Quantum square well with logarithmic central spike

Linear square-well Schrödinger equation endowed with a singular logarithmic spike in the origin is studied. The study is methodical, motivated by the problem of non-gausson states ψ_n(x), n neq 0 generated by nonlinear Schrödinger equations. Once the state-dependent self-interaction term is chosen logarithmic, sim -g ln\[ψ^*_n(x)ψ_n(x)\], the nonlinear model develops the puzzling logarithmic (i.e., weakly singular) repulsive barriers near the nodal zeros of ψ_n(x) at n neq 0. In our linearized approach the weak-coupling regime is shown reliably described by the routine Rayleigh-Schrödinger 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.