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We establish quantitative upper and lower bounds for Schrödinger operators with complex potentials that satisfy some weak form of sparsity. Our first result is a quantitative version of an example, due to S. Bögli (Commun Math Phys 352:629–639, 2017), of a Schrödinger operator with eigenvalues accumulating to every point of the essential spectrum. The second result shows that the eigenvalue bounds of Frank (Bull Lond Math Soc 43:745–750, 2011 and Trans Am Math Soc 370:219–240, 2018) can be improved for sparse potentials. The third result generalizes a theorem of Klaus (Ann Inst H Poincaré Sect A (N.S.) 38:7–13, 1983) on the characterization of the essential spectrum to the multidimensional non-selfadjoint case. The fourth result shows that, in one dimension, the purely imaginary (non-sparse) step potential has unexpectedly many eigenvalues, comparable to the number of resonances. Our examples show that several known upper bounds are sharp. [ABSTRACT FROM AUTHOR]

Recently, Ferrulli–Laptev–Safronov (2016) obtained eigenvalue estimates for an operator associated with bilayer graphene in terms of L q norms of the (possibly non-self-adjoint) potential. They proved that for 1 < q < 4 / 3 all nonembedded eigenvalues lie near the edges of the spectrum of the free operator. In this note, we prove this for the larger range 1 ≤ q ≤ 3 / 2 . The latter is optimal if embedded eigenvalues are also considered. We prove similar estimates for a modified bilayer operator with so-called trigonal warping term. Here, the range for q is smaller since the Fermi surface has less curvature. The main tool is new uniform resolvent estimates that may be of independent interest and is collected in an appendix (in greater generality than needed). [ABSTRACT FROM AUTHOR]

We analyze eigenvalues emerging from thresholds of the essential spectrum of one-dimensional Dirac operators perturbed by complex and non-symmetric potentials. In the general non-self-adjoint setting, we establish the existence and asymptotics of weakly coupled eigenvalues and Lieb-Thirring inequalities. As physical applications, we investigate the damped wave equation and armchair graphene nanoribbons. [ABSTRACT FROM AUTHOR]

We prove Lieb–Thirring type bounds for fractional Schrödinger operators and Dirac operators with complex-valued potentials. The main new ingredient is a resolvent bound in Schatten spaces for the unperturbed operator, in the spirit of Frank and Sabin [11] . [ABSTRACT FROM AUTHOR]

We proveLpand smoothing estimates for the resolvent of magnetic Schrödinger operators. We allow electromagnetic potentials that are small perturbations of a smooth, but possibly unbounded background potential. As an application, we prove an estimate on the location of eigenvalues of magnetic Schrödinger and Pauli operators with complex electromagnetic potentials. [ABSTRACT FROM PUBLISHER]

We show that in graphene, modelled by the two-dimensional Dirac operator, charge distributions with non-vanishing dipole moment have infinitely many bound states. The corresponding eigenvalues accumulate at the edges of the gap faster than any power. [ABSTRACT FROM AUTHOR]

We show that the non-embedded eigenvalues of the Dirac operator on the real line with complex mass and non-Hermitian potential V lie in the disjoint union of two disks, provided that the L-norm of V is bounded from above by the speed of light times the reduced Planck constant. The result is sharp; moreover, the analogous sharp result for the Schrödinger operator, originally proved by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For massless Dirac operators, the condition on V implies the absence of non-real eigenvalues. Our results are further generalized to potentials with slower decay at infinity. As an application, we determine bounds on resonances and embedded eigenvalues of Dirac operators with Hermitian dilation-analytic potentials. [ABSTRACT FROM AUTHOR]