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58 views

Conformal infinity in the Hawking-Hunter-Taylor-Robinson metric

I have been trying to follow some of the computations of this paper: http://arxiv.org/abs/hep-th/0408217 and particularly I couldn't derive the asymptotic form of the Kerr-AdS background (3.27) using ...
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26 views

Book Recommendations on Perturbation Theory

I am interested in studying Quantum Electrodynamics and figure I should begin by learning Perturbation theory and Asymptotic expansions. If anyone could recommend some books, or starting points for ...
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94 views

Eigenvalue problem $−\psi''(x) − (ix)^ N \psi(x) = E\psi(x)$ in complex plane

To find the eigenvalue in the complex plane of $x$ for one dimensional Schrodinger equation $$ −ψ''(x) − (ix)^ N ψ(x) = Eψ(x). $$ where $N$ can be any real number, the boundary condition $ψ(x) → 0$ ...
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0answers
28 views

Asymptotic expansion with stationary phase

I am reading Rudolf Haag's book on "Local Quantum Physics", and on page 89 he writes that the following function $f(t,\vec{x}) = \int ...
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0answers
50 views

How does the expanding of null hypersurface orthogonal geodesic congruence imply a particular result?

Sorry that I do not know how to summarize my problem in the title. First, please go to the website here (free access, even though it looks otherwise) to download the paper done by R. Sashs on ...
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1answer
418 views

Renormalization group resummation

I'm having trouble in understeanding a mathematical feature of RG, namely how it provides a way to resum the perturbation series and how that's defined mathematically. From a conceptual point of view ...
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1answer
58 views

Why is spatial conformal infinity a point

One property of spatial infinity is that all spacelike geodesics end at it. Since spacelike geodesics can have different directions, I do not understand why spatial infinity is a point. It looks more ...
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46 views

Physical background to the ODE $y'(x) + \frac{1}{x} = y(x)$

Most books on asymptotic methods start with a discussion on the ODE $y'(x) + \frac{1}{x} = y(x)$, which has solution $$y(x) = \int_0^{\infty} \frac{e^{-xt}}{1+t} \,dt. $$ A discussion on the ...
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73 views

What is the exact meaning that QED perturbative series is only asymptotic and eventually diverges at very high orders?

When I read paper PRB89, 235431 about the effective field theory of graphene, there is a statement that QED perturbative series is only asymptotic and eventually diverges at very high orders (e. g. ...
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1answer
89 views

To what extent can one recover plane waves from the Airy eigenfunctions of a linear potential as the field is turned off?

Consider a single massive particle in one dimension under the action of a static linear potential, with the hamiltonian $$ \hat H=\frac{\hat p^2}{2}+\hat{x}F_0. $$ The eigenstate at energy $E$ is, ...
3
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67 views

Linear KDV eq. asymptotics

The question arises from the book Solitons by P. G. Drazin about the linearized KDV eq. $$ u_t+u_{xxx}=0 $$ My first step was to take a Fourier transform of the equation, find that the dispersion ...
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31 views

Mathieu equation and instabilities

Consider the Mathieu equation $$ \tag 1 y'' + (A -2q\cos(2t))y = 0, $$ How does it provide instabilities for small $q << 1, A > q$? I don't understand this because as the result of ...
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1answer
75 views

Fourier transform & asymptotic expansion of Klein-Gordon equation

I am looking for an approximate analytical solution to the generalized Klein-Gordon equation \begin{equation} \frac{\partial^2{\phi}}{\partial{t^2}}+\frac{\partial^2{\phi}}{\partial{x^2}}+\phi=0 ...
2
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1answer
66 views

DOS integral when surface is not closed

According to the density of states (DOS) formula $$\rho(\varepsilon)\propto \int_{\varepsilon=\text{const}}\frac{dS}{|\nabla_k \varepsilon_k|}$$ Since there is an integral on the constant energy ...
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0answers
53 views

DOS behavior of Van Hove singularity in a line

When there are some points in momentum space give $|\nabla_k \varepsilon_k|=0$, they are called Van Hove points and give singularity in the desity of states (DOS). But what if $|\nabla_k ...
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194 views

DOS of Van Hove singularity in 2D square lattice tight binding model

For the simplest example, 2D square lattice tight binding model gives the energy band as $$\varepsilon_k=-2t(\cos k_x+\cos k_y) \, .$$ We know that $\vec{k}=(0,\pi)$ and related momentum points are ...
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101 views

How to get DOS of Van Hove singularity using quadratic dispersion relation

Take 2D case for example. We can first get the area in momentum space for all $k$ that satisfy $E_0<E(k)<E $, say $S(E)$. And the density of state (DOS) is $\rho(E)=\frac{\partial S(E)}{\partial ...
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32 views

How to get the asymptotic expression of DOS near Van Hove singularity [duplicate]

For the simplest example, 2D square lattice tight binding model gives the energy band as $$\varepsilon_k=-2t(\cos k_x+\cos k_y) \, .$$ We know that $\vec{k}=(0,\pi)$ and related momentum points are ...
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1answer
55 views

Use of termination in solving quantum harmonic oscillator, hydrogen atom etc

I can't seem to understand the use of termination to make the series solutions physically acceptable (when solving the linear harmonic oscillator etc.). So what if the series does not terminate, it's ...
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1answer
1k views

What is a 'turning point' in WKB and why does it fail at that point?

What is meant by a classical turning point in quantum mechanics and why does the WKB approximation fail at that point?
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68 views

Asymptotic Analysis of 1-D Schrödinger Equation [closed]

I'm looking to do a small personal project regarding the time independent Schrödinger equation in 1-D: $$y'' +V(x)y=Ey$$ $$y''=Q(x)y$$ where $ Q(x):=E-V(x) $. There is obviously nothing stopping ...
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35 views

Assigning an asymptotic power to the volume form?

I was reading about the covariant theory of asymptotic symmetries in this review: http://arxiv.org/abs/hep-th/0111246 I have a question about eq. (1.8), but before I ask I should describe what the ...
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0answers
71 views

Do all the spacelike curve terminate at the spatial infinity $i_0$ in the Penrose Diagram of a Schwarzchild black hole?

Let's restrict to the radial direction, so the metric can be expressed as $ds^2=-(1-r_S/r)dt^2+(1-r_S/r)^{-1}dr^2$ with $r_S$ the Schwarzchild radius. Expressed in Kruskal coordinates, the metric is ...
2
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1answer
120 views

Asymptotic freedom in QCD

From renormalization group equation $$ t \frac{d \bar{g}(t , g)}{dt} = \beta (\bar{g}(t , g)), \quad \bar{g}(1 , g) = g $$ (here $t$ is momentum scale factor, $g$ is initial coupling constant and ...
2
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1answer
304 views

A question about the asymptotic series in perturbative expansion in QFT

Related post I heard about the argument that the perturbative expansion in QFT must be asymptotic, such as http://ncatlab.org/nlab/show/perturbation+theory#DivergenceConvergence Roughly this can ...
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1answer
122 views

Derivation of $a_{j}$ coefficients in the quantum harmonic oscillator

In Griffiths' book page 53, when we derive the solution of the quantum harmonic oscillator by using the power series way, we have: $$a_{j+2} = \frac{2j+1-K}{(j+1)(j+2)}\, a_{j} .$$ And for large $j$, ...
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0answers
81 views

Help with deriving an asymptotic expression

Note: I don't know if this is the best place for this question, because it is very specific. However, I'm not sure of a better place to go (apart from one of the other SE's). If you have a ...
2
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0answers
85 views

(References) Study of Asymptotically Flat spacetimes

I am interested in studying the asymptotic structure of Minkowski spacetime in General Relativity. I believe most of the work in this area concerns the asymptotic structure of Minkowski space at null ...
4
votes
1answer
158 views

Solution of QM tasks by using asymptotics

When we solve QM tasks by solving the Schrödinger equation, such as tasks about a particle in a Morse potential, a Poschl-Teller potential and many others, we usually find approximations (lets call ...
2
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0answers
79 views

Motivation behind studying the asymptotic structures

I am trying to explain to myself the motivation behind studying the asymptotic structures at null, time-like and space-like infinities (For the purposes of this post, I will stick to four dimensional ...
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290 views

Questions on Penrose's paper - Conformal Treatment of Infinity

I have several questions. Perhaps it would be better to separate them into different posts. However, given their relative closeness to each other, I think putting it all in one place would be better. ...
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2answers
1k views

Conformal Compactification of spacetime

I have been reading Penrose's paper titled "Relativistic Symmetry Groups" where the concept of conformal compactification of a space-time is discussed. My other references have been this and this. In ...
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1answer
262 views

Zeta regularization gone bad

This may sound as a mathematical question, but it should be very familiar to physicists. I am trying to perform an expansion of the function $$f(x) = \sum_{n=1}^{\infty} \frac{K_2(nx)}{n^2 x^2},$$ for ...
4
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1answer
87 views

Tip of a spreading wave-packet: asymptotics beyond all orders of a saddle point expansion

This is a technical question coming from mapping of an unrelated problem onto dynamics of a non-relativistic massive particle in 1+1 dimensions. This issue is with asymptotics dominated by a term ...