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Results tagged with sp.spectral-theory
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user 13093
Schrodinger operators, operators on manifolds, general differential operators, numerical studies, integral operators, discrete models, resonances, non-self-adjoint operators, random operators/matrices
8
votes
First eigenvalue of the Laplacian on a regular polygon
For $N \geq 5$ it is still not known if the $N$-gon which minimizes the first eigenvalue under area constraint (which exists), is the regular one. I have done some numerical computations which suggest …
- 2,222
1
vote
Multiplicity of Laplace eigenvalues and symmetry
When minimizing numerically the eigenvalues of the Laplacian under area constraint in 2D it is observed that optimal shapes tend to have multiple eigenvalues. Take for example the simulations shown he …
- 2,222
2
votes
First eigenvalue of the Laplacian on a regular polygon
Some recent activity on the problem:
Results regarding the Hessian matrix with respect to vertex coordinates, numerical local minimality estimates and analytical bounds on the diameter of the optimal …
- 2,222
12
votes
0
answers
209
views
Classes for which the Spectrum determines a Convex Shape
Given a planar domain $\Omega \subset \Bbb{R}^2$ bounded and open we can associate to it the spectrum of the Laplace operator with Dirichlet boundary condition. It is known that there are planar domai …
CommunityBot
- 1
3
votes
1
answer
164
views
Analytical value for the first eigenvalue of a certain spherical triangle
I am testing some numerical algorithms for computing the Laplace-Beltrami eigenvalues on the sphere. One thing that came up was computing the first eigenvalue of the "equilateral" spherical triangle w …
- 44.4k
1
vote
Spectrum of Dirichlet Problem for Laplacian on a Parallelogram
This may not be what you wish for, but here is a list of the first $10$ eigenvalues calculated numerically for the rhombus with sidelength $1$:
$$\begin{array}{c}24.8982\\
52.6379\\
71.7085\\
…
- 2,222