Structure of Noncommutative Solitons: Existence and Spectral Theory

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 construct a ground state soliton for this equation and analyze its properties. In particular we arrive at ∞ and 1 estimates as well as a quasi-exponential spatial decay rate.Mathematics subject classification. 35Q40, 35Q55, 39A05.

Dynamics of Noncommutative Solitons I: Spectral Theory and Dispersive Estimates

“…We will use the definitions for W κ,τ , ǫ, and the like from [1]. Furthermore we will employ the spectral decay estimates of Corollary 1 of [1] as well as the quasi-exponential decay estimates of Theorem 2 of [2]. From the latter one can see that for λ > µ it is the case that d n λ w…”

“…In [1] we focus on a key estimate that is needed for scattering and stability, namely the decay in time of the solution, at the optimal rate. Fortunately, in the generic case, we find it is integrable, given by t −1 log −2 t. The proof of this result is rather direct, and employs the generating functions of the corresponding generalized eigenfunctions, to explicitly represent and estimate the resolvent of the hamiltonian at all energies.…”

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

“…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).…”

Dispersion Estimates for the Discrete Laguerre Operator

“…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].…”