Eigenvalue accumulation and bounds for non-selfadjoint matrix differential operators related to NLS

We establish results on the accumulation and location of the non-real spectrum of nonselfadjoint matrix differential operators arising in the study of non-linear Schrödinger equations (NLS) in Rd . In particular, without restrictions on the decay rate of the potentials to 0 at \infty, we show that the non-real spectrum cannot accumulate anywhere on the real axis. Under some weak assumptions satisfied, e.g., by Lp-potentials with p > d/2 , p ≥ 2, we prove that there are only finitely many non-real eigenvalues and that the non-real eigenvalues are located in a bounded lens-shaped region centered at the origin. Our key tool to prove this is a recent result on the existence of J-semi-definite invariant subspaces for J-selfadjoint operators in Krein spaces as well as abstract operator matrix methods.