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The Sturm Liouville theory claim that the eigenfunction of SL operator forms a orthonormal base. And I realized that most orthogonal functions are the eigenfunction of a SL operator. I wonder if the inverse proposition established: For every orthonormal base of $L^2$ , there is a Sturm-Liouville operator such that the eigenfunctions of this operator is the given base?

I guess the answer is No?

paul garrett
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Takanashi
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    https://math.stackexchange.com/questions/551480/can-we-construct-sturm-liouville-problems-from-an-orthogonal-basis-of-functions – N. S. Aug 23 '17 at 15:55
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    A very loud "no" indeed, there are lots of conditions. Already the individual functions are not at all arbitrary. For starters, they must have some smoothness or they couldn't be in the domain of a differential operator. You also can't have $f(a)=f'(a)=0$ anywhere because $f$ solves a linear second order homogeneous equation. –  Aug 26 '17 at 03:13

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