## New answers tagged eigenvalues

3
votes

Accepted

### A question about a series of solutions to an elliptic PDE in $B_R$ which is compactly convergent as $R \rightarrow +\infty$

The limit functions are the same basically by construction, there is no claim on uniqueness. Take an exhaustion $K_i \nearrow \mathbb{R}^n $ of $\mathbb{R}^n$ by compact sets. Then on $K_1$ by ...

5
votes

### Free probability: A unitary group heuristic for the relationship between additive free convolution and free compression

I am not sure whether this counts as heuristics, as it goes even deeper into free probability results, but it might give some high-level kind of idea why this result should be true. The main fact ...

6
votes

Accepted

### Free probability: A unitary group heuristic for the relationship between additive free convolution and free compression

I have a suggestion that seems to bring the two sides to be compared much closer together.
Take a block-diagonal matrix $B$ with $k$ blocks each a copy of $A$. This has the same eigenvalue measure as $...

6
votes

### Free probability: A unitary group heuristic for the relationship between additive free convolution and free compression

One can get a certain way towards this goal via a sort of "dimensional analysis". This isn't a completely satisfying heuristic argument - in particular, it only partially specifies what ...

2
votes

Accepted

### Conditions for distinct nonzero eigenvalues in product DAD for symmetric matrix A with repeated nonzero eigenvalues and diagonal matrix D

We can do this by (degenerate) first order perturbation theory. Let's take $D=1+\epsilon C$, with $\epsilon\not=0$ small and $C$ also diagonal. Then
$$
DAD = A +\epsilon(CA+AC) + O(\epsilon^2) .
$$
...

1
vote

### square matrix depending on complex value: spectral radius continous?

$\newcommand\la\lambda$If (all the entries $a_{ij}(z)$ of) the matrix $A(z)$ are continuous in $z$, then, yes, $\la_z$ is continuous in $z$.
Indeed, take any complex $z$ and any sequence $(z_k)$ ...

5
votes

Accepted

### Given an eigenvalue equation (elliptic PDE) in a ball $B_R$, prove the convergence of the first nonzero $\lambda_R$ and its eigenfunction $\phi_R$

For simplicity let me take $H$ smooth or at least $C^\alpha$ so that I don't have to worry about elliptic estimates:
Limit as $R\to\infty$:
Use
$$
\lambda_R = \inf_{\phi \in C^\infty_c(B_R)}\frac{\...

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