Expansion of eigenvalues of the perturbed discrete bilaplacian

We consider the family \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\widehat{{ H}}}_\mu := {\widehat{\varDelta }} {\widehat{\varDelta }} - \mu {\widehat{{ V}}},\qquad \mu \in {\mathbb {R}}, \end{aligned}$$\end{document}H^μ:=Δ^Δ^-μV^,μ∈R,of discrete Schrödinger-type operators in d-dimensional lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}^d$$\end{document}Zd, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{\varDelta }}$$\end{document}Δ^ is the discrete Laplacian and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{{ V}}}$$\end{document}V^ is of rank-one. We prove that there exist coupling constant thresholds \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _o,\mu ^o\ge 0$$\end{document}μo,μo≥0 such that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in [-\mu ^o,\mu _o]$$\end{document}μ∈[-μo,μo] the discrete spectrum of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{{ H}_\mu }}$$\end{document}Hμ^ is empty and for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in {\mathbb {R}}\setminus [-\mu ^o,\mu _o]$$\end{document}μ∈R\[-μo,μo] the discrete spectrum of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{{ H}_\mu }}$$\end{document}Hμ^ is a singleton \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{e(\mu )\},$$\end{document}{e(μ)}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e(\mu )<0$$\end{document}e(μ)<0 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu >\mu _o$$\end{document}μ>μo and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e(\mu )>4d^2$$\end{document}e(μ)>4d2 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu <-\mu ^o.$$\end{document}μ<-μo. Moreover, we study the asymptotics of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e(\mu )$$\end{document}e(μ) as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \searrow \mu _o$$\end{document}μ↘μo and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \nearrow -\mu ^o$$\end{document}μ↗-μo as well as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \rightarrow \pm \infty .$$\end{document}μ→±∞. The asymptotics highly depends on d and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{{ V}}}.$$\end{document}V^.


Introduction
In this paper we investigate the spectral properties of the perturbed discrete biharmonic operator in the d-dimensional cubical lattice Z d , where Δ is the discrete Laplacian and V is a is rank-one potential with a generating potential v. This model is associated to a one-particle system in Z d with a potential field v, in which the particle freely "jumps" from a node X of the lattice not only to one of its nearest neighbors Y (similar to the discrete Laplacian case), but also to the nearest neighbors of the node Y . From the mathematical point of view, the discrete bilaplacian represents a discrete Schrödinger operator with a degenerate bottom, i.e., Δ Δ is unitarily equivalent to a multiplication operator by a function e which behaves as o(| p − p 0 | 2 ) close to its minimum point p 0 . The spectral properties of discrete Schrödinger operators with non-degenerate bottom (i.e., e behaves as O(| p − p 0 | 2 ) close to its minimum point p 0 ), in particular with discrete Laplacian, have been extensively studied in recent years (see e.g. [1,2,7,8,10,11,20,21,23,26,28] and references therein) because of their applications in the theory of ultracold atoms in optical lattices [16,24,35,36]. In particular, it is well-known that the existence of the discrete spectrum is strongly connected to the threshold phenomenon [18,[20][21][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, then the corresponding three-body system has infinitely many bound states whose energies accumulate at the lower edge of the three-body essential spectrum.
Recall that the Efimov effect may appear only for certain attractive systems of particles [29]. However, recent experimental results in the theory of ultracold atoms in an optical lattice have shown that two-particle systems can have repulsive bound states and resonances (see e.g. [36]), thus, one expects the Efimov effect to hold also for some repulsive three-particle systems in Z 3 .
The strict mathematical justification of the Effect effect including the asymptotics for the number of negative eigenvalues of the three-body Hamiltonian has been successfully established in 3-space dimensions (for both R 3 and Z 3 ) see e.g., [1,4,13,19,29,31,32,34] and the references therein. In particular, 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 Z 3 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.
To the best of our knowledge, there are no published results related to the Efimov effect in lattice three-body systems in which associated (reduced) one-body Schrödinger operator has degenerate bottom.
We also recall that fourth order elliptic operators in R d in particular, the biharmonic operator, play also a central role in a wide class of physical models such as linear elasticity theory, rigidity problems (for instance, construction of suspension bridges) and in streamfunction formulation of Stokes flows (see e.g. [9,25,27] and references therein). Moreover, recent investigations have shown that the Laplace and biharmonic operators have high potential in image compression with the optimized and sufficiently sparse stored data [15]. The need for corresponding numerical simulations has led to a vast literature devoted to a variety of discrete approximations to the solutions of fourth order equations [5,12,33]. The question of stability of such models is basically related to their spectral properties and therefore, numerous studies have been dedicated to the numerical evaluation of the eigenvalues [3,6,30].
The aim of the present paper is the study of the existence and asymptotics of eigenvalues as well as threshold resonance and bound states of H μ defined in (1.1), which corresponds to the one-body Schrödinger operator with degenerate bottom. Namely, we study the discrete spectrum of H μ depending on μ and on v. For simplicity we assume the generator v of V to decay exponentially at infinity, however, we urge that our methods can also be adjusted to less regular cases (see Remark 2.6). Since the spectrum of Δ consists of [0, 2d] (see e.g., [1]), by the compactness of V and Weyl's Theorem, the essential spectrum of H μ fills the segment [0, 4d 2 ] independently of μ. Moreover, the essential spectrum does not give birth to a new eigenvalue while μ runs in some real interval [−μ o , μ o ], and it turns out as soon as μ leaves this interval through μ o resp. through −μ o , a unique negative resp. a unique positive eigenvalue e(μ) releases from the essential spectrum (Theorem 2.2). Now we are interested in the absorption rate of e(μ) as μ → μ o and μ → −μ o . The associated asymptotics are highly dependent not only on the dimension d of the lattice (as in the discrete Laplacian case [20,21]), but also values on the multiplicity 2n o and 2n o of 0 ∈ {v = 0} (if v(0) = 0) and π ∈ {v = 0} (if v( π) = 0), respectively. More precisely, depending on d and n o , e(μ) has a convergent expansion Furthermore, observing that the top e( π) = 4d 2 of the essential spectrum is nondegenerate, one expects the asymptotics of e(μ) as μ → −μ o to be similar as in the discrete Laplacian case [20,21]; more precisely, depending on d and The threshold analysis for more general class of nonlocal discrete Schrödinger operators with δ-potential of type can be found in [14], where is some strictly increasing C 1 -function and δ x0 is the Dirac's delta-function supported at 0. Besides the existence of eigenvalues, authors of [14] classify (embedded) threshold resonances and threshold eigenvalues depending on the behaviour of at the edges of the essential spectrum of − Δ and on the lattice dimension d. The eigenvalue expansions for the discrete bilaplacian with δperturbation have been established in [17] for d = 1 using the complex analytic methods.
The paper is organized as follows. In Sect. 2 after introducing some preliminaries we state the main results of the paper. In Theorem 2.2 we establish necessary and sufficient conditions for non-emptiness of the discrete spectrum of H μ , 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 particular, we study the asymptotics of e(μ) as μ → μ o and μ → −μ o as well as μ → ±∞. As discussed above in Theorems 2.4 and 2.5 we obtain expansions of e(μ) for small and positive μ − μ o and μ + μ o . In Sect. 3 we prove the main results. The main idea of the proof is to obtain a nonlinear equation (μ; z) = 0 with respect to the eigenvalue z = e(μ) of H μ and then study properties of (μ; z). Finally, 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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Preliminary and main results
Let Z d be the d-dimensional lattice and 2 (Z d ) be the Hilbert space of squaresummable functions on Z d . Consider the family is the discrete Laplacian, and V is a rank-one operator Let T d be the d-dimensional torus equipped with the Haar measure and L 2 (T d ) be the Hilbert space of square-integrable functions on T d . By F we denote the the standard Fourier transform Further we always assume that v and its Fourier image satisfy the following assumptions: There exist reals C, a > 0 and nonnegative integersn o , n o ≥ 0 such that here D j f ( p) is the j-th order differential of f at p, i.e. the j-th order symmetric tensor and π = (π, . . . , π) ∈ T d . Notice that under assumption (H1), v is analytic on T d .
Before stating the main results let us introduce the constants and Moreover, by Propositions A.1 and A.2:

Main results
First we concern with the existence of the discrete spectrum of H μ .
. and Next we study the threshold resonances of H μ . . .
We recall that in the literature the non-zero solutions of equations H μ f = 0 and H μ g = 4d 2 g not belonging to 2 (Z d ) are called the resonance states [1,2]. Now we study the rate of the convergences in (2.6).
where {c 9,n } are some real coefficients.
(b) Suppose that d is even: , then μ o = 0 and for sufficiently small and positive μ, where {C 5,n } are some real coefficients.
where {C 6,nm } are some real coefficients.
Here C v and C v are given by (2.5)

Proof of main results
In this section we prove the main results. By the Birman-Schwinger principle and the Fredholm Theorem we have

Remark 3.2
In view of Lemma 3.1 and the proof of Theorem 2.2, their assertions still hold for any v ∈ 2 (Z d ).
Proof of Theorem 2. 3 We prove only (a), the proof of (b) being similar. Repeating the proof of the continuity (resp. differentiability) of l f at z = 0 in Proposition A.1 one
Proof of Theorem 2.5 From (3.3) it follows that the map p ∈ T d → |v| 2 ( π + p) is even. Now the expansions of e(μ) at μ = −μ o can be proven along the same lines of Theorem 2.4 using Proposition A.2 with f = |v| 2 .
Since both p ∈ T d → f ( p) and p ∈ T d → f ( π + p) are even analytic functions, we can still apply Propositions A.1 and A.2 to find the expansions of z → T d f ( p)dp e( p)−z and thus, repeating the same arguments of the proofs of Theorems 2.4 and 2.5 one can obtain the corresponding expansions of e(μ).

Remark 3.4 When
| v(x)| = O(|x| 2n 0 +d+1 ) as |x| → ∞ for some n 0 ≥ 1, in view of Remark A.3, we need to solve equation (3.4) with respect to μ using only that left-hand side is an asymptotic sum (not a convergent series). This still can be done using appropriate modification of the Implicit Function Theorem for differentiable functions. As a result, we obtain only (Taylor-type) asymptotics of e(μ).
and by the analyticity of f in B π (0) ⊂ R d , the series converges absolutely in p ∈ B π (0). By the definition of ϕ, ϕ(r w) ⊂ B π (0) for any r ∈ (0, γ ) and w = (w 1 , . . . , w d ) ∈ S d−1 , where S d−1 is the unit sphere in R d . Then letting p = ϕ(r w) and using the Taylor series ϕ i (r w) = 2r w i + r 3 w 3 i 3 +