20,870 research outputs found
Fourth Order Algorithms for Solving the Multivariable Langevin Equation and the Kramers Equation
We develop a fourth order simulation algorithm for solving the stochastic
Langevin equation. The method consists of identifying solvable operators in the
Fokker-Planck equation, factorizing the evolution operator for small time steps
to fourth order and implementing the factorization process numerically. A key
contribution of this work is to show how certain double commutators in the
factorization process can be simulated in practice. The method is general,
applicable to the multivariable case, and systematic, with known procedures for
doing fourth order factorizations. The fourth order convergence of the
resulting algorithm allowed very large time steps to be used. In simulating the
Brownian dynamics of 121 Yukawa particles in two dimensions, the converged
result of a first order algorithm can be obtained by using time steps 50 times
as large. To further demostrate the versatility of our method, we derive two
new classes of fourth order algorithms for solving the simpler Kramers equation
without requiring the derivative of the force. The convergence of many fourth
order algorithms for solving this equation are compared.Comment: 19 pages, 2 figure
Exact and quasiexact solvability of second-order superintegrable quantum systems: I. Euclidean space preliminaries
We show that second-order superintegrable systems in two-dimensional and three-dimensional Euclidean space generate both exactly solvable (ES) and quasiexactly solvable (QES) problems in quantum mechanics via separation of variables, and demonstrate the increased insight into the structure of such problems provided by superintegrability. A principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable. Most past separation of variables treatments of QES problems via partial differential equations have only incorporated separability, not multiseparability. Also, we propose another definition of ES and QES. The quantum mechanical problem is called ES if the solution of Schrödinger equation can be expressed in terms of hypergeometric functions mFn and is QES if the Schrödinger equation admits polynomial solutions with coefficients necessarily satisfying a three-term or higher order of recurrence relations. In three dimensions we give an example of a system that is QES in one set of separable coordinates, but is not ES in any other separable coordinates. This example encompasses Ushveridze's tenth-order polynomial QES problem in one set of separable coordinates and also leads to a fourth-order polynomial QES problem in another separable coordinate set
New features of collective motion of intrinsic degrees of freedom. Toward a possible way to classify the intrinsic states
Three exactly solvable Hamiltonians of complex structure are studied in the
framework of a semi-classical approach. The quantized trajectories for
intrinsic coordinates correspond to energies which may be classified in
collective bands. For two of the chosen Hamiltonians the symmetry SU2xSU2 is
the appropriate one to classify the eigenvalues in the laboratory frame.
Connections of results presented here with the molecular spectrum and
Moszkowski model are pointed out. The present approach suggests that the
intrinsic states, which in standard formalisms are heading rotational bands,
are forming themselves "rotational" bands, the rotations being performed in a
fictious boson space.Comment: 33 pages, 9 figure
Exactly-solvable coupled-channel potential models of atom-atom magnetic Feshbach resonances from supersymmetric quantum mechanics
Starting from a system of radial Schr\"odinger equations with a vanishing
potential and finite threshold differences between the channels, a coupled exactly-solvable potential model is obtained with the help of a
single non-conservative supersymmetric transformation. The obtained potential
matrix, which subsumes a result obtained in the literature, has a compact
analytical form, as well as its Jost matrix. It depends on
unconstrained parameters and on one upper-bounded parameter, the factorization
energy. A detailed study of the model is done for the case: a
geometrical analysis of the zeros of the Jost-matrix determinant shows that the
model has 0, 1 or 2 bound states, and 0 or 1 resonance; the potential
parameters are explicitly expressed in terms of its bound-state energies, of
its resonance energy and width, or of the open-channel scattering length, which
solves schematic inverse problems. As a first physical application,
exactly-solvable atom-atom interaction potentials are constructed,
for cases where a magnetic Feshbach resonance interplays with a bound or
virtual state close to threshold, which results in a large background
scattering length.Comment: 19 pages, 15 figure
Invariant Modules and the Reduction of Nonlinear Partial Differential Equations to Dynamical Systems
We completely characterize all nonlinear partial differential equations
leaving a given finite-dimensional vector space of analytic functions
invariant. Existence of an invariant subspace leads to a re duction of the
associated dynamical partial differential equations to a system of ordinary
differential equations, and provide a nonlinear counterpart to quasi-exactly
solvable quantum Hamiltonians. These results rely on a useful extension of the
classical Wronskian determinant condition for linear independence of functions.
In addition, new approaches to the characterization o f the annihilating
differential operators for spaces of analytic functions are presented.Comment: 28 pages. To appear in Advances in Mathematic
Supersymmetric quantum mechanics and Painleve equations
In these lecture notes we shall study first the supersymmetric quantum
mechanics (SUSY QM), specially when applied to the harmonic and radial
oscillators. In addition, we will define the polynomial Heisenberg algebras
(PHA), and we will study the general systems ruled by them: for zero and first
order we obtain the harmonic and radial oscillators, respectively; for second
and third order PHA the potential is determined by solutions to Painleve IV
(PIV) and Painleve V (PV) equations. Taking advantage of this connection, later
on we will find solutions to PIV and PV equations expressed in terms of
confluent hypergeometric functions. Furthermore, we will classify them into
several solution hierarchies, according to the specific special functions they
are connected with.Comment: 38 pages, 20 figures. Lecture presented at the XLIII Latin American
School of Physics: ELAF 2013 in Mexico Cit
Mutual Exclusion Statistics in Exactly Solvable Models in One and Higher Dimensions at Low Temperatures
We study statistical characterization of the many-body states in exactly
solvable models with internal degrees of freedom. The models under
consideration include the isotropic and anisotropic Heisenberg spin chain, the
Hubbard chain, and a model in higher dimensions which exhibits the Mott
metal-insulator transition. It is shown that the ground state of these systems
is all described by that of a generalized ideal gas of particles (called
exclusons) which have mutual exclusion statistics, either between different
rapidities or between different species. For the Bethe ansatz solvable models,
the low temperature properties are well described by the excluson description
if the degeneracies due to string solutions with complex rapidities are taken
into account correctly. {For} the Hubbard chain with strong but finite
coupling, charge-spin separation is shown for thermodynamics at low
temperatures. Moreover, we present an exactly solvable model in arbitrary
dimensions which, in addition to giving a perspective view of spin-charge
separation, constitutes an explicit example of mutual exclusion statistics in
more than two dimensions
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