  The Swift-Hohenberg equation describes an instability which forms
finite-wavenumber patterns near onset. We study this equation posed with a
spatial inhomogeneity; a jump-type parameter that renders the zero solution
stable for $x<0$ and unstable for $x>0$. Using normal forms and spatial
dynamics, we prove the existence of a family of steady-state solutions that
represent a transition in space from a homogeneous state to a striped pattern
state. The wavenumbers of these stripes are contained in a narrow band whose
width grows linearly with the size of the jump. This represents a severe
restriction from the usual constant-parameter case, where the allowed band
grows with the square root of the parameter. We corroborate our predictions
using numerical continuation and illustrate implications on stability of
growing patterns in direct simulations.

Pattern Formation and Solitons