Non-periodic one-gap potentials in quantum mechanics

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.