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Using zero-energy states to explore how external boundaries affect the number of bound states in a quantum well

We use the condition for the existence of zero-energy eigenstates to examine how the number of bound (negative energy) states in a finite-width one-dimensional quantum well is altered by the introduction of boundaries outside of the well. We consider a variety of quantum wells including a finite square well, a triangular well, and sets of two to five Dirac delta wells. Each of these quantum wells is placed at the center of an infinite square well with variable width and the conditions for the existence of zero-energy eigenstates are determined analytically. The zero-energy conditions (ZEC) form curves in the parameter space that separate regions with different numbers of bound states. Moving across one of these curves changes the number of bound states by one. We find that, for the systems studied, introducing external boundaries changes the number of bound states by at most two. This work illustrates the usefulness of the ZEC as a tool for studying how the number of bound states in a quantum well depends on the system parameters.