Abstract: We show that all self-adjoint extensions of semi-bounded Sturm--Liouville operators with general limit-circle endpoint(s) can be obtained via an additive singular form bounded self-adjoint perturbation of rank equal to the deficiency indices, say d=1 or 2. This characterization generalizes the well-known analog for semi-bounded Sturm--Liouville operators with regular endpoints. Explicitly, every self-adjoint extension of the minimal operator can be written as <br><br>
    A_{Theta} = A_0 + B Theta B*,
<br><br>where A_0 is a distinguished self-adjoint extension and Theta is a self-adjoint linear relation in C^d. The perturbation is singular in the sense that it does not belong to the underlying Hilbert space but is form bounded with respect to A_0, i.e.  it belongs to H_{-1}(A_0). The construction of a boundary triple and compatible boundary pair for the symmetric operator ensure that the perturbation is well-defined and self-adjoint extensions are in a one-to-one correspondence with self-adjoint relations Theta.
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As an example, self-adjoint extensions of the classical symmetric Jacobi differential equation (which has two limit-circle endpoints) are obtained and their spectra is analyzed with tools both from the theory of boundary triples and perturbation theory.