We study a non-relativistic particle subject to a three-dimensional spherical potential consisting of a finite well and a radial δ − δ contact interaction at the well edge. This contact potential is defined by appropriate matching conditions for the radial functions, thereby fixing a self-adjoint extension of the non-singular Hamiltonian. Since this model admits exact solutions for the wave function, we are able to characterize and calculate the number of bound states. We also extend some well-known properties of certain spherically symmetric potentials and describe the resonances, defined as unstable quantum states. Based on the Woods-Saxon potential, this configuration is implemented as a first approximation for a mean-field nuclear model. The results derived are tested with experimental and numerical data in the double magic nuclei 132 Sn and 208 Pb with an extra neutron.

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The spherically symmetric potential a δ(r − r 0 ) + b δ (r − r 0 ) is generalised for the ddimensional space as a characterisation of a unique selfadjoint extension of the free Hamiltonian. For this extension of the Dirac delta, the spectrum of negative, zero and positive energy states is studied in d ≥ 2, providing numerical results for the expectation value of the radius as a function of the free parameters of the potential. Remarkably, only if d = 2 the δ-δ potential for arbitrary a > 0 admits a bound state with zero angular momentum.

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