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 recommendation, I’d love to hear it. Anyway, here goes: I’m reading a paper (Arxiv) in which … Read more

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 εk=−2t(coskx+cosky). We know that k=(0,π) and related momentum points are saddle points which give |∇kεk|=0 and thus some kind of singularity in density of state (DOS) since ρ(ε)∝∫ε=constdS|∇kεk|. How can I get the ln divergence for DOS near ε=0? Should … Read more

Hydrodynamic interaction between two spheres in Re≪1Re\ll 1 flow

I am studying the interaction between two spherical particles of radius a in a low Reynolds number flow. Because of linearity, I know that their respective velocities will be linear in the forces applied to them. Similarly, the force \boldsymbol{F}_j applied on one particle contributes to the velocity \boldsymbol{v}_i of the other through a term … Read more

Is electic field is always asymptotic to rαr^{\alpha} for some rational α\alpha?

Suppose you have an electric field in three dimensions created by some finite (but possibly arbitrarily high) number of point charges, each with charge equal to an integer multiple (positive or negative) of e. Now you pick a point O in a three dimensional space, and a ray →ℓ originating from O in any direction. … Read more

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 beyond all orders of a saddle point expansion (singular terms of an asymptotic series), like in the problem of the lifetime of a bound … Read more

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 them as fi(xj)) of the wave-function Ψ(xj) in equilibrium points xi. Then we substitute Ψ(xj)=ψ(xj)∏ifi(xi) into the Schrödinger equation and then in most … Read more

Asymptotic series in QFT

In QFT is said that the renormalized Dyson series is only asymptotic. But to be able to say it is necessary to know to what function of g (the coupling constant) the Dyson series is asymptotic. For example, suppose that some transition amplitude A(g) is given perturbatively by a series of powers of g. In … Read more

The use of a†(k)=−i∫d3xeikx↔∂0ϕ(x)a^\dagger(\mathbf{k}) = -i \int d^3x e^{ikx}\stackrel{\leftrightarrow}{\partial}_0 \phi(x) in the derivation of the LSZ-formula

I noticed that in Srednicki’s derivation of the LSZ-formula the expression (chapter 5) for the creation (and also later for the annihilation) operator by the field operator: a†(k)=−i∫d3xeikx↔∂0ϕ(x) is used although this expression is only valid for a free field theory whereas the LSZ-formula applies for interacting fields. He just introduces the derivation with “Let … Read more

Gauge transformation and large gauge transformation

Recently, Strominger posted his lecture notes on the infrared structure of gravity and gauge theory 1703.05448. In section 2.5, the equation (2.5.16) takes the following form e2∂zN=A(0)z|I++−A(0)z|I+− Here, e is the electric charge (squared because he used some convention), z is a complex angular coordinates on a 2 sphere at the future null infinity I+, … Read more

Divergent Energies and Analytical Continuation – Two questions on the inverted harmonic oscillator and the inverted double well

I have two questions on the general topic of energy potentials that diverge at infinity. First of all, the inverted harmonic oscillator. I found this post on Physics SE, Inverted Harmonic oscillator. The answer from fellow user Mazvolej states that “<…> The QHO does not permit analytic continuation, because it’s energies and wavefunctions depend not … Read more