Cantor and band spectra for periodic quantum graphs with magnetic fields

We provide an exhaustive spectral analysis of the two-dimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable Kronig-Penney Hamiltonian. In particular, between any two Dirichlet eigenvalues the spectrum is a Cantor set for an irrational flux, and is absolutely continuous and has a band structure for a rational flux. The Dirichlet eigenvalues can be isolated or embedded, subject to the choice of parameters. Conditions for both possibilities are given. We show that generically there are infinitely many gaps in the spectrum, and the Bethe-Sommerfeld conjecture fails in this case.Comment: Misprints correcte

Schrödinger operators with δ and δ′-potentials supported on hypersurfaces

Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman–Schwinger principle and a variant of Krein’s formula are shown. Furthermore, Schatten–von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ′-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity

DLTS spectra of silicon diodes with p+-n-junction irradiated with high energy krypton ions

p+-n-Diodes have been studied. The diodes were manufactured on wafers (thickness 460 μm, (111) plane) of uniformly phosphorus doped float-zone-grown single-crystal silicon. The resistivity of silicon was 90 Ω cm and the phosphorus concentration was 5×1013 cm−3. The diodes were irradiated with 250 MeV krypton ions. The irradiation fluence was 108 cm−2. Deep-level transient spectroscopy (DLTS) was used to examine the defects induced by high energy krypton ion implantation. The DLTS spectra were recorded at a frequency of 1 MHz in the 78–290 K temperature range. The capacity-voltage characteristics have been measured at a reverse bias voltage from 0 to −19 V at a frequency of 1 MHz. We show that the main irradiation-induced defects are A-centers and divacancies. The behavior of DLTS spectra in the 150–260 K temperature range depends essentially on the emission voltage Ue. The variation of Ue allows us to separate the contributions of different defects into the DLTS spectrum in the 150–260 K temperature range. We show that, in addition to A-centers and divacancies, irradiation produces multivacancy complexes with the energy level Et = Ec−(0.5±0.02) eV and an electron capture cross section of ~4×10–13 cm2

Spectral estimates for resolvent differences of self-adjoint elliptic operators

The concept of quasi boundary triples and Weyl functions from extension theory of symmetric operators in Hilbert spaces is developed further and spectral estimates for resolvent differences of two self-adjoint extensions in terms of general operator ideals are proved. The abstract results are applied to self-adjoint realizations of second order elliptic differential operators on bounded and exterior domains, and partial differential operators with δ-potentials supported on hypersurfaces are studied