  In this note we consider a quantum mechanical particle moving inside an
infinitesimally thin layer constrained by a parabolic well in the $x$-direction
and, moreover, in the presence of an impurity modelled by an attractive
Gaussian potential. We investigate the Birman-Schwinger operator associated to
a model assuming the presence of a Gaussian impurity inside the layer and prove
that such an integral operator is Hilbert-Schmidt, which allows the use of the
modified Fredholm determinant in order to compute the bound states created by
the impurity. Furthermore, we consider the case where the Gaussian potential
degenerates to a $\delta$-potential in the $x$-direction and a Gaussian
potential in the $y$-direction. We construct the corresponding self-adjoint
Hamiltonian and prove that it is the limit in the norm resolvent sense of a
sequence of corresponding Hamiltonians with suitably scaled Gaussian
potentials. Satisfactory bounds on the ground state energies of all
Hamiltonians involved are exhibited.
