Dirac operators with $W^{1,\mathrm{\infty }}$-potential under codimension one collapse

We study the behavior of the spectrum of the Dirac operator together with a symmetric $W^{1,\mathrm{\infty }}$-potential on spin manifolds under a collapse of codimension one with bounded sectional curvature and diameter. If there is an induced spin structure on the limit space $N$ then there are convergent eigenvalues which converge to the spectrum of a first order differential operator $D$ on $N$ together with a symmetric $W^{1,\mathrm{\infty }}$-potential. If $N$ is orientable and the dimension of the limit space is even then $D$ is the Dirac operator $D^N$ on $N$ and if the dimension of the limit space is odd, then $D=D^N\oplus -D^N$.