Dynamics of Noncommutative Solitons I: Spectral Theory and Dispersive Estimates

Abstract

Abstract. We consider the Schrödinger equation with a Hamiltonian given by a second order difference operator with nonconstant growing coefficients, on the half one dimensional lattice. This operator appeared first naturally in the construction and dynamics of noncommutative solitons in the context of noncommutative field theory. We prove pointwise in time decay estimates with the decay rate t −1 log −2 t, which is optimal with the chosen weights and appears to be so generally. We use a novel technique involving generating functions of orthogonal polynomials to achieve this estimate.

Dispersion Estimates for the Discrete Laguerre Operator

“…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.…”

“…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].…”

“…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].…”

Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator

“…(i) The case α = 0 was proven in [28]. Using a different approach, a weaker estimate in the case α = 0 was obtained in [34]. (ii) Using Lemma 4.3 (see also (4.8)), we get the somewhat stronger estimate…”

“…investigated in the recent work of Krueger and Soffer [33,34,35], dispersive estimates play a crucial role in the understanding of stability of the soliton manifolds appearing in these models (for further details see [7,16,33,34,35]). It turns out that the required dispersive decay estimates for the evolution group e −itHα lead to Bernstein-type estimates for (1.11) (see [28] and Sections 6-7 below).…”