A typical interpretation of the self-adjoint extensions for the free hamiltonian in a line segment is that you get a four parametric family of possible boundary conditions, to preserve unitarity. Some of them just "bounce" the wave, some others "teletransport" it from one wall to the other. So it is also traditional to imagine this segment as a circle where you have removed a point, and then you are in the mood of studying "point interactions" or generalisations of dirac-delta potentials. The topic resurfaces from time to time, but surely some old references can be digged starting from M. Carreau. Four-parameter point-interaction in 1d quantum systems. Journal of Physics A, 26:427, 1993. In some works, I quote also Seba and Polonyi.

Sometimes the extensions are linked to the question of the domain of definition for the operator and then to the existence of anomalies. Here Phys.Rev.D34: 674-677, 1986,
"*Anomalies in conservation laws in the Hamiltonian formalism*", revisited by the same autor, J G Esteve, later in Phys.Rev.D66:125013,2002 ( http://arxiv.org/abs/hep-th/0207164 ). These topics have been live for years in the university of Zaragoza; some related material, perhaps more about boundary conditions than about extensions, is http://arxiv.org/abs/0704.1084, http://arxiv.org/abs/quant-ph/0609023, http://arxiv.org/abs/0712.4353

This post has been migrated from (A51.SE)