Question about a property of Sturm-Liouville problems

[A_B]

Hi,

In equation (7) of
http://arxiv.org/pdf/gr-qc/9604038.pdf
they consider a Sturm-Liouville problem of the form
-(Rr')'-(8/R⁵)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

 

[jvicens]

Dear A_B,

Thank you for bringing this to our attention. It is true that the operator in equation (7) is not exactly the same as the Sturm-Liouville problem mentioned in the paper. However, as you correctly noted, multiplying the operator by a positive function does not change the sign of eigenvalues. This is a well-known property of Sturm-Liouville problems and can be proven using the Rayleigh quotient.

The Rayleigh quotient is a mathematical tool used to find the eigenvalues of a Sturm-Liouville problem. It states that the eigenvalues are the critical points of the quotient, and these critical points are either positive or negative. Therefore, multiplying the operator by a positive function does not change the sign of the eigenvalues, as it does not change the critical points of the quotient.

Additionally, the explicit solution found for the new problem also has negative eigenvalues, which further supports the conclusion that the original problem has negative eigenvalues as well. This is a common approach in solving Sturm-Liouville problems, where a simpler problem is solved in order to gain insight into the original problem.

I hope this clarifies the situation for you. If you would like to learn more about Sturm-Liouville problems and their properties, I recommend checking out the book "Introduction to Partial Differential Equations" by Peter Olver.