# Tagged Questions

Spectral theory is the study of generalized notions of eigenvalues and eigenvectors for linear operators in Banach spaces.

74 views

I'm stuck on an exercise about the spectrum of a selfadjoint operator on a Hilbert space. The problem is the following: Let $(X,\langle \cdot, \cdot\rangle)$ a Hilbert space and let $A \in B(H)$ a ...
42 views

### A Question Regarding Stone's Formula

Let $A$ be a bounded self-adjoint operator on a separable Hilbert space $\mathcal{H}$: $$A\in\mathcal{B}\left(\mathcal{H}\right)\,,\,A=A^\ast$$ Stone's formula (Reed & Simon Theorem VII.13, as an ...
11 views

### $\lim_{k \to \infty} \frac{N(\lambda_k)(2 \pi)^3}{\omega_3 \text{Vol}(\mathbb{S}^2) \lambda^{\frac{3}{2}} } =1 ~\not = 0$

I would like to use Weyl's to confirm my result from the spectrum of $\mathbb{S^2}$. So far I found that the spectrum of $\mathbb{S^2}$ is $\{k(k+1) : k \in \mathbb{N} \cup \{0\} \}$ and each ...
77 views

25 views

36 views

63 views

### How to obtain $N_{\mu, i} (\lambda)=c_n \text{vol} (Q_i) \lambda^{\frac{n}{2}}+o(\lambda^{\frac{n}{2}})$? - Weyl's law

I am trying to prove the Weyl's asymptotic law for eigenvalues. In the document Weyl's law of p. $4$, I have managed to go up to the step \tilde{\nu_k} \leq \nu_k \leq \mu_k \leq \tilde{\mu}_k \...
10 views

### Spectral bands of periodic differential operators

I am reading the book "Multidimensional Periodic Schrödinger Operator" (O. Veliev, 2015) which says on page 11: It is well-known the following statements about the spectral properties of $L_{t}(q)$...
54 views

### Find spectrum of integral operator

Let Af(x) = $\int_0^1 K(x,y)f(y)dy$, $A:L_2[0,1]\rightarrow L_2[0,1].$ Where $K(x,y) = \sinh(\min(x,y)\sinh(1-\max(x,y)).$ where $\sinh(x) = \frac{e^x - e^{-x}}{2}$ Find $\sigma(A), ||A||.$ I ...
17 views

37 views

### Explanations : $\cup_{(j,k) \in E} S(j,k) ⊃E − (1, 1)$ $∩$ $($first quadrant$)$

I am stuck on a problem for a good while now. Is there anyone could tell me rigorously why $\cup_{(j,k) \in E} S(j,k) ⊃E − (1, 1) ∩ ($first quadrant$)$ of the problem of rectangle on page $18-19$ of ...
62 views

### How could we obtain $\lim_{n \to \infty} \frac{\lambda_n}{n}=\frac{4 \pi}{ab}$?

Related to the example on the rectangle in the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$nd Ed.) page $326$, is there anyone could explain to me how is it ...
In the book Strauss W.A. Partial Differential Equations - an Introduction (Wiley, 2008, 1st Ed.) page $311$, there is a comment Now an arbitrary plane domain $D$ can be approximated by unions of ...