Abstract
Wavefunctions with rotational symmetry (i.e., zero angular momentum) in D dimensions, are called s‐waves. In quantum quadratic systems (free particle, harmonic and repulsive oscillators), their radial parts obey Schrödinger equations with a fictitious centrifugal (for integer D ≥ 4) or centripetal (for D = 2) potential. These Hamiltonians close into the three‐dimensional Lorentz algebra so(2,1), whose exceptional interval corresponds to the critical range of continuous dimensions 0 < D < 4, where they exhibit a one‐parameter family of self‐adjoint extensions in ℒ︁^2^(ℜ︁^+^). We study the characterization of these extensions in the harmonic oscillator through their spectra which – except for the Friedrichs extension – are not equally spaced, and we build their time evolution Green function. The oscillator is then contracted to the free particle in continuous‐D dimensions, where the extension structure is mantained in the limit of continuous spectra. Finally, we compute the free time evolution of the expectation values of the Hamiltonian, dilatation generator, and square radius between three distinct sets of ‘heat’‐diffused localized eigenstates. This provides a simple group‐theoretic description of the purported contraction/expansion of Gaussian‐ring s‐waves in D > 0 dimensions.