169 research outputs found
Singular components of spectral measures for ergodic Jacobi matrices
For ergodic 1d Jacobi operators we prove that the random singular components
of any spectral measure are almost surely mutually disjoint as long as one
restricts to the set of positive Lyapunov exponent. In the context of extended
Harper's equation this yields the first rigorous proof of the Thouless' formula
for the Lyapunov exponent in the dual regions.Comment: to appear in the Journal of Mathematical Physics, vol 52 (2011
Multiscale Analysis in Momentum Space for Quasi-periodic Potential in Dimension Two
We consider a polyharmonic operator H=(-\Delta)^l+V(\x) in dimension two
with , being an integer, and a quasi-periodic potential V(\x).
We prove that the absolutely continuous spectrum of contains a semiaxis and
there is a family of generalized eigenfunctions at every point of this semiaxis
with the following properties. First, the eigenfunctions are close to plane
waves at the high energy region. Second, the isoenergetic
curves in the space of momenta \k corresponding to these eigenfunctions have
a form of slightly distorted circles with holes (Cantor type structure). A new
method of multiscale analysis in the momentum space is developed to prove these
results.Comment: 125 pages, 4 figures. arXiv admin note: incorporates arXiv:1205.118
Bound States at Threshold resulting from Coulomb Repulsion
The eigenvalue absorption for a many-particle Hamiltonian depending on a
parameter is analyzed in the framework of non-relativistic quantum mechanics.
The long-range part of pair potentials is assumed to be pure Coulomb and no
restriction on the particle statistics is imposed. It is proved that if the
lowest dissociation threshold corresponds to the decay into two likewise
non-zero charged clusters then the bound state, which approaches the threshold,
does not spread and eventually becomes the bound state at threshold. The
obtained results have applications in atomic and nuclear physics. In
particular, we prove that atomic ion with atomic critical charge and
electrons has a bound state at threshold given that , whereby the electrons are treated as fermions and the mass of the
nucleus is finite.Comment: This is a combined and updated version of the manuscripts
arXiv:math-ph/0611075v2 and arXiv:math-ph/0610058v
Pauli-Fierz model with Kato-class potentials and exponential decays
Generalized Pauli-Fierz Hamiltonian with Kato-class potential \KPF in
nonrelativistic quantum electrodynamics is defined and studied by a path
measure. \KPF is defined as the self-adjoint generator of a strongly
continuous one-parameter symmetric semigroup and it is shown that its bound
states spatially exponentially decay pointwise and the ground state is unique.Comment: We deleted Lemma 3.1 in vol.
On Singularity formation for the L^2-critical Boson star equation
We prove a general, non-perturbative result about finite-time blowup
solutions for the -critical boson star equation in 3 space dimensions. Under
the sole assumption that the solution blows up in at finite time, we
show that has a unique weak limit in and that has a
unique weak limit in the sense of measures. Moreover, we prove that the
limiting measure exhibits minimal mass concentration. A central ingredient used
in the proof is a "finite speed of propagation" property, which puts a strong
rigidity on the blowup behavior of .
As the second main result, we prove that any radial finite-time blowup
solution converges strongly in away from the origin. For radial
solutions, this result establishes a large data blowup conjecture for the
-critical boson star equation, similar to a conjecture which was
originally formulated by F. Merle and P. Raphael for the -critical
nonlinear Schr\"odinger equation in [CMP 253 (2005), 675-704].
We also discuss some extensions of our results to other -critical
theories of gravitational collapse, in particular to critical Hartree-type
equations.Comment: 24 pages. Accepted in Nonlinearit
On the Geometry of Supersymmetric Quantum Mechanical Systems
We consider some simple examples of supersymmetric quantum mechanical systems
and explore their possible geometric interpretation with the help of geometric
aspects of real Clifford algebras. This leads to natural extensions of the
considered systems to higher dimensions and more complicated potentials.Comment: 18 page
Extended States for Polyharmonic Operators with Quasi-periodic Potentials in Dimension Two
We consider a polyharmonic operator H=(-\Delta)^l+V(\x) in dimension two
with , being an integer, and a quasi-periodic potential V(\x).
We prove that the spectrum of contains a semiaxis and there is a family of
generalized eigenfunctions at every point of this semiaxis with the following
properties. First, the eigenfunctions are close to plane waves
at the high energy region. Second, the isoenergetic curves in the space of
momenta \k corresponding to these eigenfunctions have a form of slightly
distorted circles with holes (Cantor type structure). A new method of
multiscale analysis in the momentum space is developed to prove these results.Comment: This is an announcement only. Text with the detailed proof is under
preparation. 11 pages, 4 figures. arXiv admin note: text overlap with
arXiv:math-ph/0601008, arXiv:0711.4404, arXiv:1008.463
Existence of the Stark-Wannier quantum resonances
In this paper we prove the existence of the Stark-Wannier quantum resonances
for one-dimensional Schrodinger operators with smooth periodic potential and
small external homogeneous electric field. Such a result extends the existence
result previously obtained in the case of periodic potentials with a finite
number of open gaps.Comment: 30 pages, 1 figur
Shape invariance through Crum transformation
We show in a rigorous way that Crum's result on equal eigenvalue spectrum of
Sturm-Liouville problems can be obtained iteratively by successive Darboux
transformations. It can be shown that all neighbouring Darboux-transformed
potentials of higher order, u_{k} and u_{k+1}, satisfy the condition of shape
invariance provided the original potential u does. We use this result to proof
that under the condition of shape invariance the n-th iteration of the original
Sturm-Liouville problem defined through shape invariance is equal to the n-th
Crum transformationComment: 26 pp, one more reference, J.-M. Sparenberg and D. Baye, J. Phys. A
28, 5079 (1995), has been added as Ref. 18 in the published version, which
has 47 ref
Heat kernel estimates and spectral properties of a pseudorelativistic operator with magnetic field
Based on the Mehler heat kernel of the Schroedinger operator for a free
electron in a constant magnetic field an estimate for the kernel of E_A is
derived, where E_A represents the kinetic energy of a Dirac electron within the
pseudorelativistic no-pair Brown-Ravenhall model. This estimate is used to
provide the bottom of the essential spectrum for the two-particle
Brown-Ravenhall operator, describing the motion of the electrons in a central
Coulomb field and a constant magnetic field, if the central charge is
restricted to Z below or equal 86
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