Hyperspherical δ-δ’ potentials

Abstract

The spherically symmetric potential a δ(r − r0) + b δ′(r − r0) is generalised for the d-dimensional 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.