Hyperspherical $\delta \text{-}{\delta }^{\prime }$ potentials

Abstract

The spherically symmetric potential $a\delta (r-r_0)+b{\delta }^{\prime }(r-r_0)$ 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 $d\ge 2$, 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 $\delta$-${\delta }^{\prime }$ potential for arbitrary $a>0$ admits a bound state with zero angular momentum.

arXiv commentSubmitted for publication to Annals of Physics, 21 pages, 3 figures

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Cited by in corpus (4)

• An approximation to the Woods-Saxon potential based on a contact interaction
• Casimir Energy for concentric $\delta$-${\delta }^{\prime }$ spheres
• Band spectra of periodic hybrid $\delta$-${\delta }^{\prime }$ structures
• Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian