  One of the most pronounced non-Hermitian phenomena is the non-Hermitian skin
effect, which refers to the exponential localization of bulk eigenstates near
the boundaries of non-Hermitian systems. Whereas non-Bloch band theory has been
developed to describe the non-Hermitian skin effect in lattice systems, its
counterpart in continuum systems still lacks a quantitative characterization.
Here, we generalize the non-Bloch band theory to non-Hermitian continuum
systems. In contrast to lattice systems for which the bulk Hamiltonian alone
determines the non-Hermitian skin effect and energy spectrum, we find for
continuum systems that the number of boundary conditions must also be included
as essential information. We show that the appropriate discretization of
continuum systems into lattice models requires matching the hopping range of
the latter to the number of boundary conditions in the former. Furthermore, in
periodic non-Hermitian continuum systems, we highlight the application of the
transfer matrix in determining the generalized Brillouin zone. Our theory
serves as a useful toolbox for investigating the rich non-Bloch physics in
non-Hermitian continuum systems, such as photonic crystals, elastic media, and
certain cold-atom systems.
