Talk:Sturm separation theorem
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editThe comment(s) below were originally left at Talk:Sturm separation theorem/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
| Needs more on applications (if there are important ones, change importance field) and perhaps history (where did Sturm publish this). Is there a connection with Sturm-Liouville theory or orthogonal polynomials? -- Jitse Niesen (talk) 03:19, 19 August 2007 (UTC) |
Last edited at 03:19, 19 August 2007 (UTC). Substituted at 02:36, 5 May 2016 (UTC)
Critique of proof
editThis proof of the theorem is faulty. The statement "Since u and v are linearly independent it follows that the Wronskian W[u,v] must satisfy W[u,v](x) != 0 for all x where the differential equation is defined" is not correct. The Wronskian can be zero at a finite number of points, it just cannot be zero everywhere. Take for example, the time-independent Schrodinger equation with an infinite square well, V(x) = 0 on 0<x<π, ∞ elsewhere. The solutions u(x)=sin(x) and v(x)=sin(3x) are clearly linearly independent, but their Wronskian is zero in the middle of the well at x=π/2. Other parts of the proof fall apart after this, but I wanted to make sure I did not misunderstand some of the assumptions here before proposing a major rewrite of this page. Rjones30 (talk) 14:19, 18 July 2017 (UTC)