V. A. Geyler

Continuity Properties of Integral Kernels Associated with Schrödinger Operators on Manifolds

Abstract.For Schrödinger operators (including those with magnetic fields) with singular scalar potentials on manifolds of bounded geometry, we study continuity properties of some related integral kernels: the heat kernel, the Green function, and also kernels of some other functions of the operator. In particular, we show the joint continuity of the heat kernel and the continuity of the Green function outside the diagonal. The proof makes intensive use of the Lippmann–Schwinger equation.

Spectral properties of a short-range impurity in a quantum dot

The spectral properties of the quantum mechanical system consisting of a quantum dot with a short-range attractive impurity inside the dot are studied in the zero-range limit. The Green function of the system is obtained in an explicit form. In the case of a spherically symmetric quantum dot, the dependence of the spectrum on the impurity position and strength of the impurity potential is analyzed in detail. The recovering of the confinement potential of the dot from the spectroscopy data is proven; the consequences of the hidden symmetry breaking by the impurity are considered. The effect of the positional disorder is analyzed.

On the number of bound states for weak perturbations of spin–orbit Hamiltonians

We give a variational proof of the existence of infinitely many bound states below the continuous spectrum for some weak perturbations of a class of spin–orbit Hamiltonians including the Rashba and Dresselhaus Hamiltonians.

Explicit Green functions for spin–orbit Hamiltonians

We derive explicit expressions for Green functions and some related characteristics of the Rashba and Dresselhaus Hamiltonians with a uniform magnetic field.

Transport properties of two-arc Aharonov-Bohm interferometers with scattering centers

A model for a broad class of Aharonov-Bohm interferometers consisting of two arcs with and without scattering centers is constructed. Explicit expressions and asymptotic relations are found for the transmission coefficient for electrons in the simplest interferometers of diverse geometry (a symmetric interferometer with scattering admixture and an Aharonov-Bohm ring with two conductors attached at a single point). The influence of the relationship between the sizes of arcs and the arrangement of potentials of scattering centers, the magnetic field flux, and the energy of electrons on transport properties of the suggested nanodevices is studied.

Magnetic Field Dependence of the Energy Gap in Nanotubes

Abstract The dependence of the energy gap in the band structure of a carbon nanotube on the flux of an external uniform magnetic field is studied in the framework of the zero‐range potential method. Taking into account a renormalization procedure for the coupling constant of the zero‐range potential, we obtain an estimate for the maximal gap between the conductivity band and the valence one; this estimate is closer to experimental data when compared to the value given by the tight‐binding approximation. It is shown numerically that decreasing the diameter of nanotubes breaks down the flux periodicity of the gap structure.

On the discrete spectrum of spin-orbit Hamiltonians with singular interactions

We give a variational proof of the existence of infinitely many bound states placed below the continuous spectrum for spin-orbit Hamiltonians (including the Rashba and Dresselhaus cases) perturbed by measure potentials, thus extending results of J. Brüning, V. Geyler, K. Pankrashkin, J. Phys. A: Math. Theor. 40 F113–F117 (2007).

FERMI SURFACES OF CRYSTALS IN A HIGH MAGNETIC FIELD

A method of building and investigation of the Fermi surfaces for three-dimensional crystals subjected to a uniform magnetic field is presented. The Hamiltonian of a charged particle in the crystal is treated in the framework of the zero-range potential theory. The dispersion relation for the Hamiltonian is obtained in an explicit form.