Non-Hermitian delocalization from Hermitian Hamiltonians

Here we show that using Galilean transformations the non-Hermitian delocalization phenomenon, which is relevant in different fields, such as bacteria population (e.g., Bacillus subtilis), vortex pinning in superconductors, and stability solutions of hydrodynamical problems discovered by Hatano and Nelson [Phys. Rev. Lett. 77, 5706 (1996)], can be obtained from solutions of the time-dependent Schrödinger equation with a Hermitian Hamiltonian. Using our approach, one avoids the numerical complications and instabilities which result form the calculations of left and right eigenfunctions of the non-Hermitian Hamiltonian which are associated with the non-Hermitian delocalization phenomenon. One also avoids the need to replace the non-Hermitian Hamiltonian $\mathcal{Ĥ}$ by a supermatrix with twice the dimension of $\mathcal{Ĥ},$ where the complex frequencies serve as variational parameters rather than eigenvalues of $\mathcal{Ĥ}.$