Dynamical polarizability of particles in triangular potential wells: exact results and Airy zero sum rules

Source abstract

Abstract Linear potentials feature in countless physical models of both experimental and theoretical relevance, a prime example being triangular quantum wells formed at material interfaces with built-in electric fields. Yet, the eigenstates contain non-elementary Airy functions Ai ( x ) with associated eigenvalues that can only be found numerically. We show here that the dynamical polarizability describing the optical response of triangular quantum wells can be computed exactly yielding simple and compact expressions. Both single- and double-sided triangular quantum wells are analysed, while imposing either Dirichlet or Neumann boundary conditions for the former. Upon Taylor-expanding the frequency dependence, the polarizabilities produce infinitely many exact sum rules among the zeros of Ai ( x ) and Ai ′ ( x ) . As an application, sum rules have been utilized to compute exact relativistic energy corrections.