  Normal-incidence transmission and dispersion properties of optical
multilayers and one-dimensional stepwise potential barriers in the
non-tunneling regime are analytically investigated. The optical paths of every
constituent layer in a multilayer structure, as well as the parameters of every
step of the stepwise potential barrier, are constrained by a generalized
quarter-wave condition. No other restrictions on the structure geometry is
imposed, i.e., the layers are arranged arbitrarily. We show that the density of
states (DOS) spectra of the multilayer or barrier in question are subject to
integral conservation rules similar to the Barnett-Loudon sum rule but ocurring
within a finite frequency or energy interval. In the optical case, these
frequency intervals are regular. For the potential barriers, only non-periodic
energy intervals can be present in the spectrum of any given structure, and
only if the parameters of constituent potential steps are properly chosen.
  Abstract The integral conservation relations derived analytically have also
been verified numerically. The relations can be used in dispersion-engineered
multilayer-based devices, e.g., ultrashort pulse compressors or ultracompact
optical delay lines, as well as to design multiple-quantum-well electronic
heterostructures with engineered DOS.
