The Existence and Stability of Noncommutative Scalar Solitons

Abstract:

 We establish existence and stability results for solitons in noncommutative scalar field theories in even space dimension 2d. In particular, for any finite rank spectral projection P of the number operator 𝒩 of the d-dimensional harmonic oscillator and sufficiently large noncommutativity parameter θ we prove the existence of a rotationally invariant soliton which depends smoothly on θ and converges to a multiple of P as θ→∞.

In the two-dimensional case we prove that these solitons are stable at large θ, if P=P _N , where P _N projects onto the space spanned by the N+1 lowest eigenstates of 𝒩, and otherwise they are unstable. We also discuss the generalisation of the stability results to higher dimensions. In particular, we prove stability of the soliton corresponding to P=P ₀ for all θ in its domain of existence.

Finally, for arbitrary d and small values of θ, we prove without assuming rotational invariance that there do not exist any solitons depending smoothly on θ.