Quantum mechanics of one-dimensional motion in a field with the singularity $\lambda|x|^{-\nu}$

The one-dimensional motion of a particle in a field with singularity $\lambda|x|^{-\nu}$, $0<\nu<2$ and $\nu=2$, $-1/4<\lambda<3/4$ is investigated quantum mechanically. A physically acceptable self-adjoint extension of the Hamiltonian is found. A perturbation theory is constructed for a confining even smooth potential. It is shown that in this case matrix elements of the perturbation and Rayleigh–Schrödinger coefficients exist only for $\nu<3/2$. A way of calculating transmission coefficients for an asymptotically free potential is found. Examples of exact solutions $\nu=1$ and $\nu=2$ are given.