Abstract
We consider the family H^μ:=Δ^Δ^-μV^,μ∈R,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\widehat{{ H}}}_\\mu := {\\widehat{\\varDelta }} {\\widehat{\\varDelta }} - \\mu {\\widehat{{ V}}},\\qquad \\mu \\in {\\mathbb {R}}, \\end{aligned}$$\\end{document}of discrete Schrödinger-type operators in d-dimensional lattice {mathbb {Z}}^d, where {widehat{varDelta }} is the discrete Laplacian and {widehat{{ V}}} is of rank-one. We prove that there exist coupling constant thresholds mu _o,mu ^oge 0 such that for any mu in [-mu ^o,mu _o] the discrete spectrum of {widehat{{ H}_mu }} is empty and for any mu in {mathbb {R}}setminus [-mu ^o,mu _o] the discrete spectrum of {widehat{{ H}_mu }} is a singleton {e(mu )}, and e(mu )<0 for mu >mu _o and e(mu )>4d^2 for mu <-mu ^o. Moreover, we study the asymptotics of e(mu ) as mu searrow mu _o and mu nearrow -mu ^o as well as mu rightarrow pm infty . The asymptotics highly depends on d and {widehat{{ V}}}.
Highlights
In this paper we investigate the spectral properties of the perturbed discrete biharmonic operatorHμ := ΔΔ − μV, μ ∈ R, (1.1)in the d-dimensional cubical lattice Zd, where Δ is the discrete Laplacian and V is a is rank-one potential with a generating potential v
From the mathematical point of view, the discrete bilaplacian represents a discrete Schrödinger operator with a degenerate bottom, i.e., ΔΔ is unitarily equivalent to a multiplication operator by a function e which behaves as o(| p − p0|2) close to its minimum point p0
The spectral properties of discrete Schrödinger operators with non-degenerate bottom (i.e., e behaves as O(| p − p0|2) close to its minimum point p0), in particular with discrete Laplacian, have been extensively studied in recent years because of their applications in the theory of ultracold atoms in optical lattices [16, 24, 35, 36]. It is well-known that the existence of the discrete spectrum is strongly connected to the threshold phenomenon [18, 20–22], which plays an role in the existence the Efimov effect in three-body systems [31, 32, 34]: if any two-body subsystem in a three-body system has no bound state below its essential spectrum and at least two two-body subsystem has a zero-energy resonance, the corresponding three-body system has infinitely many bound states whose energies accumulate at the lower edge of the three-body essential spectrum
Summary
In the d-dimensional cubical lattice Zd , where Δ is the discrete Laplacian and V is a is rank-one potential with a generating potential v. The non-degeneracy of the bottom of the (reduced) one-particle Schrödinger operator played an important role in the study of resonance states of the associated two-body system [1, 31] Another keypoint in the proof of the Efimov effect in Z3 was the asymptotics of the (unique) smallest eigenvalue of the (reduced) one-particle discrete Schrödinger operator which creates a singularity in the kernel of a Birman-Schwinger-type operator which used to obtain an asymptotics to the number of three-body bound states. Observing that the top e(π ) = 4d2 of the essential spectrum is nondegenerate, one expects the asymptotics of e(μ) as μ → −μo to be similar as in the discrete Laplacian case [20, 21]; more precisely, depending on d and no, e(μ) has a convergent expansion. In Theorem 2.2 we establish necessary and sufficient conditions for non-emptiness of the discrete spectrum of Hμ, and in case of existence, we study the location and the uniqueness, analiticity, monotonicity and convexity properties of eigenvalues e(μ) as a function of μ. In appendix Section A we obtain the asymptotics of certain integrals related to (μ; z) which will be used in the proofs of main results
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