Absence Of Discrete Spectrum Research Articles

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.

Noncommutative Bloch theory

For differential operators which are invariant under the action of an Abelian group Bloch theory is the preferred tool to analyze spectral properties. By shedding some new noncommutative light on this we motivate the introduction of a noncommutative Bloch theory for elliptic operators on Hilbert C*-modules. It relates properties of C*-algebras to spectral properties of module operators such as band structure, weak genericity of cantor spectra, and absence of discrete spectrum. It applies, e.g., to differential operators invariant under a projective group action, such as Schrödinger, Dirac, and Pauli operators with periodic magnetic field, as well as to discrete models, such as the almost Matthieu equation and the quantum pendulum.

Absence of discrete spectrum of the Schr�dinger operators of positive molecular ions

For multiatomic positive molecular ions withN electrons and chargesZ i of the nuclei it is shown that there are no bound states forZ i >γN for some γ. It is also shown that the discrete spectrum of the Schrodinger operator of a molecular ion is empty if for at least some of the nuclei the conditionZ i ≫N is satisfied, whereas the charges of the other nuclei can be small.

On the spectrum of the rayleigh piston II

Building on results of an earlier paper we study the discrete spectrum of the Rayleigh piston. We first prove the absence of discrete spectrum on the subspace of odd functions everywhere in the Lorentz regime. Then we give upper bounds on the number of discrete eigenvectors as a function of the mass ratio using a variety of methods which to some degree complement each other. We also investigate the precise degree of divergence of these bounds as the mass ratio goes to infinity respectively zero.

Absence of discrete spectrum in highly negative ions

We extend the results of [1] to fermions, i.e., we show that ifH N is the Hamiltonian forN electrons in the field of a fixed point chargeZ, then there is a constantc such thatH N has no discrete spectrum forN≧N 0=cZ 6/5.

Absence of discrete spectrum in highly negative ions

LetH N be the Hamiltonian for the Coulomb system consisting ofN particles of like charge in the field of a fixed point chargeZ. We show that if the particles are bosons, thenH N has no discrete spectrum whenN≧N 0=cZ 2 for some constantc. If the particles are fermions, thenH N is bounded below uniformly inN. These results can be extended to molecules and to other power law potentials.