29 research outputs found
Commutator Criteria for Magnetic Pseudodifferential Operators
The gauge covariant magnetic Weyl calculus has been introduced and studied in
previous works. We prove criteria in terms of commutators for operators to be
magnetic pseudo-differential operators of suitable symbol classes. The approach
is completely intrinsic; neither the statements nor the proofs depend on a
choice of a vector potential. We apply this criteria to inversion problems,
functional calculus, affiliation results and to the study of the evolution
group generated by a magnetic pseudo-differential operator.Comment: Acknowledgements adde
A microscopic derivation of the quantum mechanical formal scattering cross section
We prove that the empirical distribution of crossings of a "detector''
surface by scattered particles converges in appropriate limits to the
scattering cross section computed by stationary scattering theory. Our result,
which is based on Bohmian mechanics and the flux-across-surfaces theorem, is
the first derivation of the cross section starting from first microscopic
principles.Comment: 28 pages, v2: Typos corrected, layout improved, v3: Typos corrected.
Accepted for publication in Comm. Math. Phy
Surface Gap Soliton Ground States for the Nonlinear Schr\"{o}dinger Equation
We consider the nonlinear Schr\"{o}dinger equation , with and and with periodic in each coordinate direction. This problem
describes the interface of two periodic media, e.g. photonic crystals. We study
the existence of ground state solutions (surface gap soliton ground
states) for . Using a concentration compactness
argument, we provide an abstract criterion for the existence based on ground
state energies of each periodic problem (with and ) as well as a more practical
criterion based on ground states themselves. Examples of interfaces satisfying
these criteria are provided. In 1D it is shown that, surprisingly, the criteria
can be reduced to conditions on the linear Bloch waves of the operators
and .Comment: definition of ground and bound states added, assumption (H2) weakened
(sign changing nonlinearity is now allowed); 33 pages, 4 figure
Localization on quantum graphs with random vertex couplings
We consider Schr\"odinger operators on a class of periodic quantum graphs
with randomly distributed Kirchhoff coupling constants at all vertices. Using
the technique of self-adjoint extensions we obtain conditions for localization
on quantum graphs in terms of finite volume criteria for some energy-dependent
discrete Hamiltonians. These conditions hold in the strong disorder limit and
at the spectral edges
Classical scattering with long range forces
We discuss the classical two-body scattering problem for potentials which decrease at infinity like r âÎą , 1â§Îą>0. We prove existence and uniqueness theorems for scattering orbits parametrized by their asymptotic data. Wave operators are constructed and their properties discussed. We also discuss and prove cluster properties of the S -operator.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46509/1/220_2005_Article_BF01646193.pd
Stable directions for small nonlinear Dirac standing waves
We prove that for a Dirac operator with no resonance at thresholds nor
eigenvalue at thresholds the propagator satisfies propagation and dispersive
estimates. When this linear operator has only two simple eigenvalues close
enough, we study an associated class of nonlinear Dirac equations which have
stationary solutions. As an application of our decay estimates, we show that
these solutions have stable directions which are tangent to the subspaces
associated with the continuous spectrum of the Dirac operator. This result is
the analogue, in the Dirac case, of a theorem by Tsai and Yau about the
Schr\"{o}dinger equation. To our knowledge, the present work is the first
mathematical study of the stability problem for a nonlinear Dirac equation.Comment: 62 page
SchrĂśdinger operators with δ and δâ˛-potentials supported on hypersurfaces
Self-adjoint SchrĂśdinger operators with δ and δâ˛-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 BirmanâSchwinger principle and a variant of Kreinâs formula are shown. Furthermore, Schattenâvon Neumann type estimates for the differences of the powers of the resolvents of the SchrĂśdinger operators with δ and δâ˛-potentials, and the SchrĂśdinger 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 SchrĂśdinger 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