“Fall to the center” in quantum mechanics

“Fall to the center” is studied for attractive potentials that are singular as $r\to 0$. In this case, specification of the Hamiltonian $H$ is not sufficient to determine uniquely the physical quantities, i.e., the energy levels, the scattering length, the $S$ matrix, etc. In order to make the problem mathematically correct, one must also introduce a further constant $\gamma$, into the theory. Physically, $\gamma$ is the scattering phase at the point $r=0$; phenomenologically, it takes into account, the cutoff of the potential at small distances. Mathematically, the specification of $\gamma$ determines the choice of the self-adjoint extension of the formally Hermitian operator $H$. A number of specific potentials are studied for which the Schrödinger equation can be solved analytically. These examples are used to show how the different physical quantities depend on $\gamma$. Fall to the center is discussed for the case of the one-dimensional $N$-body problem.