I've read for a regular Sturm-Liouville problem for each eigenvalue there corresponds unique eigenfunction. For periodic Sturm Liouville problem Which of the following are true? Each eigenvalue of (periodic Sturm Liouville problem) corresponds to 1. one eigenfunction 2. two eigenfunctions 3. two linearly independent eigenfunctions 4. two orthogonal eigenfunctions What are the Properties of eigenvalues and eigenfunctions of periodic Sturm Liouville problem? Are these depend on boundary conditions or same for all periodic Sturm Liouville problems?
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This isn't an answer, but evidently I don't have enough points to leave a "comment." I don't have a reference on Sturm-Liouville problems handy. However option (2) doesn't make sense. If there is one eigenfunction, there are infinitely many for the same eigenvalue: just take your eigenfunction $u$ and multiply by any non-zero real $\alpha$. Also, (3) implies (4): if there are two linearly independent eigenfunctions, you can produce two orthogonal functions (spanning the same subspace) using a Gram-Schmidt type argument.
Hope this clarifies a couple things.