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2 votes
0 answers
159 views

On Fredholm alternative for Neumann conditions

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and $f \in L^2(\Omega)$. It is well known that if $\lambda$ is a Dirichlet Laplacian eigenvalue, then the equation $$\begin{cases} -\Delta ...
student's user avatar
  • 1,350
1 vote
0 answers
400 views

Calculating frequency of sound of ringing metal coin

I would like to reproduce the results of Manas - The music of gold: Can gold counterfeited coins be detected by ear?, but it skips a lot of steps, and the mathematics behind it is a bit advanced for ...
Black Carrot's user avatar
4 votes
1 answer
501 views

Question on expansion into Neumann eigenfunctions

Let $\Omega$ be an open bounded domain with a boundary $\partial\Omega$. Consider the following Neumann eigenvalue problem for Laplacian: find $(\phi_n,\lambda_n)\in H^1(\Omega)\times \mathbb{R}$ \...
user118240's user avatar
1 vote
0 answers
116 views

Eigenvalues of elliptic operator analytic with respect to a parameter

I am interested when one can say the eigenvalues of an elliptic operator are real analytic with respect to a parameter. In particular I have seen many people say the first eigenvalue is analytic but ...
Math604's user avatar
  • 1,385
4 votes
1 answer
336 views

Fundamental gap for Schrödinger operator

Consider $ \Omega$ a smooth bounded domain in $ \mathbb R^N$. I am interested in the gap between the first and second eigenvalues of the operator $ -\Delta + V(x)$. Let $ \phi_1>0$ and $ \phi_2$ ...
Math604's user avatar
  • 1,385
1 vote
0 answers
74 views

Fundamental gap for Neumann BVP with potential

I am sure this is extremely well known but I have been digging a bit and I can't find what I need. Consider $B$ to be the unit ball in $ \mathbb R^N$ and consider the eigenvalue problem \begin{cases}...
Math604's user avatar
  • 1,385