16 research outputs found
Nonlocally-induced (quasirelativistic) bound states: Harmonic confinement and the finite well
Nonlocal Hamiltonian-type operators, like e.g. fractional and
quasirelativistic, seem to be instrumental for a conceptual broadening of
current quantum paradigms. However physically relevant properties of related
quantum systems have not yet received due (and scientifically undisputable)
coverage in the literature. In the present paper we address
Schr\"{o}dinger-type eigenvalue problems for , where a kinetic term
is a quasirelativistic energy operator of mass particle. A potential we assume
to refer to the harmonic confinement or finite well of an arbitrary depth. We
analyze spectral solutions of the pertinent nonlocal quantum systems with a
focus on their -dependence. Extremal mass regimes for eigenvalues and
eigenfunctions of are investigated: (i) spectral affinity
("closeness") with the Cauchy-eigenvalue problem () and (ii) spectral affinity with the nonrelativistic eigenvalue
problem (). To this end we generalize to
nonlocal operators an efficient computer-assisted method to solve
Schr\"{o}dinger eigenvalue problems, widely used in quantum physics and quantum
chemistry. A resultant spectrum-generating algorithm allows to carry out all
computations directly in the configuration space of the nonlocal quantum
system. This allows for a proper assessment of the spatial nonlocality impact
on simulation outcomes. Although the nonlocality of might seem to stay in
conflict with various numerics-enforced cutoffs, this potentially serious
obstacle is kept under control and effectively tamed.Comment: 23 pages, 16 figure
Fractional-order operators: Boundary problems, heat equations
The first half of this work gives a survey of the fractional Laplacian (and
related operators), its restricted Dirichlet realization on a bounded domain,
and its nonhomogeneous local boundary conditions, as treated by
pseudodifferential methods. The second half takes up the associated heat
equation with homogeneous Dirichlet condition. Here we recall recently shown
sharp results on interior regularity and on -estimates up to the boundary,
as well as recent H\"older estimates. This is supplied with new higher
regularity estimates in -spaces using a technique of Lions and Magenes,
and higher -regularity estimates (with arbitrarily high H\"older estimates
in the time-parameter) based on a general result of Amann. Moreover, it is
shown that an improvement to spatial -regularity at the boundary is
not in general possible.Comment: 29 pages, updated version, to appear in a Springer Proceedings in
Mathematics and Statistics: "New Perspectives in Mathematical Analysis -
Plenary Lectures, ISAAC 2017, Vaxjo Sweden
Thermalization of Lévy Flights: Path-Wise Picture in 2D
We analyze two-dimensional (2D) random systems driven by a symmetric Lévy stable noise which in the presence of confining potentials may asymptotically set down at Boltzmann-type thermal equilibria. In view of the Eliazar-Klafter no-go statement, such dynamical behavior is plainly incompatible with the standard Langevin modeling of Lévy flights. No explicit path-wise description has been so far devised for the thermally equilibrating random motion we address, and its formulation is the principal goal of the present work. To this end we prescribe a priori the target pdf ρ∗ in the Boltzmann form ~exp[] and next select the Lévy noise (e.g., its Lévy measure) of interest. To reconstruct random paths of the underlying stochastic process we resort to numerical methods. We create a suitably modified version of the time honored Gillespie algorithm, originally invented
in the chemical kinetics context. A statistical analysis of generated sample trajectories allows us to infer a surrogate pdf dynamics which sets down at a predefined target, in consistency with the associated kinetic (master) equation
Levy processes in bounded domains: Path-wise reflection scenarios and signatures of confinement
We discuss an impact of various (path-wise) reflection-from-the barrier
scenarios upon confining properties of a paradigmatic family of symmetric
-stable L\'{e}vy processes, whose permanent residence in a finite
interval on a line is secured by a two-sided reflection. Depending on the
specific reflection "mechanism", the inferred jump-type processes differ in
their spectral and statistical characteristics, like e.g. relaxation
properties, and functional shapes of invariant (equilibrium, or asymptotic
near-equilibrium) probability density functions in the interval. The analysis
is carried out in conjunction with attempts to give meaning to the notion of a
reflecting L\'{e}vy process, in terms of the domain of its motion generator, to
which an invariant pdf (actually an eigenfunction) does belong.Comment: 20 pp, 8 figures, Text amendments, Abstract and Section I modifie