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In equation (7) of

http://arxiv.org/pdf/gr-qc/9604038.pdf

they consider a Sturm-Liouville problem of the form

-(Rr')'-(8/R^{5})r = ω²r.

with R(x) a positive function on (0, +∞) and r(x) the eigenfunction with eigenvalue ω². The goal is to show that there are negative eigenmodes.

Equation (7) in this paper is not the sturm-liouville problem above. The operator is multiplied from the left by R(x) in order to allow an explicit solution to be found. This explicit solution to the new problem turn out to have negative eigenvalue, and it is concluded that the original problem also has a negative mode. I wonder why this can be done. Perhaps multiplication of the operator by a positive function does not change the sign of eigenvalues?

Can someone see what's going on here? Perhaps point me to a reference/proof for the relevant property of Sturm-Liouville problems?

Thanks,

A_B

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# Question about a property of Sturm-Liouville problems

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