $w$ is the correct linear combination of $M$ and $U$ to match the BCs.

For example if we want the eigenfunctions $w$ to decay for $z\to\pm\infty$ then $w=U$ is the only choice, because $U\sim z^{-a}$ for large $z$ ($M$ explodes), and then $a$ has to be positive for this to be a proper decay. So I would naively integrate on $a\in\mathbb{R}_+$ in this case.

I also tried to look for this type of integral in tables but haven't found anything useful.

Does anybody have any ideas how to make this integral precise? Thanks.

## 1 Answer

What you are looking for is called Weyl's limit point(LP)/limit circle(LC) classification. An end point is LC if both solutions are square integrable (w.r.t . the weight function) and LP otherwise. If both endpoints are LC then you need boundary conditions and you will have a complete set of eigenfunctions. Otherwise the spectrum might have a continuous component and the eigenfunction expansion will be an integral transform. The Kummer equation is LC near $0$ for $0<b<2$ and LP otherwise. $\infty$ is always LP. However, I don't know the precise spectrum of this equation. Near $0$ you can take the Kummer function which is entire with respect to the spectral parameter, hence there is a corresponding integral transform $$\hat{f}(a) = \int_0^\infty M(a,b,x) f(x) W(x) dx$$ and you can go back via $$f(x) = \int_{-\infty}^\infty M(a,b,x) \hat{f}(a) d\rho(a)$$ where $\rho$ is the associated singular spectral measure.

For general background concerning LC/LP you could e.g. look at

http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/index.html

Depending on your background this might be however quite technical. Moreover, it only covers the case where the left endpoint $0$ is regular. For the general case you will need

http://arxiv.org/abs/1007.0136