Energy-momentum spectrum of some two-particle lattice Schrödinger Hamiltonians

Abstract

We determine the excitation spectrum of some one and two-particle Z(d) lattice Schrödinger Hamiltonians. They occur as approximate Hamiltonians for the low-lying energy-momentum spectrum of diverse infinite lattice nonlinear quantum systems. A unitary staggering transformation relates the low-energy-momentum spectrum to the high-energy-momentum spectrum of the transformed operators. A feature for the one-particle repulsive delta function Hamiltonian is that, in addition to the continuous band spectrum, there is a bound state above the band, and the repulsive case spectrum and scattering can be obtained from the attractive potential case by staggering. For the two-particle pair potential Hamiltonian, there are commuting self-adjoint energy-momentum operators, and we determine the joint spectrum. For the case of a lambda delta pair potential, and equal particle masses, for arbitrarily small /lambda/, lambda < 0, and d >or = 3, there is no bound state for small system momentum, but a bound state exists below the band if the momentum is large. We find that the binding energy is an increasing function of the system momentum. The existence of this bound state is in contrast with the continuum case, where the Birman-Schwinger bound excludes negative-energy bound states for small couplings; this bound state is absent if the two masses are different. Other spectral results are also obtained for the large coupling case. An eigenfunction expansion that uses products of plane waves in the sum and difference coordinates is used to obtain the spectral results.

On the Spectrum of an Hamiltonian in Fock Space. Discrete Spectrum Asymptotics

“…For instance, due to the fact that the discrete analogue of the Laplacian (or its generalizations) is not rotationally invariant, the Hamiltonian of a system does not separate into two parts, one relating to the center-of-mass motion and the other one to the internal degrees of freedom. In particular, the Efimov effect exists only for the zero value of the three-particle quasi-momentum K ∈ T 3 (see, e.g., [3,4,6,17,20,21,25] for relevant discussions and [10,11,16,25,26,28,31,38,40] for the general study of the low-lying excitation spectrum for quantum systems on lattices).…”

Spectral properties of a two-particle hamiltonian on a d-dimensional lattice

“…Lattice two-particle Hamiltonians have been investigated in [1][2][3]. In [1], the problem of the two-particle bound states for the transfer-matrix in a wide class of Gibbs fields on the lattices in the high temperature domains of (t 1), as well in [2] the appearance of bound state levels standing in a definite distance from the essential spectrum has been shown for some quasi-momenta values.…”

“…In [1], the problem of the two-particle bound states for the transfer-matrix in a wide class of Gibbs fields on the lattices in the high temperature domains of (t 1), as well in [2] the appearance of bound state levels standing in a definite distance from the essential spectrum has been shown for some quasi-momenta values. The spectral properties of the two-particle operator depending on total quasi-momentum have been studied in [3].…”

"Espectro de excitação para modelos quânticos na rede"

“…Em particular, foi mostrado em [9] e [16], que espectro do operador de Schrödinger não relativístico com um potencial delta na origem…”

“…Os operadores P Destacamos ainda que só estamos considerando o caso de massas iguais, pois em nosso modelo bosônico as partículas são idênticas. Uma análise para hamiltonianos quânticos mais gerais com potencial δ é feita em [16].…”