Magnetic layers with periodic point perturbations

We study spectral properties of a spinless quantum particle confined to an infinite planar layer with hard walls which interacts with a periodic lattice of point perturbations and a homogeneous magnetic field perpendicular to the layer. It is supposed that the lattice cell contains a finite number of impurities and the flux through the cell is rational. Using the Landau-Zak transformation, we convert the problem into investigation of the corresponding fiber operators which is performed by means of Krein's formula. This yields an explicit description of the spectral bands which may be absolutely continuous or degenerate, depending on the parameters of the model.Comment: LaTeX 2e, 30 pages; with minor revisions, to appear in Rep. Math. Phy

Berry phase in magnetic systems with point perturbations

We study a two-dimensional charged particle interacting with a magnetic field, in general non-homogeneous, perpendicular to the plane, a confining potential, and a point interaction. If the latter moves adiabatically along a loop the state corresponding to an isolated eigenvalue acquires a Berry phase. We derive an expression for it and evaluate it in several examples such as a homogeneous field, a magnetic whisker, a particle confined at a ring or in quantum dots, a parabolic and a zero-range one. We also discuss the behavior of the lowest Landau level in this setting obtaining an explicit example of the Wilczek-Zee phase for an infinitely degenerated eigenvalue.Comment: LaTeX, 26 page

Model of tunnelling through periodic array of quantum dots

Several explicitly solvable models of electron tunnelling in a system of single and double two-dimensional periodic arrays of quantum dots with two laterally coupled leads in a homogeneous magnetic field are constructed. First, a model of single layer formed by periodic array of zero-range potentials is described. The Landau operator (the Schrodinger operator with a magnetic field) with point-like interactions is the system Hamiltonian. We deal with two types of the layer lattices: square and honeycomb. The periodicity condition gives one an invariance property for the Hamiltonian in respect to magnetic translations group. The consideration of double quantum layer reduces to the replacement of the basic cell for the single layer by a cell including centers of different layers. Two variants of themodel for the double layer are suggested: with direct tunneling between the layers and with the connecting channels (segments in the model) between the layers. The theory of self-adjoint extensions of symmetric operators is a mathematical background of the model. The third stage of the construction is the description of leads connection. It is made by the operator extensions theory method too. Electron tunneling from input lead to the output lead through the double quantum layer is described. Energy ranges with extremely small (practically, zero) transmission were found. Dependencies of the transmission coefficient (particularly, “zero transmission bands” positions) on the magnetic field, the energy of electron and the distance between layers are investigated. The results are compared with the corresponding single-layer transmission

Electron transport in a two-terminal Aharonov-Bohm ring with impurities

Electron transport in a two-terminal Aharonov-Bohm ring with a few short-range scatterers is investigated. An analytical expression for the conductance as a function of the electron Fermi energy and magnetic flux is obtained using the zero-range potential theory. The dependence of the conductance on positions of scatterers is studied. We have found that the conductance exhibits asymmetric Fano resonances at certain energies. The dependence of the Fano resonances on magnetic field and positions of impurities is investigated. It is found that collapse of the Fano resonances occurs and discrete energy levels in the continuous spectrum appear at certain conditions. An explicit form for the wave function corresponding to the discrete level is obtained.Comment: 25 pages (one-column), 8 figure

Continuity properties of integral kernels associated with Schroedinger operators on manifolds

For Schroedinger operators (including those with magnetic fields) with singular (locally integrable) 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.Comment: 38 pages, major revision; to appear in Annales Henri Poincare (2007

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 investigated 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 the strength of the impurity potential is analyzed in detail. It is proven that the confinement potential of the dot can be recovered from the spectroscopy data. The consequences of the hidden symmetry breaking by the impurity are considered. The effect of the positional disorder is studied.Comment: 30 pages, 6 figures, Late