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

TL;DR

This paper generalizes the spherically symmetric delta-delta prime potential to higher dimensions, analyzing its spectral properties and revealing unique bound state conditions in two dimensions.

Contribution

It introduces a selfadjoint extension of the free Hamiltonian for the delta-delta prime potential in d-dimensional space and studies its spectral characteristics.

Findings

Only in 2D does the potential admit a zero angular momentum bound state for positive a.

Numerical results show the expectation value of the radius varies with potential parameters.

Spectrum includes negative, zero, and positive energy states depending on parameters.

Abstract

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