The discrete spectrum of the Laplace operator on the fundamental domain of the modular group and the Chebyshev psi-function

2019 ◽  
Vol 83 (5) ◽  
pp. 1066-1079 ◽  
Author(s):  
D. A. Popov
Keyword(s):  
Discrete Spectrum ◽  
Laplace Operator ◽  
Fundamental Domain ◽  
Modular Group ◽  
Psi Function ◽  
The Laplace Operator

scholarly journals A zeta function connected with the eigenvalues of the Laplace-Beltrami operator on the fundamental domain of the modular group

1984 ◽  
Vol 96 ◽  
pp. 167-174 ◽  
Author(s):  
Akio Fujii
Keyword(s):  
Zeta Function ◽  
Discrete Spectrum ◽  
Complex Plane ◽  
Fundamental Domain ◽  
Modular Group ◽  
Beltrami Operator

Let ; … run over the eigenvalues of the discrete spectrum of the Laplace-Beltrami operator on L2(H/yΓ), where H is the upper half of the complex plane and we take Γ = PSL(2, Z). It is well known that Let a be a positive number. Here we are concerned with the zeta function defined by

scholarly journals An upper bound for λ1 for Γ(q) and Γ0(q)

1990 ◽  
Vol 33 (2) ◽  
pp. 241-250
Author(s):  
C. J. Mozzochi
Keyword(s):  
Discrete Spectrum ◽  
Laplace Operator ◽  
Trace Formula ◽  
Positive Constant ◽  
Upper Bound ◽  
Selberg Trace Formula ◽  
Absolute Positive Constant ◽  
Image Position ◽  
The Laplace Operator

Under the assumption of the Selberg conjecture I establish by means of the Selberg trace formula the following:Theorem. Let Γ denote Γ(q) or Γ0(q), q square-free. Let Δq denote the Laplace operator on L2(Γ\H), and let Σq denote its discrete spectrum. Then there exists an absolute positive constant A such that for q≧A

scholarly journals Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

2020 ◽  
Vol 18 (1) ◽  
pp. 1518-1530
Author(s):  
Xuesen Qi ◽  
Ximin Liu
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Coefficient Function ◽  
First Eigenvalue ◽  
Initial Hypersurface ◽  
The Mean ◽  
The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

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