120 research outputs found
Higher order Schrodinger and Hartree-Fock equations
The domain of validity of the higher-order Schrodinger equations is analyzed
for harmonic-oscillator and Coulomb potentials as typical examples. Then the
Cauchy theory for higher-order Hartree-Fock equations with bounded and Coulomb
potentials is developed. Finally, the existence of associated ground states for
the odd-order equations is proved. This renders these quantum equations
relevant for physics.Comment: 19 pages, to appear in J. Math. Phy
Approximation by point potentials in a magnetic field
We discuss magnetic Schrodinger operators perturbed by measures from the
generalized Kato class. Using an explicit Krein-like formula for their
resolvent, we prove that these operators can be approximated in the strong
resolvent sense by magnetic Schrodinger operators with point potentials. Since
the spectral problem of the latter operators is solvable, one in fact gets an
alternative way to calculate discrete spectra; we illustrate it by numerical
calculations in the case when the potential is supported by a circle.Comment: 16 pages, 2 eps figures, submitted to J. Phys.
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
Magnetic transport in a straight parabolic channel
We study a charged two-dimensional particle confined to a straight
parabolic-potential channel and exposed to a homogeneous magnetic field under
influence of a potential perturbation . If is bounded and periodic along
the channel, a perturbative argument yields the absolute continuity of the
bottom of the spectrum. We show it can have any finite number of open gaps
provided the confining potential is sufficiently strong. However, if
depends on the periodic variable only, we prove by Thomas argument that the
whole spectrum is absolutely continuous, irrespectively of the size of the
perturbation. On the other hand, if is small and satisfies a weak
localization condition in the the longitudinal direction, we prove by Mourre
method that a part of the absolutely continuous spectrum persists
Rigorous Real-Time Feynman Path Integral for Vector Potentials
we will show the existence and uniqueness of a real-time, time-sliced Feynman
path integral for quantum systems with vector potential. Our formulation of the
path integral will be derived on the transition probability amplitude via
improper Riemann integrals. Our formulation will hold for vector potential
Hamiltonian for which its potential and vector potential each carries at most a
finite number of singularities and discontinuities
Ballistic transport in random magnetic fields with anisotropic long-ranged correlations
We present exact theoretical results about energetic and dynamic properties
of a spinless charged quantum particle on the Euclidean plane subjected to a
perpendicular random magnetic field of Gaussian type with non-zero mean. Our
results refer to the simplifying but remarkably illuminating limiting case of
an infinite correlation length along one direction and a finite but strictly
positive correlation length along the perpendicular direction in the plane.
They are therefore ``random analogs'' of results first obtained by A. Iwatsuka
in 1985 and by J. E. M\"uller in 1992, which are greatly esteemed, in
particular for providing a basic understanding of transport properties in
certain quasi-two-dimensional semiconductor heterostructures subjected to
non-random inhomogeneous magnetic fields
Selection of the ground state for nonlinear Schroedinger equations
We prove for a class of nonlinear Schr\"odinger systems (NLS) having two
nonlinear bound states that the (generic) large time behavior is characterized
by decay of the excited state, asymptotic approach to the nonlinear ground
state and dispersive radiation. Our analysis elucidates the mechanism through
which initial conditions which are very near the excited state branch evolve
into a (nonlinear) ground state, a phenomenon known as {\it ground state
selection}.
Key steps in the analysis are the introduction of a particular linearization
and the derivation of a normal form which reflects the dynamics on all time
scales and yields, in particular, nonlinear Master equations.
Then, a novel multiple time scale dynamic stability theory is developed.
Consequently, we give a detailed description of the asymptotic behavior of the
two bound state NLS for all small initial data. The methods are general and can
be extended to treat NLS with more than two bound states and more general
nonlinearities including those of Hartree-Fock type.Comment: Revision of 2001 preprint; 108 pages Te
On the AC spectrum of one-dimensional random Schroedinger operators with matrix-valued potentials
We consider discrete one-dimensional random Schroedinger operators with
decaying matrix-valued, independent potentials. We show that if the l^2-norm of
this potential has finite expectation value with respect to the product measure
then almost surely the Schroedinger operator has an interval of purely
absolutely continuous (ac) spectrum. We apply this result to Schroedinger
operators on a strip. This work provides a new proof and generalizes a result
obtained by Delyon, Simon, and Souillard.Comment: (1 figure
Are N=1 and N=2 supersymmetric quantum mechanics equivalent?
After recalling different formulations of the definition of supersymmetric
quantum mechanics given in the literature, we discuss the relationships between
them in order to provide an answer to the question raised in the title.Comment: 15 page
Convergence and completeness for square-well Stark resonant state expansions
In this paper we investigate the completeness of the Stark resonant
eigenstates for a particle in a square-well potential. We find that the
resonant state expansions for target functions converge inside the potential
well and that the existence of this convergence does not depend on the depth of
the potential well. By analyzing the asymptotic form of the terms in these
expansions we prove some results on the relation between smoothness of target
functions and the rate of convergence of the corresponding resonant state
expansion
- …