In quantum mechanics the one-dimensional motion of a free particle
may be modified by a suitable, phenomenologically motivated
coordinate-dependence either in its effective mass (controlled by
the medium and altering the kinetic-energy operator $K$) and/or in
an external field [characterized by the potential-energy term
$V=V(x)$]. The authors consider one of the simplest special choices
of the corresponding generic Hamiltonian $H = K+V$ [cf. Eq. (1)].
They restrict their attention to the special model with a point
interaction in the origin [cf. Eq. (2)] and with the most schematic
step-shaped form of the mass $m=m(x)$ [cf. Eq. (4)]. Having accepted
the two sample self-adjoint extensions of their Hamiltonian [based
on the matching of wave functions, cf. sections 2 and 4,
respectively] they finally discuss the existence [requiring,
unfortunately, a correlation between $m(x)$ and $V(x)$] and the
closed-form construction of the corresponding bound state.

MR2552015 Gadella, M.; Heras, F. J. H.; Negro, J.; Nieto, L. M. A
delta well with a mass jump. J. Phys. A 42 (2009), no. 46, 465207,
11 pp. 81Q05