6,393 research outputs found
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
Second Order Superintegrable Systems in Three Dimensions
A classical (or quantum) superintegrable system on an n-dimensional
Riemannian manifold is an integrable Hamiltonian system with potential that
admits 2n-1 functionally independent constants of the motion that are
polynomial in the momenta, the maximum number possible. If these constants of
the motion are all quadratic, the system is second order superintegrable. Such
systems have remarkable properties. Typical properties are that 1) they are
integrable in multiple ways and comparison of ways of integration leads to new
facts about the systems, 2) they are multiseparable, 3) the second order
symmetries generate a closed quadratic algebra and in the quantum case the
representation theory of the quadratic algebra yields important facts about the
spectral resolution of the Schr\"odinger operator and the other symmetry
operators, and 4) there are deep connections with expansion formulas relating
classes of special functions and with the theory of Exact and Quasi-exactly
Solvable systems. For n=2 the author, E.G. Kalnins and J. Kress, have worked
out the structure of these systems and classified all of the possible spaces
and potentials. Here I discuss our recent work and announce new results for the
much more difficult case n=3.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
High-order numerical methods for 2D parabolic problems in single and composite domains
In this work, we discuss and compare three methods for the numerical
approximation of constant- and variable-coefficient diffusion equations in both
single and composite domains with possible discontinuity in the solution/flux
at interfaces, considering (i) the Cut Finite Element Method; (ii) the
Difference Potentials Method; and (iii) the summation-by-parts Finite
Difference Method. First we give a brief introduction for each of the three
methods. Next, we propose benchmark problems, and consider numerical tests-with
respect to accuracy and convergence-for linear parabolic problems on a single
domain, and continue with similar tests for linear parabolic problems on a
composite domain (with the interface defined either explicitly or implicitly).
Lastly, a comparative discussion of the methods and numerical results will be
given.Comment: 45 pages, 12 figures, in revision for Journal of Scientific Computin
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