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The left-definite Hilbert spaces for singular Sturm-Liouville problems of the limit-circle type are explicitly constructed. The construction only uses an explicit form of the leftdefinite boundary conditions, together with a principle solution and a non-principle solution of the differential equation involved.

We consider the zeros of the solution 0 α (t) = E α (−t α ), 1 < α < 2, of the fractional oscillation equation in terms of the Mittag-Leffler function, and give a wholly and clarified description for these zeros. We find that the number of zeros can be any finite number: 1, 2, 3, 4, . . . , not necessarily an odd number. When the number of zeros of 0 α (t) is an even number, 0 α (t) has a critical zero. All of the values of α for which 0 α (t) has an even number of zeros constitute a countable set S. For each α ∈ (1, 2) \ S, 0 α (t) has an odd number of zeros. These results are a supplement and a perfecting for the existed related documents. We also show that the eigenvalue problems are related with the zeros of the Mittag-Leffler functions.MSC 2010 : Primary 26A33; Secondary 33E12, 34A08, 44A10

We prove that the potential q(x) of an indefinite Sturm-Liouville problem on the closed interval [a, b] with the indefinite weight function w(x) can be determined uniquely by three spectra, which are generated by the indefinite problem defined on [a, b] and two right-definite problems defined on [a, 0] and [0, b], where point 0 lies in (a, b) and is the turning point of the weight function w(x).