Spectral theory for transition operators of one-dimensional symmetric Lévy process killed upon hitting the origin is studied. Under very mild assumptions, an integral-type formula for eigenfunctions is obtained, and eigenfunction expansion of transition operators and the generator is proved. As an application, and the primary motivation, integral fomulae for the transition density and the distribution of the hitting time of the origin are given in terms of the eigenfunctions.
"Spectral theory for symmetric one-dimensional Lévy processes killed upon hitting the origin." Electron. J. Probab. 17 1 - 29, 2012. https://doi.org/10.1214/EJP.v17-2013