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We study multipoint scatterers with zero-energy bound states in three dimensions. We present examples of such scatterers with multiple zero eigenvalue or with strong multipole localization of zero-energy bound states.
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We study the transmission eigenvalues for the multipoint scatterers of the Bethe-Peierls-Fermi-Zeldovich-Beresin-Faddeev type in dimensions $d=2$ and $d=3$. We show that for these scatterers: 1) each positive energy $E$ is a transmission eigenvalue (in the strong sense) of infinite multiplicity; 2) each complex $E$ is an interior transmission eigenvalue of infinite multiplicity. The case of dimension $d=1$ is also discussed.
For given two unitary and self-adjoint operators on a Hilbert space, a spectral mapping theorem was proved in cite{HiSeSu}. In this paper, as an application of the spectral mapping theorem, we investigate the spectrum of a one-dimensional split-step quantum walk. We give a criterion for when there is no eigenvalues around $pm 1$ in terms of a discriminant operator. We also provide a criterion for when eigenvalues $pm 1$ exist in terms of birth eigenspaces. Moreover, we prove that eigenvectors from the birth eigenspaces decay exponentially at spatial infinity and that the birth eigenspaces are robust against perturbations.
We devise a supersymmetry-based method for the construction of zero-energy states in graphene. Our method is applied to a two-dimensional massless Dirac equation with a hyperbolic scalar potential. We determine supersymmetric partners of our initial system and derive a reality condition for the transformed potential. The Dirac potentials generated by our method can be used to approximate interactions that are experimentally realizable.
We consider a three-body one-dimensional Schrodinger operator with zero range potentials, which models a positive impurity with charge $kappa > 0$ interacting with an exciton. We study the existence of discrete eigenvalues as $kappa$ is varied. On one hand, we show that for sufficiently small $kappa$ there exists a unique bound state whose binding energy behaves like $kappa^4$, and we explicitly compute its leading coefficient. On the other hand, if $kappa$ is larger than some critical value then the system has no bound states.
We construct explicit bound state wave functions and bound state energies for certain $N$--body Hamiltonians in one dimension that are analogous to $N$--electron Hamiltonians for (three-dimensional) atoms and monatomic ions.
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