Quasi-exact solvability of a hyperbolic intermolecular potential induced by an effective mass step (Q539356)

Consider the stationary Schrödinger equation \[ {1\over 2m(x)}\psi''(x)- {m'(x)\over 2m^2(x)} \psi'(x)+ (E- V(x))\psi(x)= 0\tag{\(*\)} \] with \[ \begin{gathered} m(x)= 1+ 1+\tanh(x),\\ V(x; a,b,c,d)= a\tanh(x)+ b\tanh^2(x)+ c\tanh^3(x)+ d,\\ E= b-a+ d-c.\end{gathered} \] The author establishes explicit solutions of \((*)\) and derives conditions for their \(L^2\)-normalizability.