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Mapping between Hamiltonians with attractive and repulsive potentials on a lattice

Through a simple and exact analytical derivation, we show that for a particle on a lattice, there is a one-to-one correspondence between the spectra in the presence of an attractive potential $\hat{V}$ and its repulsive counterpart $-\hat{V}$. For a Hermitian potential, this result implies that the number of localized states is the same in both, attractive and repulsive, cases although these states occur above (below) the band-continnum for the repulsive (attractive) case. For a $\mP\mT$-symmetric potential that is odd under parity, our result implies that in the $\mP\mT$-unbroken phase, the energy eigenvalues are symmetric around zero, and that the corresponding eigenfunctions are closely related to each other.