Twisted sections and deficiency indices

This restriction is stronger than rotation and $z$-translation invariance alone. Such invariance can also permit measurable $k_z$-dependent boundary conditions. Those extensions, longitudinally nonlocal conditions, and non-product extensions are not classified here.

The rotation characters $e^{\mathrm{i}(n+\beta )\gamma }$ are distinct. Thus $B_{\perp }$ is diagonal in $n$. Fixing a reference length ${\ell }_0>0$, the deficient radial domains have boundary coefficients of the form

The coefficients can equivalently be defined by limiting Wronskians. The corresponding Green boundary form shows that the self-adjoint one-mode conditions are:

• $\nu =0$: the Friedrichs condition $a=0$, or

  $${\mathsf{L}}_L:f(r)=a\log (r/L)+o(1),0<L<\infty ;$$

• $0<\nu <1$: the Friedrichs condition $a=0$, or

  $${\mathsf{E}}_{\lambda }:b=\lambda a,\lambda \in \mathbb{R}.$$

  We also write $\lambda =\infty$ for Friedrichs.

Indeed, for $0<\nu <1$ the boundary form is a nonzero real multiple of ${\overline{a}}_f{b}_g-{\overline{b}}_f{a}_g$, so its one-dimensional self-adjoint subspaces have exactly this real projective parametrisation. The logarithmic case is analogous.