We introduced some contact potentials that can be written as a linear combination of the Dirac delta and its first derivative, the $delta$-$delta$ interaction. After a simple general presentation in one dimension, we briefly discuss a one dimensional periodic potential with a $delta$-$delta$ interaction at each node. The dependence of energy bands with the parameters (coefficients of the deltas) can be computed numerically. We also study the $delta$-$delta$ interaction supported on spheres of arbitrary dimension. The spherical symmetry of this model allows us to obtain rigorous conclusions concerning the number of bound states in terms of the parameters and the dimension. Finally, a $delta$-$delta$ interaction is used to approximate a potential of wide use in nuclear physics, and estimate the total number of bound states as well as the behaviour of some resonance poles with the lowest energy.
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