More on the Asymmetric Infinite Square Well: Energy Eigenstates with   Zero Curvature
  We extend the standard treatment of the asymmetric infinite square well to include solutions that have zero curvature over part of the well. This type of solution, both within the specific context of the asymmetric infinite square well and within the broader context of bound states of arbitrary piecewise-constant potential energy functions, is not often discussed as part of quantum mechanics texts at any level. We begin by outlining the general mathematical condition in one-dimensional time-independent quantum mechanics for a bound-state wave function to have zero curvature over an extended region of space and still be a valid wave function. We then briefly review the standard asymmetric infinite square well solutions, focusing on zero-curvature solutions as represented by energy eigenstates in position and momentum space. 
L. P. Gilbert, M. Belloni, M. A. Doncheski, and R. W. Robinett