Quantum square-well with logarithmic central spike

Singular repulsive barrier V (x) = −gln(|x|) inside a square-well is interpreted and studied as a linear analog of the state-dependent interaction ℒeff(x) = −gln[ψ∗(x)ψ(x)] in nonlinear Schrodinger equation. In the linearized case, Rayleigh–Schrodinger perturbation theory is shown to provide a closed-form spectrum at sufficiently small g or after an amendment of the unperturbed Hamiltonian. At any spike strength g, the model remains solvable numerically, by the matching of wave functions. Analytically, the singularity is shown regularized via the change of variables x = expy which interchanges the roles of the asymptotic and central boundary conditions.