Inverse Spectral Theorem - L10 - Frederic Schuller
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when asking about specific steps in Schuller's treatment of projection-valued measures and Borel measures, or citing his blackboard derivations.
Recommended by 1 people across 1 different months, 2017 to 2022.
What it is
A 1h54m lecture from Frederic Schuller's Lectures on Quantum Theory series, covering the inverse spectral theorem for self-adjoint operators on a Hilbert space. It builds on projection-valued measures and complex measures defined by polarization, and is delivered at a blackboard by Schuller.
What people said
Recommends user8469759 · 2017-08 · Mathematics Stack Exchange · on “Inverse spectral theorem, clarification from a video.”
I'm following some of the lectures of professor Schuller. I'm watching this video [this video] specifically At about minute 23 he defines the PVM (Projected Valued Measure) as a map $$ P_A : \sigma(\Theta_\mathbb{R}) \rightarrow \mathcal{L}(\mathcal{H}) $$ with the specific expression $$ \langle\psi,P_A(\Omega)\phi\rangle=\int \chi_\Omega d\mu_{\psi,\phi}^A \;\;\forall\psi, \phi \in \mathcal{H} $$ Here $\mathcal{H}$ is an hilbert space, $A$ is a self-adjoint operator in $\mathcal{H}$, while $\mu_{\psi,\phi}^A$ is a complex valued measure constructed by polarization of a real valued measure $\mu_{\phi}^A$ (these concepts are expressed a bit earlier in the same lecture, but all the constructions ar
Cites amilton moreira · 2019-11 · Mathematics Stack Exchange · on “Borel measure generator”
Let $\mu$ be a Borel measure in $R$ . In this lecture [this video] it is said that if we specify the mesure to half open intervals $I=(-\infty,a]$ than we can specify to al other Borel sets. How can we proof that?
Cites Antoni Parellada · 2022-05 · Mathematics Stack Exchange · on “Proof that the covariant derivative is orthogonal to the tangent plane”
Well... It wasn't an equal sign, it was a minus. Naturally, the joke is on me for falling for it, but there is something to be said about the penmanship and the Austrian blackboard tradition [this video] . It turns out that the expression intended is $$\vec S^\gamma \nabla_\alpha \vec S_\beta= \vec S^\gamma \cdot \frac{\partial \vec S_\beta}{\partial S^\alpha}-\Gamma^\omega_{\alpha \beta} \vec S_\omega\cdot \vec S^\gamma=\vec S^\gamma \cdot \color{blue}{\frac{\partial \vec S_\beta}{\partial S^\alpha}}-\Gamma^\omega_{\alpha \beta} \delta^\gamma_\omega=0$$ and $$\Gamma^\gamma_{\alpha\beta}=\vec S^\gamma\cdot \color{blue}{\frac{\partial\vec S_\alpha}{\partial S^\beta}}$$ with the blue expressions be