  Non-Hermitian skin effect (NHSE) is a distinctive phenomenon in non-Hermitian
systems, characterized by a significant accumulation of eigenstates at system
boundaries. While well-understood in one dimension via non-Bloch band theory,
unraveling the NHSE in higher dimensions faces formidable challenges due to the
diversity of open boundary conditions or lattice geometries and inevitable
numerical errors. Key issues, including higher-dimensional non-Bloch band
theory, geometric dependency, spectral convergence and stability, and a
complete classification of NHSE, remain elusive. In this work, we address these
challenges by presenting a geometry-adaptive non-Bloch band theory in arbitrary
dimensions, through the lens of spectral potential. Our formulation accurately
determines the energy spectra, density of states, and generalized Brillouin
zone for a given geometry in the thermodynamic limit (TDL), revealing their
geometric dependencies. Furthermore, we systematically classify the NHSE into
critical and non-reciprocal types using net winding numbers. In the critical
case, we identify novel scale-free skin modes residing on the boundary. In the
nonreciprocal case, the skin modes manifest in various forms, including normal
or anomalous corner modes, boundary modes or scale-free modes. We reveal the
non-convergence and instability of the non-Bloch spectra in the presence of
scale-free modes and attribute it to the non-exchangeability of the
zero-perturbation limit and the TDL. The instability drives the energy spectra
towards the Amoeba spectra in the critical case. Our findings provide a unified
non-Bloch band theory governing the energy spectra, density of states, and
generalized Brillouin zone in the TDL, offering a comprehensive understanding
of NHSE in arbitrary dimensions.
