756 research outputs found
Spectral Asymptotics for Waveguides with Perturbed Periodic Twisting
We consider the twisted waveguide , i.e. the domain obtained
by the rotation of the bounded cross section
of the straight tube at angle
which depends on the variable along the axis of . We study the spectral
properties of the Dirichlet Laplacian in , unitarily equivalent
under the diffeomorphism to the operator
, self-adjoint in . We assume that where is a -periodic function, and
decays at infinity. Then in the spectrum of the unperturbed
operator there is a semi-bounded gap ,
and, possibly, a number of bounded open gaps . Since decays at infinity, the essential spectra of
and coincide. We investigate the asymptotic
behaviour of the discrete spectrum of near an arbitrary
fixed spectral edge . We establish necessary and quite
close sufficient conditions which guarantee the finiteness of in a neighbourhood of . In the
case where the necessary conditions are violated, we obtain the main asymptotic
term of the corresponding eigenvalue counting function. The effective
Hamiltonian which governs the the asymptotics of near could be represented as a
finite orthogonal sum of operators of the form , self-adjoint in ; here, is a
constant related to the so-called effective mass, while is
-periodic function depending on and .Comment: 29 pages, typos corrected, to appear in Journal of Spectral Theor
The Fate of the Landau Levels under Perturbations of Constant Sign
We show that the Landau levels cease to be eigenvalues if we perturb the 2D
Schr\"odinger operator with constant magnetic field, by bounded electric
potentials of fixed sign. We also show that, if the perturbation is not of
fixed sign, then any Landau level may be an eigenvalue of arbitrary large,
finite or infinite, multiplicity of the perturbed problem.Comment: This version corrects an error in the statement of Theorem 2. To
appear in International Mathematics Research Notice
Discrete Spectrum of Quantum Hall Effect Hamiltonians I. Monotone Edge Potential
We consider the unperturbed operator ,
self-adjoint in . Here is a magnetic potential which generates a
constant magnetic field , and the edge potential is a non-decreasing
non constant bounded function depending only on the first coordinate
of . Then the spectrum of has a band structure and is
absolutely continuous; moreover, the assumption implies the existence of infinitely many spectral gaps for .
We consider the perturbed operators where the electric
potential is non-negative and decays at infinity. We
investigate the asymptotic distribution of the discrete spectrum of in
the spectral gaps of . We introduce an effective Hamiltonian which governs
the main asymptotic term; this Hamiltonian involves a pseudo-differential
operator with generalized anti-Wick symbol equal to . Further, we restrict
our attention on perturbations of compact support and constant sign. We
establish a geometric condition on the support of which guarantees the
finiteness of the eigenvalues of in any spectral gap of . In the
case where this condition is violated, we show that, generically, the
convergence of the infinite series of eigenvalues of (resp. ) to the
left (resp. right) edge of a given spectral gap, is Gaussian.Comment: 32 page
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