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

  • Christiane Tretter

    Universität Bern, Switzerland
Eigenvalue accumulation and bounds for non-selfadjoint matrix differential operators related to NLS cover
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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 Rd\mathbb{R}^d. In particular, without restrictions on the decay rate of the potentials to 00 at \infty, we show that the non-real spectrum cannot accumulate anywhere on the real axis. Under some weak assumptions satisfied, e.g., by LpL_p-potentials with p>d2p>\frac d2, p2p\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 J\mathcal{J}-semi-definite invariant subspaces for J\mathcal{J}-selfadjoint operators in Krein spaces as well as abstract operator matrix methods.