  We consider the problem of verifying the existence of $H^1$ ground states of
the 1D nonlinear Schr\"odinger equation for an interface of two periodic
structures: $$-u" +V(x)u -\lambda u = \Gamma(x) |u|^{p-1}u \ {on} \R$$ with
$V(x) = V_1(x), \Gamma(x)=\Gamma_1(x)$ for $x\geq 0$ and $V(x) = V_2(x),
\Gamma(x)=\Gamma_2(x)$ for $x<0$. Here $V_1,V_2,\Gamma_1,\Gamma_2$ are
periodic, $\lambda <\min\sigma(-\tfrac{d^2}{dx^2}+V)$, and $p>1$. The article
[T. Dohnal, M. Plum and W. Reichel, "Surface Gap Soliton Ground States for the
Nonlinear Schr\"odinger Equation," \textit{Comm. Math. Phys.} \textbf{308},
511-542 (2011)] provides in the 1D case an existence criterion in the form of
an integral inequality involving the linear potentials $V_{1},V_2$ and the
Bloch waves of the operators $-\tfrac{d^2}{dx^2}+V_{1,2}-\lambda$. We choose
here the classes of piecewise constant and piecewise linear potentials
$V_{1,2}$ and check this criterion for a set of parameter values. In the
piecewise constant case the Bloch waves are calculated explicitly and in the
piecewise linear case verified enclosures of the Bloch waves are computed
numerically. The integrals in the criterion are evaluated via interval
arithmetic so that rigorous existence statements are produced. Examples of
interfaces supporting ground states are reported including such, for which
ground state existence follows for all periodic $\Gamma_ {1,2}$ with $\esssup
\Gamma_{1,2}>0$.
