Analytically solvable quasi-one-dimensional Kronig-Penney model

Abstract

We generalize the textbook Kronig-Penney model to realistic conditions for a quantum-particle moving in the quasi-one-dimensional (quasi-1D) waveguide, where motion in the transverse direction is confined by a harmonic trapping potential. Along the waveguide, the particle scatters on an infinite array of regularized delta potentials. Our starting point is the Lippmann-Schwinger equation, which for quasi-1D geometry can be solved exactly, based on the analytical formula for the quasi-1D Green's function. We study the properties of eigen-energies as a function of particle quasi-momentum, which form band structure, as in standard Kronig-Penney model. We test our model by comparing it to the numerical calculations for an atom scattering on an infinite chain of ions in quasi-1D geometry. The agreement is fairly good and can be further improved by introducing energy-dependent scattering length in the regularized delta potential. The energy spectrum exhibits the presence of multiple overlapping bands resulting from excitations in the transverse direction. At large lattice constants, our model reduces to standard Kronig-Penney result with one-dimensional coupling constant for quasi-1D scattering, exhibiting confinement-induced resonances. In the opposite limit, when lattice constant becomes comparable to harmonic oscillator length of the transverse potential, we calculate the correction to the quasi-1D coupling constant due to the quantum interference between scatterers. Finally, we calculate the effective mass for the lowest band and show that it becomes negative for large and positive scattering lengths.