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

Abstract

We establish results on the accumulation and location of the non-real spectrum of non-selfadjoint matrix differential operators arising in the study of non-linear Schrödinger equations (NLS) in $\mathbb{R}^d$. 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 $L_p$-potentials with $p>\frac d2$, $p\ge 2$, 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 $\mathcal{J}$-semi-definite invariant subspaces for $\mathcal{J}$-selfadjoint operators in Krein spaces as well as abstract operator matrix methods.