  We describe a broad class of bounded non-periodic potentials in
one-dimensional stationary quantum mechanics having the same spectral
properties as periodic potentials. The spectrum of the corresponding
Schroedinger operator consists of a finite or infinite number of allowed bands
separated by gaps. In this letter we consider the simplest class of potentials,
whose spectra consist of an interval on the negative semiaxis and the entire
positive axis. The potentials are reflectionless, and a particle with positive
energy moves freely in both directions. The potential is constructed as a limit
of Bargmann potentials and is determined by a Riemann-Hilbert problem, which is
equivalent to a pair of singular integral equations that can be efficiently
solved using numerical techniques.
