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

Interfaces Supporting Surface Gap Soliton Ground States in the 1D Nonlinear Schroedinger Equation