Regular representation of the groupSU 3: Eigenfunctions and eigenvalues of the beltrami-laplace operator defined on the 4-dimensional manifold in the euclidean spaceR 8

1969 ◽  
Vol 59 (2) ◽  
pp. 303-307
Author(s):  
R. S. Liotta ◽  
L. Triolo
Keyword(s):  
Laplace Operator ◽  
Regular Representation ◽  
Dimensional Manifold ◽  
Beltrami Laplace Operator

SPECTRAL ASYMPTOTICS OF THE LAPLACE OPERATOR ON MANIFOLDS WITH CYLINDRICAL ENDS

1995 ◽  
Vol 06 (06) ◽  
pp. 911-920 ◽  
Author(s):  
L.B. PARNOVSKI
Keyword(s):  
Laplace Operator ◽  
Scattering Matrix ◽  
Spectral Asymptotics ◽  
First Eigenvalue ◽  
Dimensional Manifold ◽  
The First Eigenvalue ◽  
Counting Functions ◽  
The Laplace Operator

Let M be an n-dimensional manifold with cylindrical ends. We consider the sum of the counting functions of the discrete (Nd(λ)) and continuous spectra of M, the latter beingdefined as [Formula: see text] where T(ν) is the scattering matrix and µ1 is the first eigenvalue of the cylinder’s section. Using the modification of the Colin de Verdière cut-off Laplacian, we prove the followingasymptotic formula: [Formula: see text] where |M0| is the regularized volume of |M|, and Cn is the Weyl constant.

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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