Threshold of discrete Schrödinger operators with delta potentials on n-dimensional lattice

Abstract

Eigenvalue behaviors of Schrödinger operator defined on n-dimensional lattice with n + 1 delta potentials is studied. It can be shown that lower threshold eigenvalue and lower threshold resonance are appeared for n ≥ 2, and lower super-threshold resonance appeared for n = 1.where δ xs is the Kronecker delta.

On the spectrum of Schrödinger-type operators on two dimensional lattices

“…We remark that when H 0 = − ∆ in 2 (Z 2 ), the complete classification of the discrete spectrum of H a,b (1) in terms of a and b has been established recently in [11,26]. Our results generalize these existence results for a more general class of Hopping matrices (see Figure 1 below).…”

“…In the current paper, as we consider the discrete spectrum above the essential spectrum, we are interested in high energy resonances. Among various definitions, we mainly follow to [1,3,34,38]: in the momentum representation, a resonance of energy e max , where e max is the top of the essential spectrum, is a nonzero solution f of the eigenvalue equation H a,b (µ)f = e max f belonging to L 1 (T 2 ) \ L 2 (T 2 ) ; see also [6,7,11,22] and the references therein for other notions of resonances. Using the momentum representation, in Theorem 3.5 below we completely characterize the threshold eigenfunctions and threshold resonances, also finding them explicitly.…”

“…H a,b (µ), and so, we study the spectrum of H a,b (µ) reduced to these subspaces. As in [26,11] we use the Fredholm determinants theory [33,32] to find an implicit equation for the eigenvalues of H a,b (µ) . These equations will be the main tool in identifying the coupling constant thresholds and also in obtaining the convergent expansions of the eigenvalues.…”

Bose-Hubbard models with on-site and nearest-neighbor interactions: Exactly solvable case

“…As far as we know for continuous two-body Schrödinger operators R 2 there are no analogous examples and results. In the discrete case similar results for the number of eigenvalues of oneparticle Schrödinger operators in Z d with zero-range on-site and nearest-neighbor interactions have been obtained, for instance, in [12] for d = 3 with attractive potential field, in [16] for d = 1, and in [5] for all d ≥ 1 considering only negative eigenvalues.…”

The number and location of eigenvalues of the one particle discrete Schr¨odinger operators

“…Few-body Hamiltonians [29], among such models may be viewed as the simplest version of the corresponding Bose-Hubbard model involving a finite number of particles of a certain type. The few-body lattice Hamiltonians have been intensively studied over the past several decades [2,3,4,5,10,11,17,19,20,21,22,23,27,25,30].…”