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Trial And Error.

The picture is a portrait of a famous Mathematician who was often criticized for not being rigorous enough in his new inventions.

"Anyone who has never made a mistake has never tried anything new." - Albert Einstein


50m
comment Two questions in spectral theory: the spectrum of the Fourier transform and the Hamiltonian of the hydrogen atom.
You originally asked us to show that the spectrum of the Hamiltonian for the Hydrogen atom has spectrum $[0,\infty)$, and I told you in comments that was false, and I hinted for you to look at the bound states of Hydrogen. You questioned me on that, and seemed to feel that the spectrum of the operator was somehow different than the classical Physics spectrum. So I produced an eigenfunction with eigenvalue $\lambda < 0$ associated with the ground state of Hydroen, thereby proving that it would be impossible to show the thing you asked us to show. I think that was a very good answer.
20h
comment Introduction to Toeplitz operators
I don't know where to tell you to start. Learn how to solve Wiener-Hopf integral equations, and you'll know the history. Maybe that's a good springboard. You need Calculus, Complex Analysis, Fourier and Laplace transforms, especially in the Complex plane. Find an applied Math book with some Wiener-Hopf equations.
21h
comment Fredholm index for 1-d Schroedinger operator
Note: If $H$ is selfadjoint, then $\mathcal{N}(H)=\mathcal{R}(H)^{\perp}$. So, if you know the range of $H$ is of finite co-dimension, then the index of $H$ has to be $0$.
23h
comment Properties of the Double Layer Potential
Yes, that's what I understand. Nice problem to post.
23h
comment Properties of the Double Layer Potential
For your problem, $W_{\nu}^{i}$ is constant inside $\Omega$, and another constant outside. That makes the limit constant, depending on whether you're coming from the inside or from the outside.
1d
comment Properties of the Double Layer Potential
You're differentiating with respect to $y$.
1d
comment Properties of the Double Layer Potential
$\nabla_{y}((|x-y|^{2})^{-1/2})=(-1/2)(|x-y|^{2})^{-3/2}2(y-x)=-|x-y|^{-3}(y-x)$‌​.
1d
comment Properties of the Double Layer Potential
I'm always messing up the negative signs. This is the correct arugment, once all the signs are correct. And, it's rigorous, without a bunch of $\delta$ function hand-waving. In fact, the argument extends considerably. Keep this in mind: the outward normal on $B_{\epsilon}$ points toward the inside of the sphere because the outside of $\Omega\setminus B_{\epsilon}$ is inside the sphere, and that's where we're applying the Divergence Theorem.
1d
comment Fredholm index for 1-d Schroedinger operator
That's the real beauty of Sturm-Liouville Theory: you can write down the resolvent operator $(H-\lambda I)^{-1}$ as a function of $\lambda$, and invoke classical ODE theory. The PDE formulations require abstract techniques, but Sturm-Liouville you can write down, at least well enough to feel like you know exactly what you're doing. To write down general solutions, all you need is (a) the classical eigenfunctions of the ODE which exist for all $\lambda\in\mathbb{C}$ and (b) variation of parameters. Periodic is messier to satisfy the endpoint conditions, but it's basically algebra.
1d
comment Properties of the Double Layer Potential
You need that $x\notin\Omega$ to apply the divergence theorem on $\Omega$. The divergence theorem does not allow singularities inside $\Omega$. That's also the reason why you must cirumvent the singularity when $x$ is inside $\Omega$. The integral which gives you $4\pi$ is a surface integral. The surface area over $|x|=\epsilon$ is $4\pi \epsilon^{2}$. And you have a function inside that is $1/|x|^{2}$. So everything cancels perfectly to give you a constant.
1d
answered Introduction to Toeplitz operators
1d
comment Fredholm index for 1-d Schroedinger operator
I think everything is okay without periodicity of $V$. $V \in L^{\infty}[a,b]$ with $V$ real should be enough so that $Hf = -f''+Vf$ is Fredholm with index $0$ on $L^{2}[a,b]$. The domain would be the set of $f$ which are twice absolutely continuous on $(a,b)$, with $f$, $f'$ continuous on $[a,b]$ and with $Hf \in L^{2}[a,b]$ (and, of course periodic $f$, $f'$.) I think you might be able to relax conditions to allow a real $V \in L^{1}[a,b]$, but I'd have to check on that.
1d
comment Two questions in spectral theory: the spectrum of the Fourier transform and the Hamiltonian of the hydrogen atom.
Showing that your conjecture is false IS an answer.
1d
revised Two questions in spectral theory: the spectrum of the Fourier transform and the Hamiltonian of the hydrogen atom.
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1d
comment Two questions in spectral theory: the spectrum of the Fourier transform and the Hamiltonian of the hydrogen atom.
What does the spectral mapping theorem have to do with it?
1d
revised Complex Fourier Series and using the square norm
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1d
answered Why are $L^p$-spaces so ubiquitous?
1d
revised Two questions in spectral theory: the spectrum of the Fourier transform and the Hamiltonian of the hydrogen atom.
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1d
answered Two questions in spectral theory: the spectrum of the Fourier transform and the Hamiltonian of the hydrogen atom.
1d
comment Two questions in spectral theory: the spectrum of the Fourier transform and the Hamiltonian of the hydrogen atom.
To make the calculation easier, use $e^{-ar}$ and write the Laplacian in spherical coordinates so that you can simplify to a second-order differential operator in $r$. Find $a$ so that $e^{-ar}$ is an eigenfunction, and determine the eigenvalue. I'm sure you did this in Physics, but probably forgot.