# Tagged Questions

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

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### Spectrum of the circle - Clarification of certain points

I have a bit of difficulty as doing the problem Eigenvalues of the circle over the Laplacian operator. Since $g$ is a periodic function, do I have to use the Fourier series? If so, how could I do that?...
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### Eigenvalues of the circle over the Laplacian operator

I would like to find the spectrum of a circle. It seems that there are no boundary conditions, but I'm not quite certain. How could I find the spectrum of a circle over the Laplacian operator?
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### Green's function on the unit interval $[0,1]$

I would like to find the Green's function on the unit interval $[0,1]$. We have to solve the linear equation $$\frac{d^2f(x)}{dx}=g(x)$$ with boundary condition $f(0)=a$, $f(1)=b$. So the Green's ...
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### What does 'mode' mean in this context?

The spectrum of an operator $\mathcal{L}$ is the disjoint union of two sets: the point spectrum that consists of all isolated eigenvalues with finite multiplicity and its complement, which we call the ...
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### Laplacian on the straight circle

I would like to well understand the notion of Green's function. I know that a Green's function, $G(x,s)$, of a linear differential operator $L = L(x)$ acting on distributions over a subset of the ...
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### Trace of the heat operator $Z(t)=\sum_{m,n=1}^{\infty}e^{-\frac{\alpha_{m,n}^2}{r_0^2}t}$

I know that the spectrum of the disk of radius $r_0$ is $\lambda_{m,n}=\frac{\alpha_{m,n}^2}{r_0^2}$, where $\alpha_{m,n}$ is the n-th root of the Bessel's function of order $m$. I have to find the ...
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### Good numerical method for finding the eigenvalues and eigenfunctions of the Dirichlet-Laplacian?

Let us confine ourselves to 2D. What is the best numerical method for solving the eigenvalues and eigenfunctions of the Dirichlet-Laplacian operator? Possibly, it depends on the shape of the domain? ...
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### Triangle with the lowest laplacian eigenvalue under the Dirichlet boundary condition

Let us fix the area of the triangle. Which triangle has the lowest Laplacian eigenvalue? The equilateral one?
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### Riemann mapping theorem - Personal project

I would like to work on the Riemann mapping theorem this summer. Does anyone could give me some good references linked to this objective. For your information, I now currently finishing a degree in ...
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### Functional calculus: Does $A$ commute with $e^{iA^2}$?

Let $A$ be an unbounded self-adjoint operator. Is it then true that $A$ commutes with $e^{iA^2}$? This sounds natural to me but I have no clue whether this is true in general. The problem is that we ...
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### An expository reference on spherical harmonics on $S^n$.

I am looking for a thorough reference which explains how to compute the spherical harmonics on $S^n$ and how to upper and lower-bound their values. About the first part of my query, I am not so much ...
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### Show that any conjugate pair of complex numbers (with non-zero imaginary part) cannot be the spectrum of any 2x2 matrix with real, nonnegative entries [duplicate]

My professor showed me this in her office today but I didn't like her method and wanted to use another method. So, I computed the characteristic polynomial of some arbitrary $2 \times 2$ matrix ...
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### Spectrum of Linear Operator

The spectrum of a linear operator on a finite dimensional space is pure point spectrum, that is, both continuous and residual spectrums are empty. Or we can say that on a finite dimensional space, ...
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### The Hilbert-Schmidt Theorem for Compact, Self-adjoint operators

Suppose $T$ is a compact, self-adjoint operator on a Hilbert space $\mathcal{H}$. Then there exists an orthonormal set $\{e_n\}_{n=1}^{\infty}$ of eigenvalues of $T$ such that every $x \in \mathcal{H}$...
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### Spectrum of the Laplace operator with Neumann condition on intervals

Let $-\Delta f=-f''$ be the Laplace operator on $[0,l]$ with domain consisting of functions on $[0,l]$ which have are in $H^2([0,l])$ and satisfy $f'(0)=f'(l)=0$. Then $-\Delta$ is self-adjoint and ...
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### Smallest closed unital subalgebra containing an element has connected resolvent.

Suppose I have $A$ a commutative, unital Banach algebra. Let $C$ be the smallest closed unital subalgebra containing $a\in{}A$ (i.e. the closure of polynomials in $a$). I want to show that the ...
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### Proving that the point spectrum of $T$ is not empty

This problem comes from Rudin's Functional Analysis, Chapter 10. It's the problem 15. Suppose $X$ is a Banach space, $T\in\mathcal{B}(X)$ is compact, and $\|T^n\|\geq 1$ for all $n\geq 1$. ...
I am not sure that I explained to myself the missing details in the proof right, so please check my explanations. (The proof is taken from "$C^*$ -Algebras by Example"-Davidson) First, I don't know ...