Mapping between Hamiltonians with attractive and repulsive potentials on a lattice

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.

6 pages, 1 figure

References in corpus (4)

• Repulsively bound atom pairs in an optical lattice
• Exponentially Fragile PT-Symmetry in Lattices with Localized Eigenmodes
• Two-particle states in the Hubbard model
• Scattering resonances and two-particle bound states of the extended Hubbard model

Cited by in corpus (15)

• Optical waveguide arrays: quantum effects and PT symmetry breaking
• $\mathcal{P}\mathcal{T}$-breaking threshold in spatially asymmetric Aubry-Andre Harper models: hidden symmetry and topological states
• The robust $\mP\mT$-symmetric chain and properties of its Hermitian counterpart
• Degrees and signatures of broken PT-symmetry in (non-uniform) lattices
• PT-restoration via increased loss-gain in PT-symmetric Aubry-Andre model
• Hanbury Brown and Twiss Correlations of Anderson Localized Waves
• Origin of maximal symmetry breaking in even PT-symmetric lattices
• Tunable waveguide lattices with non-uniform parity-symmetric tunneling
• Dynamics, disorder effects, and PT-symmetry breaking in waveguide lattices with localized eigenstates
• Observation of two PT transitions in an electric circuit with balanced gain and loss
• PT-symmetry breaking with divergent potentials: lattice and continuum cases
• Parity-time symmetric systems with memory
• Non-Hermiticity in quantum nonlinear optics through symplectic transformations
• Disorder effects in tunable waveguide arrays with parity-symmetric tunneling
• PT-symmetry enabled stable modes in multi-core fiber