I'm solving a quantum mechanical problem, and the quantization condition requires me to solve the equation $$ U\left(\frac12(\ell+1-E), \ell+1, r^2\right) = 0, $$ where $U(a,b,z)$ is the confluent hypergeometric function of the second kind, $r>0$ is a fixed positive real number, and $\ell\geq0$ is a fixed non-negative integer; I am looking for the sequence $E_1,E_2,E_3,\ldots$ of zeros on the energy parameter $E$, as a function of the other two parameters $r$ and $\ell$.

The best I've found so far is to use the asymptotics of the zeros, DLMF (13.9.16), as the starting point in Mathematica's FindRoot function:

Block[{ℓ = 0, a},
 (*a=1/2 (1-e+ℓ);*)
      {r, (ℓ + 1 - 2 a)} /. FindRoot[
        HypergeometricU[a, 1 + ℓ, r^2]/Gamma[1 - a] == 0
        , {a, -n - 2/π r Sqrt[n] - (2 r^2)/π^2 + (ℓ + 1)/2 + 1/4}
      , {r, 0.01, 8, 0.2}]
    , {n, 1, 10}]
   , ImageSize -> 700
   , Frame -> True
   , GridLines -> All
   , FrameLabel -> {"Radius of inner boundary", "Energy eigenvalue"}
   ] /. {Line[pts_] -> {PointSize[0.01], Line[pts], Point[pts]}}

Here $a= (\ell+1-E)/2$ for convenience, generally negative, and I've added a factor of $1/\Gamma(1-a)$ to counteract the exponential growth of $U$ as a function of $E$. This works quite well for middling parameters:

These are the various zeros (i.e. different $E_n$'s as different lines) for $\ell=0$ as a function of $r$ between $0$ and $8$.

However, if I use nonzero $\ell$s (for low $r$'s) or I expand to a bigger box in $r$, this method starts bugging out:

Mathematica graphics

This brings me to my question: Are there good standard methods for solving this root-finding problem? How can I stabilize it in the cases where it's bugging out?

If those solutions are already implemented in Mathematica, then all the better!

  • $\begingroup$ One option is to find the eigenvalues of your (approximated) operator instead of finding the roots of the determinant. Another option that you could try is to approximate your function using a polynomial and find the roots of that instead, using Chebfun, for example. $\endgroup$
    – nicoguaro
    Sep 4, 2019 at 13:20
  • 1
    $\begingroup$ When I plot HypergeometricU[-e,1,r^2]/(Gamma[1+e]Exp[r^2/2]), the function $U$ itself appears to be computed incorrectly for $r=10,e>75$, so I think this failure is not due to the guess but due to HypergeometricU. $\endgroup$
    – Kirill
    Sep 5, 2019 at 17:38
  • $\begingroup$ Some limited experiments I did indicates that a preliminary application of the Kummer transformation on the Tricomi function reduces (but does not completely banish) the unruly behavior. At worst, you might have to resort to arbitrary precision. $\endgroup$
    – J. M.
    Nov 14, 2019 at 16:33

1 Answer 1


It looks like the eigenvalues depend smoothly on your $r$ parameter. Since the problem seems to be easy to solve for small values of $r$, you can compute the eigenvalues for a set of radii $r_1,r_2,r_3$, and then extrapolate each of these eigenvalues using a quadratic function to some $r^\ast>r_3$. This extrapolation should then be an excellent starting guess for the root finding problem at $r^\ast$.


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