Mapping between Hamiltonians with attractive and repulsive potentials on a lattice

TL;DR

This paper establishes an exact analytical mapping between the spectra of particles on a lattice with attractive and repulsive potentials, revealing symmetry properties and state correspondences, especially in the context of Hermitian and -symmetric potentials.

Contribution

It provides a novel, exact analytical derivation of the spectral correspondence between attractive and repulsive potentials on a lattice, including implications for -symmetric systems.

Findings

Spectra in attractive and repulsive potentials are in one-to-one correspondence.

Number of localized states is the same for both potential types.

In -symmetric, parity-odd potentials, eigenvalues are symmetric around zero.

Abstract

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.