ASYMPTOTICS OF THE EIGENVALUES OF THE LAPLACIAN AND QUASIMODES. A SERIES OF QUASIMODES CORRESPONDING TO A SYSTEM OF CAUSTICS CLOSE TO THE BOUNDARY OF THE DOMAIN

1973 ◽  
Vol 7 (2) ◽  
pp. 439-466 ◽  
Author(s):  
V F Lazutkin
Keyword(s):  
Eigenvalues Of The Laplacian ◽  
Asymptotics Of The Eigenvalues

scholarly journals Extrema of Low Eigenvalues of the Dirichlet–Neumann Laplacian on a Disk

2010 ◽  
Vol 62 (4) ◽  
pp. 808-826
Author(s):  
Eveline Legendre
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Neumann Boundary ◽  
Mixed Boundary Conditions ◽  
First Eigenvalue ◽  
Bounded Number ◽  
Eigenvalues Of The Laplacian ◽  
Neumann Laplacian ◽  
Dirichlet And Neumann Conditions ◽  
Second Eigenvalue

AbstractWe study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet–Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact 1-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.

Jacobi Matrices and the Spectrum of the Neumann Operator on a Family of Riemann Surfaces

1993 ◽  
Vol 45 (4) ◽  
pp. 709-726
Author(s):  
Julian Edward
Keyword(s):  
Harmonic Function ◽  
Riemann Surfaces ◽  
Smooth Manifold ◽  
Hypergeometric Functions ◽  
Normal Derivative ◽  
Jacobi Matrices ◽  
Uniform Bounds ◽  
Neumann Operator ◽  
Direct Sum ◽  
Asymptotics Of The Eigenvalues

AbstractThe Neumann operator is an operator on the boundary of a smooth manifold which maps the boundary value of a harmonic function to its normal derivative. The spectrum of the Neumann operator is studied on the curves bounding a family of Riemann surfaces. The Neumann operator is shown to be isospectral to a direct sum of symmetric Jacobi matrices, each acting on l2(ℤ). The Jacobi matrices are shown to be isospectral to generators of bilateral, linear birth-death processes. Using the connection between Jacobi matrices and continued fractions, it is shown that the eigenvalues of the Neumann operator must solve a certain equation involving hypergeometric functions. Study of the equation yields uniform bounds on the eigenvalues and also the asymptotics of the eigenvalues as the curves degenerate into a wedge of circles.

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