2,053 research outputs found
Finite element approximation for the fractional eigenvalue problem
The purpose of this work is to study a finite element method for finding
solutions to the eigenvalue problem for the fractional Laplacian. We prove that
the discrete eigenvalue problem converges to the continuous one and we show the
order of such convergence. Finally, we perform some numerical experiments and
compare our results with previous work by other authors.Comment: 20 pages, 6 figure
Spectral resolution in hyperbolic orbifolds, quantum chaos, and cosmology
We present a few subjects from physics that have one in common: the spectral
resolution of the Laplacian.Comment: 24 pages. Contribution to the TSL Expository Lecture Series V
"Computational Physical Sciences 2006", Universiti Putra Malaysi
Quantum mechanics in fractional and other anomalous spacetimes
We formulate quantum mechanics in spacetimes with real-order fractional
geometry and more general factorizable measures. In spacetimes where
coordinates and momenta span the whole real line, Heisenberg's principle is
proven and the wave-functions minimizing the uncertainty are found. In spite of
the fact that ordinary time and spatial translations are broken and the
dynamics is not unitary, the theory is in one-to-one correspondence with a
unitary one, thus allowing us to employ standard tools of analysis. These
features are illustrated in the examples of the free particle and the harmonic
oscillator. While fractional (and the more general anomalous-spacetime) free
models are formally indistinguishable from ordinary ones at the classical
level, at the quantum level they differ both in the Hilbert space and for a
topological term fixing the classical action in the path integral formulation.
Thus, all non-unitarity in fractional quantum dynamics is encoded in a
contribution depending only on the initial and final state.Comment: 22 pages, 1 figure. v2: typos correcte
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