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There are numerous ways to estimate the eigenvalues of a radial Schrödinger equation, see http://arxiv.org/abs/math-ph/0703040 as an example. Anyhow, the formulas only cover the Schrödinger equations on a semi-infinite (truncated) integration interval, for example $$ y''(x) + \left( E + \frac{1}{x}\right)y(x) = 0 \, \, \text{ where } x \in (0, \infty). $$

Is it possible to construct formulas which estimate the eigenvalues $E$ of (for example) the following Schrödinger equation

$$ y''(x) + \left( E + \frac{1}{x}\right)y(x) = 0 \, \, \text{ where } x \in (0, 1]? $$

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  • $\begingroup$ Don't you use the same basis functions as in the semi-infinite case? $\endgroup$ – DanielSank Mar 14 '15 at 17:20
  • $\begingroup$ @Qmechanic I don't see why this should have the homework-and-exercises tag $\endgroup$ – David Z Mar 14 '15 at 17:36
  • $\begingroup$ @David Z: Why not? $\endgroup$ – Qmechanic Mar 14 '15 at 17:56
  • $\begingroup$ This isn't a homework/excercise because the reader isn't asking a specific answer, but the reader is asking a general question hence the "for example" statement. Wouldn't something like Newton's method be used here? $\endgroup$ – John M Mar 14 '15 at 18:12
  • $\begingroup$ @Qmechanic it doesn't read like a homework-like question. It sounds like a question asked out of general curiosity, not a question asked for the educational value. The process one would go through to answer this question seems unlikely to be something one would specifically study, and the answer seems more interesting than the process. $\endgroup$ – David Z Mar 14 '15 at 18:19
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The answer is yes: Obviously it is possible to solve radially symmetric Schroedinger equations, including the specialized treatment (radial component only) that you ask about. However, you will need a well-defined problem to meaningfully begin something as rigorous as an analytical (or numerical) treatment.

The differential equation you have given for the radial wavefunction is not well-defined for the restricted range of your radial corrdinate $x$ because that restricts only the radii you allow it to be used with. In addition, you need to incorporate information how the wavefunction gets restricted to that range. A natural way to achieve this is to introduce an energy barrier that is infinitely high for all radii equal to or larger than $1$, but nonexistent in the range of radii of interest to you. This affects the behavior of the differential equation and the values of the radial wavefunction. Note that this is a choice, namely one to calculate for a specific system with this physical reason for the restricted radial range of wavefunctions. Unless you are looking for answers so general as to cover all possible choices and hence apply to just about any differential equation, you will have to make some such choice first.

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