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We address an eigenvalue problem for the ultrarelativistic (Cauchy) operator$(-\Delta )^{1/2}$, whose action is restricted to functions that vanish beyondthe interior of a unit sphere in three spatial dimensions. We provide highaccuracy spectral datafor lowest eigenvalues and eigenfunctions of thisinfinite spherical well problem. Our focus is on radial and orbital shapes ofeigenfunctions. The spectrum consists of an ordered set of strictly positiveeigenvalues which naturally splits into non-overlapping, orbitally labelled$E_{(k,l)}$ series. For each orbital label $l=0,1,2,...$ the label $k =1,2,...$enumerates consecutive $l$-th series eigenvalues. Each of them is$2l+1$-degenerate. The $l=0$ eigenvalues series $E_{(k,0)}$ are identical withthe set of even labeled eigenvalues for the $d=1$ Cauchy well:$E_{(k,0)}(d=3)=E_{2 k}(d=1)$. Likewise, the eigenfunctions $\psi_{(k,0)}(d=3)$ and $\psi_{2k }(d=1)$ show affinity. We have identified the genericfunctional form of eigenfunctions of the spherical well which appear to becomposed of a product of a solid harmonic and of a suitable purely radialfunction. The method to evaluate (approximately) the latter has been found tofollow the universal pattern which effectively allows to skip all, sometimesinvolved, intermediate calculations (those were in usage, while computing theeigenvalues for $l \leq 3$).

Ultrarelativistic bound states in the spherical well