  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.
