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Perturbation Theory for Singular Potentials in Quantum Mechanics

We study perturbation theory in certain quantum mechanics problems in which the perturbing potential diverges at some points, even though the energy eigenvalues are smooth functions of the coefficient of the potential. We discuss some of the unusual techniques which are required to obtain perturbative expansions of the energies in such cases. These include a point-splitting prescription for expansions around the Dirichlet (fermionic) limit of the $δ$-function potential, and performing a similarity transformation to a non-Hermitian potential in the Calogero-Sutherland model. As an application of the first technique, we study the ground state of the $δ$-function Bose gas near the fermionic limit.