  We construct a family of hermitian potentials in 1D quantum mechanics that
converges in the zero-range limit to a $\delta$ interaction with an
energy-dependent coupling. It falls out of the standard four-parameter family
of pointlike interactions in 1D. Such classification was made by requiring the
pointlike interaction to be hermitian. But we show that although our
Hamiltonian is hermitian for the standard inner product when the range of the
potential is finite, it becomes hermitian for a different inner product in the
zero-range limit. This inner product attributes a finite probability (and not
probability density) for the particle to be exactly located at the position of
the potential. Such pointlike interactions can then be used to construct
potentials with a finite support with an energy-dependent coupling.
