Embedded eigenvalues of generalized Schrodinger operators

We provide examples of operators T(D)+V in L2(Rd) with decaying potentials that have embedded eigenvalues. The decay of the potential depends on the curvature of the (Fermi) surfaces of constant kinetic energy T. We make the connection to counterexamples in Fourier restriction theory

Dipoles in graphene have infinitely many bound states

© 2014 AIP Publishing LLC. 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

Eigenvalue estimates for bilayer graphene

© 2019, Springer Nature Switzerland AG. 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)

Estimates on complex eigenvalues for Dirac operators on the half-line

We derive bounds on the location of non-embedded eigenvalues of Dirac operators on the half-line with non-Hermitian L 1-potentials. The results are sharp in the non-relativistic or weak-coupling limit. In the massless case, the absence of discrete spectrum is proved under a smallness assumption

Schrödinger operators with complex sparse potentials

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.</p

Eigenvalue bounds for Dirac and fractional Schrödinger operators with complex potentials

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]

Sharp spectral estimates for the perturbed Landau Hamiltonian with L^p potentials

We establish a sharp uniform estimate on the size of the spectral clusters of the Landau Hamiltonian with (possibly complex-valued) Lp potentials as the cluster index tends to infinity

Improved eigenvalue bounds for Schrödinger operators with slowly decaying potentials

We extend a result of Davies and Nath (J Comput Appl Math 148(1):1–28, 2002) on the location of eigenvalues of Schrödinger operators with slowly decaying complex-valued potentials to higher dimensions. In this context, we also discuss various examples related to the Laptev and Safronov conjecture (Laptev and Safronov in Commun Math Phys 292(1):29–54, 2009)

Sharp spectral bounds for complex perturbations of the indefinite Laplacian

© 2020 Elsevier Inc. We derive quantitative bounds for eigenvalues of complex perturbations of the indefinite Laplacian on the real line. Our results substantially improve existing results even for real potentials. For L1-potentials, we obtain optimal spectral enclosures which accommodate also embedded eigenvalues, while our result for Lp-potentials yield sharp spectral bounds on the imaginary parts of eigenvalues of the perturbed operator for all p∈[1,∞). The sharpness of the results are demonstrated by means of explicit examples

Weak coupling limit for Schrödinger-type operators with degenerate kinetic energy for a large class of potentials

We improve results by Frank, Hainzl, Naboko, and Seiringer (J Geom Anal 17(4):559–567, 2007) and Hainzl and Seiringer (Math Nachr 283(3):489–499, 2010) on the weak coupling limit of eigenvalues for Schrödinger-type operators whose kinetic energy vanishes on a codimension one submanifold. The main technical innovation that allows us to go beyond the potentials considered in Frank, Hainzl, Naboko, and Seiringer (2007), Hainzl and Seiringer (2010) is the use of the Tomas–Stein theorem