I am trying to solve the radial Schroedinger equation in the form: \begin{align} \frac{\partial^2 P}{\partial r^2} + 2 \left(E + \frac{Z}{r} - \frac{l(l+1)}{2r^2}\right) P = 0 \end{align} Where $E = \frac{1}{2n^2}$, $n = 1,2,3, \dotsc$, $Z=1$ and $l = 0,1,2, \dotsc, n-1$. In the derivation of the radial equation one finds that the solutions can have the following asymptotic behavior: \begin{align} \lim_{r \rightarrow 0} P(r) &= \begin{cases} r^{-l} \\ r^{l+1} \end{cases} \\ \lim_{r\rightarrow \infty} P(r) &= \begin{cases} e^{\lambda r} \\ e^{-\lambda r} \end{cases}, \lambda = \sqrt{-2E} = \frac{1}{n} \end{align}


To find the physical (regular) solution, which is normalizable, one usually proceeds with the ansatz $P(r) = r^{l+1} e^{-\lambda r} F(r)$, $\lambda = \sqrt{-2E} = \frac{1}{n}$. This leads to a Kummer differential equation for F(z):

\begin{align} &z F''(z) + (c-z)F'(z) - a F(z) = 0\\ &a = (l+1)-n, c = 2(l+1), z=\frac{2}{n} r \end{align}

Solutions to this equation are called confluent hypergeometric functions or Kummer functions of the first M(a,c,z) and second kind U(a,c,z) (which is sometimes also referred to as Tricomi's functions).

I am concerned with the irregular solution, which should not be normalizable at it diverges at the origin (and for large r). However, the chosen ansatz leads to negative or zero parameter $a$. From my understanding, for example the irregular solution corresponding to the 1s-orbital $(n=1, l=0)$ should be expressible as $Q(r) := r e^{-r} U(0, 2, 2r)$. But $U(0, 2, 2r) = 1$. Hence, this is not a linearly independent solution to $X(r) = r e^{-r} M(0,2,2r) = r e^{-r}$.

  • How can one construct the second linear independent solution to this parameters of the Kummer equation?
  • I thought the choice of the ansatz, using the asymptotic behaviour, should be arbitrary (despite a particular choice might be advantageous) as in the end there is a second order ode which has two solutions. Other choices lead to equally problematic (?!) parameters, except the choice $P(r) = r^{l+1} e^{-\lambda r} F(r)$, for which one gets $a=n+l+1$, $b=2(l+1)$ and $z = -\frac{2}{n}r$. For this choice I found a irregular solution, however, the negative argument is kind of problematic. Despite not being implemented in a lot of numerical packages to evaluate the U function, consulting Wolfram Alpha or for example the mpmath package in python the U function is imaginary in the regime of interest (1s: $U(2,2,-2) \approx -0.3 + i 1.155$). I consulted Morse & Feshbach [1] for an expression, but the derivation of a formula for integer $c$ starts with formula (5.3.59) under the restriction $0<\phi<\pi$ which I interpreted as being defined as $z=|z| e^{i\phi}$ and, hence, the exclusion of negative arugments (?!). The formula in the end leads to something similar to DLMF 13.2.9. where I can not find such restrictions.
    • What is the right expression for irregular function U(a,c,z) in this parameter regime of the Kummer equation?
    • Why is it complex for integer a (I suspect the logarithm) and how does it satisfy the Wronskian $PQ' - P' Q = 1$ with this non zero imaginary part (I did a short test and it seemed not to hold, but maybe I was not careful enough)?

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