Eigenfunctions of the Beltrami-Laplace operator on a hyperboloid of one sheet

1970 ◽  
Vol 7 (2) ◽  
pp. 152-157
Author(s):  
V. -B. K. Rogov
Keyword(s):  
Laplace Operator ◽  
Beltrami Laplace Operator

Diffusion in κ-deformed space and spectral dimension

2016 ◽  
Vol 31 (09) ◽  
pp. 1650056 ◽  
Author(s):  
V. Anjana
Keyword(s):  
Diffusion Equation ◽  
Laplace Operator ◽  
Deformation Parameter ◽  
Minkowski Spacetime ◽  
Spectral Dimension ◽  
Topological Dimension ◽  
High Energies ◽  
Large Diffusion ◽  
Beltrami Laplace Operator

In this paper, we derive the expression for spectral dimension using a modified diffusion equation in the [Formula: see text]-deformed spacetime. We start with the Beltrami–Laplace operator in the [Formula: see text]-Minkowski spacetime and obtain the deformed diffusion equation. From the solution of this deformed diffusion equation, we calculate the spectral dimension which depends on the deformation parameter “[Formula: see text]” and also on an integer “[Formula: see text]”, apart from the topological dimension. Using this, we show that, for large diffusion times the spectral dimension approaches the usual topological dimension whereas spectral dimension diverges to [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text] at high energies.

Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds

1949 ◽  
Vol 1 (3) ◽  
pp. 242-256 ◽  
Author(s):  
S. Minakshisundaram ◽  
Å. Pleijel
Keyword(s):  
Differential Equation ◽  
Riemannian Manifold ◽  
Laplace Operator ◽  
Riemannian Manifolds ◽  
Line Element ◽  
Metric Tensor ◽  
Local Coordinates ◽  
Image Position ◽  
The Laplace Operator ◽  
Beltrami Laplace Operator

Let V be a connected, compact, differentiable Riemannian manifold. If V is not closed we denote its boundary by S. In terms of local coordinates (xi), i = 1, 2, … Ν, the line-element dr is given by where gik (x1, x2, … xN) are the components of the metric tensor on V We denote by Δ the Beltrami-Laplace-Operator and we consider on V the differential equation (1) Δu + λu = 0.

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