349,775 research outputs found
The Step-Harmonic Potential
We analyze the behavior of a quantum system described by a one-dimensional
asymmetric potential consisting of a step plus a harmonic barrier. We solve the
eigenvalue equation by the integral representation method, which allows us to
classify the independent solutions as equivalence classes of homotopic paths in
the complex plane. We then consider the propagation of a wave packet reflected
by the harmonic barrier and obtain an expression for the interaction time as a
function of the peak energy. For high energies we recover the classical
half-period limit.Comment: 19 pages, 7 figure
Preconditioning harmonic unsteady potential flow calculations
This paper considers finite element discretisations of the Helmholtz equation and its generalisation arising from harmonic acoustics perturbations to a non-uniform steady potential flow. A novel elliptic, positive definite preconditioner, with a multigrid implementation, is used to accelerate the iterative convergence of Krylov subspace solvers. Both theory and numerical results show that for a model 1D Helmholtz test problem the preconditioner clusters the discrete system's eigenvalues and lowers its condition number to a level independent of grid resolution. For the 2D Helmholtz equation, grid independent convergence is achieved using a QMR Krylov solver, significantly outperforming the popular SSOR preconditioner. Impressive results are also presented on more complex domains, including an axisymmetric aircraft engine inlet with non-stagnant mean flow and modal boundary conditions
Efimov Trimers in a Harmonic Potential
We study the Efimov effect in a harmonic oscillator in the hyperspherical
formulation, and show how a reduced model allows for a description that is a
generalization of the Efimov effect in free space and leads to results that are
easily interpreted. Efimov physics may be observed by varying the value of the
scattering length, since in the regime where the trimers have a mixed harmonic
oscillator and Efimov character, the inelastic properties of these states are
still manageable. The model also allows for the study of non-universal Efimov
trimers by including the effective range scattering parameter. While we find
that in a certain regime the effective range parameter can take over the role
of the three-body parameter, interestingly, we obtain a numerical relationship
between these two parameters different from what was found in other models.Comment: 15 pages, 9 figure
Nonrelativistic conformal field theories
We study representations of the Schr\"odinger algebra in terms of operators
in nonrelativistic conformal field theories. We prove a correspondence between
primary operators and eigenstates of few-body systems in a harmonic potential.
Using the correspondence we compute analytically the energy of fermions at
unitarity in a harmonic potential near two and four spatial dimensions. We also
compute the energy of anyons in a harmonic potential near the bosonic and
fermionic limits.Comment: 26 pages, 9 figures; added a comment on the convergence of epsilon
expansion
Flows and Decompositions of Games: Harmonic and Potential Games
In this paper we introduce a novel flow representation for finite games in
strategic form. This representation allows us to develop a canonical direct sum
decomposition of an arbitrary game into three components, which we refer to as
the potential, harmonic and nonstrategic components. We analyze natural classes
of games that are induced by this decomposition, and in particular, focus on
games with no harmonic component and games with no potential component. We show
that the first class corresponds to the well-known potential games. We refer to
the second class of games as harmonic games, and study the structural and
equilibrium properties of this new class of games. Intuitively, the potential
component of a game captures interactions that can equivalently be represented
as a common interest game, while the harmonic part represents the conflicts
between the interests of the players. We make this intuition precise, by
studying the properties of these two classes, and show that indeed they have
quite distinct and remarkable characteristics. For instance, while finite
potential games always have pure Nash equilibria, harmonic games generically
never do. Moreover, we show that the nonstrategic component does not affect the
equilibria of a game, but plays a fundamental role in their efficiency
properties, thus decoupling the location of equilibria and their payoff-related
properties. Exploiting the properties of the decomposition framework, we obtain
explicit expressions for the projections of games onto the subspaces of
potential and harmonic games. This enables an extension of the properties of
potential and harmonic games to "nearby" games. We exemplify this point by
showing that the set of approximate equilibria of an arbitrary game can be
characterized through the equilibria of its projection onto the set of
potential games
Transitionless quantum drivings for the harmonic oscillator
Two methods to change a quantum harmonic oscillator frequency without
transitions in a finite time are described and compared. The first method, a
transitionless-tracking algorithm, makes use of a generalized harmonic
oscillator and a non-local potential. The second method, based on engineering
an invariant of motion, only modifies the harmonic frequency in time, keeping
the potential local at all times.Comment: 11 pages, 1 figure. Submitted for publicatio
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