Universal Spectral Moment Theorem And Its Applications In Non-Hermitian Systems

The high sensitivity of the spectrum to boundary conditions poses significant obstacles for understanding bulk physical observables in non-Hermitian systems. In this paper, we propose the spectral moments in the tight-binding Hamiltonian with finite hopping range as intrinsic bulk quantities, independent of boundary conditions. Utilizing the invariance of the spectral moments, we demonstrate that in the continuum limit under open-boundary conditions, the upper and lower bounds of the imaginary parts of the spectrum remain constant regardless of the boundary geometry. This leads to the conclusion that the outer boundary of the continuum spectrum is insensitive to the shape of boundary geometry. We further establish a criterion for the reality of the spectrum and employ it to determine the $mathcal{PT}$-symmetry-breaking threshold in $mathcal{PT}$-symmetric systems in arbitrary dimensions. Finally, we show that in the long-time limit, the universal dynamical behavior under open boundary conditions converges to that of an infinite system, remarkably different from the behavior under periodic boundary conditions.