4,681 research outputs found
A nodal domain theorem for integrable billiards in two dimensions
Eigenfunctions of integrable planar billiards are studied - in particular,
the number of nodal domains, , of the eigenfunctions are considered. The
billiards for which the time-independent Schr\"odinger equation (Helmholtz
equation) is separable admit trivial expressions for the number of domains.
Here, we discover that for all separable and non-separable integrable
billiards, satisfies certain difference equations. This has been
possible because the eigenfunctions can be classified in families labelled by
the same value of , given a particular , for a set of quantum
numbers, . Further, we observe that the patterns in a family are similar
and the algebraic representation of the geometrical nodal patterns is found.
Instances of this representation are explained in detail to understand the
beauty of the patterns. This paper therefore presents a mathematical connection
between integrable systems and difference equations.Comment: 13 pages, 5 figure
The Solution of Elliptic Difference Equations by Semi-Explicit Iterative Techniques
In [8], the author discusses an iterative scheme for
solving a difference analogue for the elliptic differential equation ∇•α∇u = f on two-dimensional rectangular regions with Dirichlet boundary conditions. It is shown there that a semi-explicit technique involving the inversion only
of the Peaceman-Rachford [10] alternating-direction operators for the Laplacian gives convergence in O(h^(-2) log h^(-1)) operations
Quantum Spectra of Triangular Billiards on the Sphere
We study the quantal energy spectrum of triangular billiards on a spherical
surface. Group theory yields analytical results for tiling billiards while the
generic case is treated numerically. We find that the statistical properties of
the spectra do not follow the standard random matrix results and their peculiar
behaviour can be related to the corresponding classical phase space structure.Comment: 18 pages, 5 eps figure
Conforming finite element methods for the clamped plate problem
Finite element methods for solving biharmonic boundary value problems are considered. The particular problem discussed is that of a clamped thin plate. This problem is reformulated in a weak, form in the Sobolev space Techniques for setting up conforming trial
Functions are utilized in a Galerkin technique to produce finite element solutions. The shortcomings of various trial function formulations are discussed, and a macro—element approach to local mesh refinement using rectangular elements is given
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