  We determine approximate eigenvalues and eigenfunctions shapes for bound
states in the $3D$ shallow spherical ultrarelativistic well. Existence
thresholds for the ground state and first excited states are identified, both
in the purely radial and orbitally nontrivial cases. This contributes to an
understanding of how energy may be stored or accumulated in the form of bound
states of Schr\"odinger - type quantum systems that are devoid of any mass.
