Dispersion Estimates for the Discrete Laguerre Operator

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This operator appeared recently in the study of radial waves in (2 + 1)dimensional noncommutative scalar field theory [1,13] and has attracted further interest in [9,[19][20][21]. More precisely, (1.1) is the linear part in the nonlinear Schrödinger equation (NLS) iψ(t, n) = H 0 ψ(t, n) − |ψ(t, n)| 2σ ψ(t, n), σ ∈ N, (t, n) ∈ R + × N 0 , (1.3) investigated in the recent work of Krueger and Soffer [19][20][21]. Also H 0 appeared in the discrete nonlinear Klein-Gordon equation (NLKG) [9,13]. In turn, the dynamics of noncommutative solitons in the context of noncommutative field theory (see, e.g., [6,13,23] for reviews) can be reduced to the study of discrete NLKG and NLS equations. In contrast to asymptotic metastability for the NLKG solitons conjectured in [9], it is expected that the NLS solitons are asymptotically stable (see [21, §8]). In this connection, let us emphasize that dispersive estimates for the linear part (1.1) as well as for its perturbations are an important ingredient in the standard procedure for the proof of asymptotic stability for nonlinear PDEs (see [7,8,26,27]). In fact, the case of Jacobi operators with asymptotically constant coefficients has already attracted a lot of attention and we refer to [10,11] and the references therein (see also [17] for discrete Dirac-type operators).
To formulate our results, we recall the weighted spaces Of course, the case σ = 0 corresponds to the usual p 0 = p spaces without weight. Then, in [20,Theorem 2] it was shown that for σ ≥ 3. This is in contrast to the case of the discrete Laplacian , where one has (cf. e.g., [11]) The purpose of the present note is to improve (1.4) by showing that the weights are not necessary, that is, it holds for σ ≥ 0. Even more, we are able to compute this norm explicitly: THEOREM 1.1. The following equality holds.
This result in turn is based on the following explicit expression for the kernel of the evolution group e −it H 0 given in terms of Jacobi polynomials (see [4,25] for the definition and basic properties): where is the Jacobi polynomial.
As it was already mentioned, the understanding of the dynamics of (1.1), which is the linear part of (1.3), is of crucial importance in the study of (1.3) and the NLKG equations. Furthermore, the understanding of the free evolution (1.1) is a necessary prerequisite for a successful development of scattering theory.
The proof of Theorems 1.1 and 1.2 is given in the next section and it is based on the fact that every element of the kernel of e −it H 0 is a Laplace transform of a product of two Laguerre polynomials (Lemma 2.3). Now notice that the required decay estimate will follow once we have a uniform estimate for the Jacobi polynomials P which is clearly insufficient for our purposes. It is somewhat surprising that the required estimate follows from the unitarity of the so-called Wigner d-matrix (Jacobi polynomials appear as matrix elements for the irreducible representations of SU(2), see [4,30]). Even more, to the best of our knowledge, the analytic proof of this estimate was obtained only recently by Haagerup and Schlichtkrull in [16]. Let us also mention one more dispersive estimate which follows from the Haagerup-Schlichtkrull inequality for Jacobi polynomials [16, Theorem 1.1] (see (2.16) below).
42 such that the following inequality holds for all t > 0.
Note that the estimate (1.10) does not provide an optimal decay rate in t; however, it gives an additional decay of the coefficients of the kernel in n and m. Let us mention that the Haagerup-Schlichtkrull inequality (2.16) was derived in order to obtain uniform bounds on a complete set of matrix coefficients for the irreducible representations of SU(2) with a decay rate d −1/4 in the dimension d of the representation. Furthermore, the Bernstein (see (2.17)) and the Haagerup-Schlichtkrull estimates were used in [15,22], respectively, for establishing the absence of the approximation property of Haagerup and Kraus [14] for SL(3, R) and Sp(2, R).
On the other hand, in the follow-up paper [18], we investigate the decay estimate for generalized Laguerre operators H α (tri-diagonal matrices associated with generalized Laguerre polynomials L (α) n , see [25]), where the coefficient α can be seen as a measure of the delocalization of the field configuration and it is related to the planar angular momentum [2] (α = 0 corresponds to radial waves in (2 + 1)dimensional noncommutative scalar field theory). It turned out that the optimal dispersive decay estimate leads to new Bernstein-type inequalities for Jacobi polynomials. All these connections are mathematically very appealing and we hope that the present note will stipulate further research in this direction.
To end this section, let us briefly outline the content of the paper. Before proving the main result, we collect the basic spectral properties of the operator H 0 in Theorem 2.1 and also present its proof based on spectral theory of Jacobi operators. Next, in Lemma 2.3 we represent the kernel of the evolution group e −it H 0 by means of the Laguerre polynomials and then prove our main results Theorems 1.1, 1.2, 1.3. Finally, in Lemma 2.6 we present another representation of the kernel of e −it H 0 , which might be of independent interest. In particular, it allows to obtain a simple proof of (1.4) for σ ≥ 1/2.

Proof of the Main Results
We start with a precise definition of the operator H 0 . Let D : 2 (N 0 ) → 2 (N 0 ) be the multiplication operator given by For a sequence u = {u n } n≥0 we define the difference expression τ : u → τ u by setting Then the operator H 0 associated with the Jacobi matrix (1.2) is defined by where D max = {u ∈ 2 (N 0 ) : τ u ∈ 2 (N 0 )}. Note that 2 1 (N 0 ) ⊂ D max ; however, simple examples (take u = {(−1) n /(n + 1)} n≥0 ) show that the inclusion is strict.
The spectral properties of H 0 were derived in [9,20], however, without using the well-developed spectral theory for Jacobi operators [28]. We collected them in the following theorem and give a short proof using this connection.
(iii) The spectrum of H 0 is purely absolutely continuous and coincides with R + = [0, ∞).
Proof. (i) Self-adjointness clearly follows from the Carleman test (see, e.g., [3], [ (ii) Notice that the polynomials of the first kind for H 0 are given by are the Laguerre polynomials. Indeed (see, e.g., [25, Chapter V]), they satisfy the following recursion relations and the orthogonality relations Therefore, (2.9) and (i) imply that dρ(λ) = 1 R + (λ)e −λ dλ is the spectral measure of H 0 , that is, H 0 is unitarily equivalent to a multiplication operator in L 2 (R + , dρ) (cf. e.g., [28,Theorem 2.12]). It remains to note that the corresponding Weyl function is the Stieltjes transform of the measure dρ (cf. e.g., [28,Chapter 2]). (iiii) The claim immediately follows from (ii). Indeed, as it was already mentioned, H 0 is unitarily equivalent to a multiplication operatorH acting in L 2 (R + , dρ), where dom(H ) = L 2 (R + ; λ 2 dρ) Hence the spectra as well as the spectral types of both operators coincide [28, Eq. (2.106)]). It remains to note that the spectrum ofH is purely absolutely continuous and coincides with R + = [0, ∞).
It follows from (2.12) that every element of the kernel of the operator e −it H 0 is the Laplace transform of a product of the corresponding Laguerre polynomials and hence one can compute them explicitly: Proof of Theorem 1.2. Recall that (see [12, (4.11.35 whenever Re( p) > 0 with 2 F 1 is the hypergeometric function (see [24,Chapter 15 Finally, we recall the connection with the Jacobi polynomials [25, (4.22.1)] which establishes (1.7). We note some special cases: (ii) In the case m = 1 we have where P n (z) = 1 2 n (n + 1) d n dz n (z 2 − 1) n are the Legendre polynomials [25].
Noting that the Fourier transform of a product of two functions is equal to the convolution of their Fourier transforms and using (2.12), we end up with (2.19).
Remark 2.7. Noting that and then estimating the convolution, one can show by using induction that