We introduce the self-adjoint Hamiltonian of a particle in dimension one subject to a point or delta interaction. We characterize the spectrum, compute proper, and generalized eigenfunctions and prove the eigenfunction expansion theorem. These results are applied to obtain the long time behavior of the unitary group generated by the Hamiltonian and to study the scattering problem in full detail. Finally, we consider the case of two point interactions and we study eigenvalues and eigenvectors in the semiclassical regime, showing the typical behavior occurring in the double well problem.
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