Quantum square-well with logarithmic central spike

Abstract

Singular repulsive barrier V (x) = −g ln(|x|) inside a square well is interpreted and studied as a linear analogue of the state-dependent interaction L ef f (x) = −g ln[ψ * (x)ψ(x)] in nonlinear Schrödinger equation. In the linearized case, Rayleigh-Schrödinger perturbation theory is shown to provide a closed-form spectrum at the 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 = exp y which interchanges the roles of the asymptotic and central boundary conditions. 1 znojil@ujf.cas.cz 2 semoradova@ujf.cas.cz

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