  We initiate the study of the conformal bootstrap using Sturm-Liouville
theory, specializing to four-point functions in one-dimensional CFTs. We do so
by decomposing conformal correlators using a basis of eigenfunctions of the
Casimir which are labeled by a complex number alpha. This leads to a systematic
method for computing conformal block decompositions. Analyzing bootstrap
equations in alpha space turns crossing symmetry into an eigenvalue problem for
an integral operator K. The operator K is closely related to the Wilson
transform, and some of its eigenfunctions can be found in closed form.
