Magnetic-Sparseness and Schrödinger Operators on Graphs

Ju l 2 00 6 Schrödinger operators on zigzag graphs

We consider the Schrödinger operator on zigzag graphs with a periodic potential. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe all compactly supported eigenfunctions with the same eigenvalue. We define a Lyapunov function, which is analytic on some Riemann surfa...

Asymptotics and Gaps in the Spectra of Magnetic Schrödinger Operators

In this paper, we study an L version of the semiclassical approximation of magnetic Schrödinger operators with invariant Morse type potentials on covering spaces of compact manifolds. In particular, we are able to establish the existence of an arbitrary large number of gaps in the spectrum of these operators, in the semiclassical limit as the coupling constant μ goes to zero.

Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators

We study two-dimensional magnetic Schrödinger operators with a magnetic field that is equal to b > 0 for x > 0 and −b for x < 0. This magnetic Schrödinger operator exhibits a magnetic barrier at x = 0. The unperturbed system is invariant with respect to translations in the ydirection. As a result, the Schrödinger operator admits a direct integral decomposition. We analyze the band functions of ...

Locating the Spectrum for Magnetic Dirac and Schrödinger Operators

Convergence of Schrödinger Operators

For a large class, containing the Kato class, of real-valued Radon measures m on R the operators −∆ + ε∆ + m in L(R, dx) tend to the operator −∆ +m in the norm resolvent sense, as ε tends to zero. If d ≤ 3 and a sequence (μn) of finite real-valued Radon measures on R converges to the finite real-valued Radon measure m weakly and, in addition, supn∈N μ ± n (R) < ∞, then the operators −∆ + ε∆ + μ...