The Riemann Hypothesis for Selberg's Zeta-Function and the Asymptotic Behavior of Eigenvalues of the Laplace Operator

1978 ◽  
Vol 236 ◽  
pp. 209 ◽  
Author(s):  
Burton Randol
Keyword(s):  
Asymptotic Behavior ◽  
Zeta Function ◽  
Laplace Operator ◽  
Riemann Hypothesis ◽  
The Laplace Operator ◽  
Selberg’S Zeta Function

scholarly journals Zeta function determinant of the Laplace operator on theD-dimensional ball

1996 ◽  
Vol 179 (1) ◽  
pp. 215-234 ◽  
Author(s):  
M. Bordag ◽  
B. Geyer ◽  
K. Kirsten ◽  
E. Elizalde
Keyword(s):  
Zeta Function ◽  
Laplace Operator ◽  
Dimensional Ball ◽  
The Laplace Operator

scholarly journals ON THE ASYMPTOTIC BEHAVIOR OF EIGENVALUES OF THE BOUNDARY VALUE PROBLEM WITH A PARAMETER

2017 ◽  
Vol 21 (6) ◽  
pp. 135-140
Author(s):  
A.V. Filinovskiy
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Dirichlet Problem ◽  
Laplace Operator ◽  
Smooth Boundary ◽  
Robin Boundary Condition ◽  
Simple Eigenvalue ◽  
Robin Problem ◽  
The Laplace Operator ◽  
Robin Boundary

The paper presents the investigation of an eigenvalue problem for the Laplace operator with Robin boundary condition in a bounded domain with smooth boundary. The case of boundary condition containing a real parameter is con- sidered. It is proved that multiplicity of the eigenvalue to the Robin problem for all values of the parameter greater than some number does not exceed the mul- tiplicity of the corresponding eigenvalue to the Dirichlet problem for the Laplace operator. For simple eigenvalue of the Dirichlet problem the convergence of eigen- function of the Robin problem to the eigenfunction of the Dirichlet problem for unlimited increase of the parameter is proved. The formula for derivative on the parameter for eigenvalues of the Robin problem is established. This formula is used to justify the asymptotic expansions of eigenvalues of the Robin problem for large positive values of the parameter.

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