Timeline for Characterization of the Hamiltonian's spectrum in quantum mechanics
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7 events
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| Apr 6, 2023 at 12:57 | comment | added | MathMath | @ChristianRemling so, when physicists solve Schrödinger's equation $H\psi = E\psi$ to find the eigenvalues of the Hamiltonian, what they have in mind is the characterization of the discrete and essential spectrum? Here I am using your definition of discrete spectrum, i.e. the isolated eigenvalues of finite multiplicity. | |
| Mar 24, 2023 at 16:56 | comment | added | Christian Remling | I assume you have a physics background and then the whole thing inevitably gets quite confusing because many (almost all, I really want to say) physicists are sloppy with the terminology, for example routinely use "continuous spectrum" (defined as $\sigma_{ac}\cup\sigma_{sc}$ for the mathematician) when they really mean the essential spectrum. Also, they often implicitly assume that essential spectrum always is ac spectrum, which is very wrong. | |
| Mar 24, 2023 at 16:54 | comment | added | Christian Remling | @MathMath: That's because $\sigma_{d},\sigma_{ess}$ (unlike $\sigma_{ac}$ etc.) are not directly related to dynamical behavior, other than the trivial remark that the points in $\sigma_d$ are eigenvalues, so correspond to bound states. However, in $\sigma_{ess}$, all three types of spectrum (pp, sc, ac) are possible. The $\sigma_d$, $\sigma_{ess}$ decomposition indeed mainly has another purpose: $\sigma_{ess}$ is stable under compact perturbations. (Your definition of $\sigma_d$ is not quite the usual one: the discrete spectrum contains the isolated eigenvalues of finite multiplicity.) | |
| Mar 23, 2023 at 1:01 | comment | added | MathMath | @ChristianRemling thank you for the lecture notes. I skimmed over it and they look really well-written. It seems, however, that you only consider the $\sigma_{pp}(H)$, $\sigma_{ac}(H)$ and $\sigma_{s}(H)$ decomposition, right? This is somehow part of the question because, as you do not consider the essential and discrete spectrum in your notes, many other references only consider these, instead of the former. I still do not understand why choosing one decomposition over the other one. Do they serve for different purposes? | |
| Mar 20, 2023 at 19:26 | comment | added | Willie Wong | A classic physical interpretation of the relevance of the Lebesgue decomposition is the RAGE theorem. In Teschl's "Mathematical Methods of Quantum Mechanics" (which is freely available on his website) this can be found in Chapter 5. // There's one mistake I'd like to point out: scattering states (especially as you defined it) are not $L^2$ functions (especially easy to see when you work on Schrodinger operators on the line with good potentials), so they don't have a corresponding eigenvector. | |
| Mar 20, 2023 at 17:04 | comment | added | Christian Remling | From a mathematical point of view, you got many of the small details wrong. Chapter 13 of my functional analysis lecture notes discusses many of these issues, see here: math.ou.edu/~cremling/teaching/ln.html | |
| Mar 20, 2023 at 16:43 | history | asked | MathMath | CC BY-SA 4.0 |