Asymptotics of eigenvalues of the zero-range perturbation of the discrete bilaplacian
aa r X i v : . [ m a t h . SP ] N ov ASYMPTOTICS OF EIGENVALUES OF THE ZERO-RANGEPERTURBATION OF THE DISCRETE BILAPLACIAN
SHOKHRUKH YU. KHOLMATOV AND MARDON PARDABAEV
Abstract.
We consider the family b h µ := b ∆ b ∆ − µ b v , µ ∈ R , of discrete Schr¨odinger-type operators in one-dimensional lattice Z , where b ∆ is thediscrete Laplacian and b v is of zero-range. We prove that for any µ = 0 the discretespectrum of b h µ is a singleton { e ( µ ) } , and e ( µ ) < µ > e ( µ ) > µ < . Moreover, we study the properties of e ( µ ) as a function of µ, in particular, wefind the asymptotics of e ( µ ) as µ ց µ ր . Introduction
In this paper we study the spectral properties of the family b h µ := b h − µ b v , µ ∈ R , of self-adjoint bounded discrete Schr¨odinger-type operators in the Hilbert space ℓ ( Z )of square-summable complex-valued functions defined on the one-dimensional lattice Z . Here b h is discrete bilaplacian, i.e. b h := b ∆ b ∆, where b ∆ b f ( x ) = b f ( x ) − b f ( x + 1) + b f ( x − , b f ∈ ℓ ( Z ) , is the discrete Laplacian, and b v is a rank-one operator b v b f ( x ) = ( b f (0) , x = 0 , , x = 0 . This model can be considered as the discrete Schr¨odinger operator on Z , associated to asystem of one particle whose dispersion relation has a degenerate bottom.The spectral theory of discrete Schr¨odinger operators with non-degenerate bottom inparticular, with discrete Laplacian, have been extensively studied in recent years (see e.g.[1, 2, 3, 8, 9, 12, 13] and references therein) because of their applications in the theoryof ultracold atoms in optical lattices [5, 11, 15, 16]. For these models the appearanceof weakly coupled bound states has been sufficiently well-understood and sufficient andnecessary conditions for the existence of discrete spectrum in terms of the coupling constanthave been established. This conditions naturally leads to the coupling constant thresholdphenomenon [7]: consider − ∆ + λV with V short range at a value λ , where someeigenvalue e ( λ ) is absorbed into the continuous spectrum as λ ց λ , and conversely, forany ε > , as λ ր λ + ε the continuous spectrum gives birth to a new eigenvalue. Thisphenomenon and the absorption rate of eigenvalues as λ → λ have been established fordiscrete Schr¨odinger operators with non-degenerate bottom, for example, in [8, 9, 10] andcontinuous Schr¨odinger operators, for example, in [6, 7, 14]. Date : November 20, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Discrete bilaplacian, essential spectrum, discrete spectrum, eigenvalues, asymp-totics, expansion . n the case of Schr¨odinger operators with degenerate bottom some sufficient conditionsfor existence of bound states have been recently observed in [4]. However, to the bestof our knowledge the results related to the absorption rate of eigenvalues have not beenpublished yet.Recall that σ ( b ∆ ) = σ ess ( b ∆ ) = [0 , , thus, from the spectral theory it follows that σ ( b h ) = σ ess ( b h ) = [0 , , and hence, by the compactness of b v and Weyl’s Theorem, σ ess ( b h µ ) = σ ess ( b h ) = [0 , µ ∈ R . Let F : ℓ ( Z ) → L ( T ) , F b f ( p ) = 1 √ π X x ∈ Z b f ( x ) e ixp be the standard Fourier transform with the inverse F − : L ( T ) → ℓ ( Z ) , F − f ( x ) = 1 √ π Z T f ( p ) e − ixp dp. The operator h µ = F b h µ F − is called the momentum representation of b h µ . Note that h µ = h − µ v , (1.1)where h := F b h F − is the multiplication operator in L ( T ) by e ( q ) := (1 − cos q ) , and v := F b v F − is the rank-one integral operator v f ( p ) = 1 √ π Z T f ( q ) dq. The main result of the paper is the following.
Theorem 1.1.
For any µ = 0 the operator h µ has a unique eigenvalue e ( µ ) outside theessential spectrum with the associated eigenfunction f µ ( p ) = 1 e ( q ) − e ( µ ) . (1.2) Moreover: (a) e ( µ ) < for any µ > and e ( µ ) > for any µ < the function µ ∈ R \ { } 7→ e ( µ ) is real-analytic strictly decreasing, strictly convexin ( −∞ , and strictly concave in (0 , + ∞ ) with asymptotics lim µ →±∞ e ( µ ) µ = − and lim µ ր e ( µ ) − µ = 18 , lim µ ց e ( µ ) µ / = − / ;(c) there exists γ > such that ( e ( µ ) − / = − µ √ − µ − µ √ − X n ≥ C n µ n (1.3) for any µ ∈ ( − γ, and ( − e ( µ )) / = µ / / + µ − / µ /
288 + X n ≥ D n µ (2 n +1) / (1.4) or any µ ∈ (0 , γ ) , where { C n } and { D n } are some real coefficients. Proof of the main results
We start with the following lemma in which we get an equation for eigenvalues of h µ . Lemma 2.1. z ∈ C \ [0 , is an eigenvalue of h µ if and only if ∆( µ ; z ) = 0 , where ∆( µ ; z ) := 1 − µ π Z T dq e ( q ) − z . Proof.
Suppose that z ∈ C \ [0 ,
4] is an eigenvalue of h µ with an associated eigenfunction0 = f ∈ L ( T ) . Then from h µ f = zf it follows that f ( p ) = C e ( p ) − z , (2.1)where C := µ π Z T f ( q ) dq. (2.2)Inserting the representation (2.1) of f in (2.2) we get C = Cµ π Z T dq e ( q ) − z . (2.3)Since C = 0 (otherwise f = 0 by (2.1)), from (2.3) it follows that ∆( µ ; z ) = 0 . Conversely, suppose that ∆( µ ; z ) = 0 for some z ∈ C \ [0 , , and set f ( p ) := 1 e ( p ) − z . Then f ∈ L ( T ) \ { } and( h µ − z ) f ( p ) = 1 − µ π Z T dq e ( q ) − z = ∆( µ ; z ) = 0 , i.e. z is an eigenvalue of h µ with associated eigenvector f. (cid:3) Notice that z ∈ C \ [0 , ∆( µ ; z )is analytic for any µ ∈ R and and ∂∂z ∆( µ ; z ) = − µ Z T dq ( e ( q ) − z ) . Thus, for any µ > µ < , the function ∆( µ ; · ) is strictly decreasing resp.strictly increasing in R \ [0 , . Moreover,lim z →±∞ ∆( µ ; z ) = 1 . (2.4) Lemma 2.2.
For any z ∈ R \ [0 , , √ π Z T dq e ( q ) − z = − sign( z ) p z ( z − (cid:18) r zz − (cid:19) / , (2.5) where sign( z ) is the sign of z, i.e. equal to if z > , to if z = 0 and − if z < . Proof.
We establish (2.5) for z <
0; one can prove it for z > z = − α for α > . Using the Euler formula e iq = cos q + i sin q, werewrite the integral as I ( z ) := Z T dq e ( q ) − z = Z T dq (cid:0) − e iq + e − iq (cid:1) + α = Z T e iq dq (cid:0) e iq − e iq + 1 (cid:1) + 4 α e iq o that I ( z ) = − i Z T e iq de iq ( e iq − + 4 α e iq . Changing variables ξ = e iq we see that I ( z ) = − i Z | ξ | =1 ξdξ ( ξ − + 4 α ξ . Since the only zeros ξ := (1 − αA ) + i ( Aα − α ) and ξ of the fourth-order polynomial( ξ − + 4 α ξ with real coefficients belong to the ball {| ξ | < } , where A := s √ α + α , (2.6)by the Residue Theorem for analytic functions, I ( z ) = 8 π (cid:16) ξ ξ − + 8 α ξ + ξ ξ − + 8 α ξ (cid:17) = 4 π Re ξ ( ξ − + 2 α ξ . Since ( ξ − + 4 α ξ = 0 , one has ( ξ − = − ξ ξ − , thus, I ( z ) = − πα Re ξ − ξ + 1 . Using the definition (2.6) of A, by the direct computation one finds ξ − ξ + 1 = − A + iA √ α α, therefore, I ( z ) = 2 Aπα √ α , and (2.5) follows. (cid:3) Proof of Theorem 1.1.
Rewriting (2.5) as1 √ π Z T dq e ( q ) − z = − sign( z ) p | z | p | z − | qp | z − | + p | z | , (2.7)we observe that lim z ր Z T dq e ( q ) − z = + ∞ , lim z ց Z T dq e ( q ) − z = −∞ . (2.8)Since ∆ ≥ µ ; z ) ∈ [0 , + ∞ ) × (4 , + ∞ ) and for ( µ ; z ) ∈ ( −∞ , × ( −∞ , , by (2.8), (2.4) and thestrict monotonicity and the analyticity of z ∈ R \ [0 , ∆( µ ; z ) , for any µ = 0 thereexists a unique e ( µ ) ∈ R \ [0 ,
4] such that ∆( µ ; e ( µ )) = 0 . (a) By Lemma 2.1, e ( µ ) is the unique eigenvalue of h µ with the associated eigenfunc-tion (1.2). Moreover, from (2.9) it follows that e ( µ ) < µ > e ( µ ) > µ < . (b) By the Implicit Function Theorem, the function µ ∈ R \{ } 7→ e ( µ ) is real-analytic.Moreover, computing the derivatives of the implicit function e ( µ ) we find: e ′ ( µ ) = − µ Z T dq e ( q ) − e ( µ ) (cid:16) Z T dq ( e ( q ) − e ( µ )) (cid:17) − , µ = 0 , (2.10)thus, using µ ( e ( q ) − e ( µ )) > e ′ ( µ ) < , i.e. e ( · ) is strictly decreasing in R \ { } . Differentiating (2.10) once more and using µ R T dq e ( q ) − e ( µ ) = 1 we get e ′′ ( µ ) = 2 e ′ ( µ ) µ − µe ′ ( µ ) Z T dq ( e ( q ) − e ( µ )) (cid:18)Z T dq ( e ( q ) − e ( µ )) (cid:19) − ! . herefore, e ′′ ( µ ) > e ( · ) is strictly convex) for µ < e ′′ ( µ ) < e ( · ) isstrictly concave) for µ > . By (2.5), e ( µ ) solves √ − µ p e ( µ ) p e ( µ ) − qp e ( µ ) − p e ( µ ) for µ < √ µ p − e ( µ ) p − e ( µ ) qp − e ( µ ) + p − e ( µ ) for µ > . (2.12)The strict monotonicity of e ( · ) and (2.11) and (2.12) imply thatlim µ →−∞ e ( µ ) = + ∞ , lim µ ր e ( µ ) = 4 , lim µ ց e ( µ ) = 0 , lim µ → + ∞ e ( µ ) = −∞ , (2.13)hence, e ( · ) has a jump at µ = 0 . In particular, from (2.11) and (2.13) we obtainlim µ →−∞ e ( µ ) − µ = 1 and lim µ ր p − e ( µ ) − µ = 1 √ µ ր p e ( µ ) = 12 √ . Analogously, by (2.12) and (2.13),lim µ → + ∞ − e ( µ ) µ = 1 and lim µ ց p − e ( µ ) µ = 1 √ µ ց p − e ( µ ) = 12 . Now we establish (1.3). Setting α := α ( µ ) = p e ( µ ) − α = − µ / (1 + α ) / (cid:16)(cid:16) α (cid:17) / + α (cid:17) / . (2.14)By (b), µ ∈ ( −∞ , α ( µ ) is strictly decreasing. In view of (2.13) there exists γ > α ∈ (0 ,
1) for any µ ∈ ( − γ , . Using(1 + x ) / = 1 + x X n ≥ ( − n − n n ! n − Y j =0 (1 + 2 j ) x n (2.15)for | x | < , one has (cid:16)(cid:16) α (cid:17) / + α (cid:17) / = (cid:16) α α X n ≥ ( − n − n n ! n − Y j =0 (1 + 2 j ) α n (cid:17) / =1 + α α
16 + X m ≥ ( − m − · m m ! m − Y j =0 (1 + 2 j ) α m + X n ≥ ( − n − n n ! n − Y j =0 (1 + 2 j ) (cid:16) α α X m ≥ ( − m − m m ! m − Y j =0 (1 + 2 j ) α m (cid:17) n . Thus, using (1 + x ) − / = 1 + X n ≥ ( − n n n ! n − Y j =0 (3 + 4 j ) x n for | x | < α ) / (cid:16)(cid:16) α (cid:17) / + α (cid:17) / = 1 − α
16 + 13 α
512 + X n ≥ c n α n , (2.16)where { c n } are some real coefficients. Using (2.16) we rewrite (2.14) as α = − µ / (cid:16) − α
16 + 13 α
512 + X n ≥ c n α n (cid:17) (2.17) or µ ∈ ( − γ , . Setting α = − µ / (1 + u ) we can represent (2.17) as F ( u, µ ) = 0 , where F ( u, µ ) := u − µ (1 + u )16 · / − µ (1 + u ) − X n ≥ c n n/ µ n (1 + u ) n is real-analytic in a neighborhood of ( u, µ ) = (0 , ,F (0 ,
0) = 0 , F u (0 ,
0) = 1 . Hence, by the Analytic Implicit Function Theorem, there exist γ > u = u ( µ ) given by a series u ( µ ) = P n ≥ a n µ n with real coefficients { a n } andabsolutely convergent for | µ | < γ such that F ( u ( µ ) , µ ) ≡ | µ | < γ. Inserting α = − µ / (1 + u ) and u ( µ ) = P n ≥ a n µ n in (2.17) one finds inductively a = √ ,a = and so on. This implies (1.3).Now we prove (1.4). Set µ := λ and e ( µ ) = − α , where α := α ( λ ) > λ > . Then (2.12) is rewritten as √ λ α √ α qp α + α . (2.18)We solve this equation with respect to α. To this aim first we rewrite it the right handside of (2.18) as an absolutely convergent series of α, and then from the Implicit FunctionTheorem in analytical case we deduce that α has a convergent power series in λ and thecoefficients of the series will be found inductively from the series in α .By the strict decrease of α ( · ) and asymptotics (2.13) there exists a unique γ > | α ( λ ) | < λ ∈ (0 , γ ) . Recalling for | x | < x ) − / = 1 + X n ≥ ( − n n n ! n − Y j =0 (1 + 4 j ) x n , and (1 + x ) − / = 1 + X n ≥ ( − n n n ! n − Y j =0 (1 + 2 j ) x n , as well as (2.15) we get √ α ) / = 1 + X n ≥ ( − n n n ! n − Y j =0 (1 + 4 j ) α n . and α √ α = α (cid:16) α (cid:17) − / = α X n ≥ ( − n · n n ! n − Y j =0 (1 + 2 j ) α n +2 so that s α √ α =1 + α X n ≥ ( − n · n n ! n − Y j =0 (1 + 2 j ) α n +2 + X n ≥ ( − n − n n ! n − Y j =0 (1 + 2 j ) (cid:16) α X m ≥ ( − m · m m ! m − Y j =0 (1 + 2 j ) α m +2 (cid:17) n . This implies √ α ) / s r α α = 1 + α − α
32 + X n ≥ C n α n , here C n , n = 3 , , . . . , are real coefficients. Hence for λ ∈ (0 , γ ) the equation (2.18) isrepresented as 2 α = λ (cid:16) α − α
32 + X n ≥ C n α n (cid:17) . (2.19)Let α = λ (2 − / + u ) we rewrite (2.19) as2 / u +2 / u + 23 u = λ (2 − / + u ) − λ (2 − / + u )
96 + X n ≥ C n λ n (2 − / + u ) n . (2.20)For µ := λ , (2.20) is represented as F ( u, µ ) = 0 , where F ( u, µ ) := 2 / u +2 / u + 23 u − µ (2 − / + u )
12 + µ (2 − / + u ) − X n ≥ C n µ n (2 − / + u ) n is analytic in a neighborhood of ( u, µ ) = (0 , , and satisfies F (0 ,
0) = 0 and F u (0 ,
0) = 2 / > . Hence, by the Implicit Function Theorem in analytical case there exists γ > u = u ( µ ) given by the absolutely convergent series u ( µ ) = P n ≥ a n µ n with { a n } ⊂ R for | µ | < γ and F ( u ( µ ) , µ ) ≡ µ ∈ ( − γ, γ ) . Inserting u ( µ ) = P n ≥ a n µ n in (2.20) we find the coefficients a k inductively: a = 124 , a = − / α ( λ ) = 2 / λ + 124 λ − / λ + X n ≥ a n λ n +1 . Now the definitions of α and λ imply (1.4). (cid:3) Acknowledgments
The first author acknowledges support from the Austrian Science Fund (FWF) projectM 2571-N32.
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E-mail address , Sh. Kholmatov: [email protected] (Mardon Pardabaev)
Samarkand State University, University boulevard 3, 140104Samarkand (Uzbekistan)
E-mail address , M. Pardabaev: p [email protected] [email protected]