Quantum square well with logarithmic central spike

Linear square-well Schr"{o}dinger equation endowed with a singular logarithmic spike in the origin is studied. The study is methodical, motivated by the problem of non-gausson states ${\psi }_n(x)$, $n\mathrm{\ne }0$ generated by nonlinear Schr"{o}dinger equations. Once the state-dependent self-interaction term is chosen logarithmic, $\sim -g\ln [{\psi }_n^{\ast }(x){\psi }_n(x)]$, the nonlinear model develops the puzzling logarithmic (i.e., weakly singular) repulsive barriers near the nodal zeros of ${\psi }_n(x)$ at $n\mathrm{\ne }0$. In our linearized approach the weak-coupling regime is shown reliably described by the routine Rayleigh-Schr"{o}dinger perturbation theory. It even provides the first-order picture of the spectrum in closed-form. Beyond the weak-coupling regime an amendment of the unperturbed Hamiltonian is recommended. Finally, an analytic insight into the nature of the singularity at $x=0$ is obtained, in a non-perturbative setting, after the change of variables $x=\exp y$.