  We consider the existence of localized modes corresponding to eigenvalues of
the periodic Schr\"{o}dinger operator $-\partial_x^2+ V(x)$ with an interface.
The interface is modeled by a jump either in the value or the derivative of
$V(x)$ and, in general, does not correspond to a localized perturbation of the
perfectly periodic operator. The periodic potentials on each side of the
interface can, moreover, be different. As we show, eigenvalues can only occur
in spectral gaps. We pose the eigenvalue problem as a $C^1$ gluing problem for
the fundamental solutions (Bloch functions) of the second order ODEs on each
side of the interface. The problem is thus reduced to finding matchings of the
ratio functions $R_\pm=\frac{\psi_\pm'(0)}{\psi_\pm(0)}$, where $\psi_\pm$ are
those Bloch functions that decay on the respective half-lines. These ratio
functions are analyzed with the help of the Pr\"{u}fer transformation. The
limit values of $R_\pm$ at band edges depend on the ordering of Dirichlet and
Neumann eigenvalues at gap edges. We show that the ordering can be determined
in the first two gaps via variational analysis for potentials satisfying
certain monotonicity conditions. Numerical computations of interface
eigenvalues are presented to corroborate the analysis.
