The mathematical analogue of the Bohr correspondence principle, which states that in the $\hbar\to 0$ limit quantum mechanics should approach classical mechanics. For many evolution equations, this suggests that the high frequency limit of partial differential equations should be able to be approximated by dynamical systems (ordinary differential equations). Semiclassical analysis aims at making these intuitions precise.
Questions tagged [semiclassical-analysis]
15 questions
4
votes
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Intuition for the differences between two notions of quantum ergodicity: One given by weak-* convergence and one by pseudodifferential operators
Consider the two notions of quantum ergodicity of the Laplacian operator $\Delta$ . ( Phase space ): $\Delta$ is said to be quantum ergodic (in the phase space) in a compact Riemannian manifold if there exists a density one index set of the natural…
Epsilon Away
- 1,100
3
votes
1 answer
Proof of first step theorem 7.7.1. in Lars Hormander's "The Analysis of Linear Partial Differential Operators I"
I'm struggling to follow the "obvious" k=0 step in the proof of this theorem: THEOREM 7.7.1. Let $K\subset\mathbb{R}^n$ be a compact set, $X$ an open neighborhood containing $K$ and $j, k$ non-negative integers. If $u\in C^{k}_c(K), f\in C^{k+1}(X)$…
3
votes
1 answer
Wavefront set of coherent state
Given that $$\psi_{(x_0,\xi_0)}(x)=(\pi h)^\frac{-n}{4}e^{\frac{i}{h}(x-\frac{x_0}{2}).\xi_0}e^{-\frac{1}{2h}(x-x_0)^2}$$ Then the wavefront set $\text{WS}(\psi_{(x_0,\xi_0)})=\{(x_0,\xi_0)\}$ In the textbook of semi-classical analysis they give…
Hoang Skadi
- 43
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votes
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Reference request for the Liouville theorem (Liouville measure & Hamiltonians) on Riemannian manifolds
I am looking for a reference that states and proves Liouville's theorem regarding the existence of invariant measure(s) (i.e. Liouville measure(s)) on energy shells of some nice enough observables. To be precise, I am reading the lecture notes about…
Epsilon Away
- 1,100
2
votes
1 answer
$O_{\mathcal{S}'\rightarrow \mathcal{S}}(h^\infty)$
What does the notation: $$O_{\mathcal{S}'\rightarrow \mathcal{S}}(h^\infty)$$ mean in Zworski's semiclassical analysis? (To be clear, I know what $O(h^\infty)$ means in the context of normed and semi-normed spaces.)
goonfiend
- 2,904
2
votes
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About semiclassical analysis and resource request
I was going through pseudodifferential operators when I got to know about "Semiclassical Analysis". Looking at some resources, I think this subject is mostly (or exactly) related to Mathematics for physics (or maybe I am wrong). I want to learn it…
Himanshu
- 129
2
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Spectrum of the Weyl quantized operator $\mathrm{Op}\left(\sqrt{\frac{x^2+p^2}{2}}\right)$
Consider a 1D phase space whose generic points are denoted as $(x,p)$ . We know that the Weyl quantization $\mathrm{Op}\left(\frac{x^2+p^2}{2}\right)$ is the harmonic oscillator Hamiltonian, whose spectrum is $\left\{n+\frac12:n\in\mathbb Z_{\geq…
Laplacian
- 2,418
2
votes
1 answer
Asymptotic expansions with compactly supported terms and smoothing operators
Suppose that we have a pseudodifferental operator $A\in \Psi^m(\mathbb{R}^n)$ with symbol $a\in S^m(\mathbb{R}^n\times\mathbb{R}^n)$ , and $a$ has an asymptotic expansion $a\sim \sum\limits_{j=0}^\infty a_j.$ Suppose further that each $a_j$ is…
user900940
- 105
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How to justify interchanging the integral with the quantization operator?
For reference, I am reading Lemma 2.2 here . Suppose that $D\subseteq \mathbb{R}^n$ is non-empty bounded domain with piece-wise smooth boundary. Let $(u_j)$ be a sequence of Dirichlet eigenfunctions on $D$ with corresponding eigenvalues $$ 0 <…
Quoka
- 3,141
1
vote
0 answers
Semiclassical analysis - Stationary phase lemma in practice
Let $\phi(x) = a(x)\exp\left({i \cfrac{\xi_0.x}{h}}\right) $ , with $\xi_0 \in \mathbb R^d$ fixed and $a \in C^{\infty}_c(\mathbb R^d)$ . Let $f \in C^{\infty}_c(\mathbb R^{2d}) $ . I want to show that $$ (Op^W_h(f)\phi)(x) = b(x,h)\exp\left({i…
zarathustra
- 191
1
vote
0 answers
Understanding why $\partial_t\kappa_t = H_q = (\kappa_t)_*H_q$ for smooth symbol $q$, symplectomorphism $\kappa$ and Hamiltonian vector field $H$
I am reading Zworski's book on semiclassical analysis and I can't make sense of the following opening paragraph the of chapter 11: Quantum And Classical Dynamics . We assume for this section that $\{q_t\}_{0 \leq t \leq T}$ denotes a family of…
Wasradin
- 1,683
1
vote
1 answer
Creating a bounded operator from an unbounded symbol ? - discussion around the Calderon-Vaillancourt theorem
Calderon-Vaillancourt theorem states that any quantization of a bounded symbol (in the sense of this article for example) is a bounded operator on the $L^2(\mathbb{R^n})$ space. I was wondering if there was any way of going back from a…
Vincent
- 13
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vote
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Approximating integral involving associated Laguerre polynomial
I need to numerically evaluate the following integral $$\sqrt{\frac{n!(n+1)!}{(n+\alpha)!(n+1+\alpha)!}}\int_0^\infty \frac{1}{\sqrt{x+c}}x^\alpha e^{-x}L_n^\alpha(x)L_{n+1}^\alpha(x)\;\mathrm dx$$ where $c\ge 0$ , and $n,\alpha$ are integers. I…
MMa
- 439
1
vote
1 answer
Is there a practical formula for quantization on arbitrary subsets of $\mathbb{R}^n$
On $\mathbb{R}^n$ , there is a convenient formula for quantization in terms of the Fourier transform. More precisely, given a (sufficiently nice) function $a :\mathbb{R}^{2n}\to\mathbb{C}$ , $$ \operatorname{Op}_h(a)u(x) = \frac{1}{(2\pi…
Quoka
- 3,141
1
vote
1 answer
Applying propagator to Laplace eigenfunctions
Let $\Omega \subseteq \mathbb{R}^n$ be a non-empty domain with piece-wise smooth boundary and consider a Dirichlet eigenfunction $\varphi$ on $\Omega$ . That is, $\varphi : \Omega \to \mathbb{C}$ is a non-trivial solution to $\Delta \varphi =…
Quoka
- 3,141