A question concerning the semi-classical approximation.

[Matheux]

Hello.
First of all I am not physicist, so plz don't expect too much knowledge
from me....
I am mainly interested in oscillatory integrals and in section 3 of
this paper :

http://uk.arxiv.org/abs/math.SP/0502568

the author is using an approximation via oscillatory integrals related
to a classical dynamics.
So what is the validity of such an approximation???

Also in which kind of physical situation is this approximation
relevant??
For example I doubt that someone can measure the spectrum of an
operator in a laboratory....
But, perhaps, am I simply ignorant...

Sincerely,
ES.

[Pindare]

Matheux a écrit :

> the author is using an approximation via oscillatory integrals related
> to a classical dynamics.
> So what is the validity of such an approximation???

Oscillatory integrals pop up naturally when one studies the hbar--->0
limit of the Green function, which is the Fourier transform of the
"propagator" K, i.e. of something of the form K(q_a,q_b,t)=Sum_j
A_j(q_a,q_b,t)*exp(iR_j(q_a,q_b,t)/hbar), where j labels classical
trajectories going from q_a to q_b in time t.

Such a form is called WKB ansatz or BKW ansatz or Van Vleck ansatz
depending on the author. It is a generalization of the 1D case, and the
phase functions R_j are Hamilton's principal function from classical
dynamics (the correct expression of K is a bit more complicated, some
crucial topological phases arise).

For a mathematical point of view on this, see the introduction chapter
of the old book of Guillemin & Sternberg available at
http://www.ams.org/online_bks/surv14/

There are also very recent and clear lecture notes by Zworski and Evans
at http://math.berkeley.edu/~zworski/semiclassical.pdf

A physics approach of all this can be found in chapter 28 of the book
of Cvitanovic-et-al at
http://www.nbi.dk/ChaosBook/stable/chapters/Webbook.pdf


> Also in which kind of physical situation is this approximation
> relevant??

The semiclassical limit and this kind of propagator is by definition
relevant when hbar is small compared to the characteristic actions
involved in the system. Bluntly this means considering highly excited
states. In practise (i.e. when forgeting theorems and doing actual
physics computations) one can sometimes obtain fairly good
approximations even down to the ground state. It depends a lot on the
detailed structure of the classical system (its geometry and dynamics,
in particular the actual values of the actions of short periodic orbits
-- something which can only be found numerically in non-trivial system)

[Pindare]

Matheux a écrit :

> the author is using an approximation via oscillatory integrals related
> to a classical dynamics.
> So what is the validity of such an approximation???

Oscillatory integrals pop up naturally when one studies the hbar--->0
limit of the Green function, which is the Fourier transform of the
"propagator" K, i.e. of something of the form K(q_a,q_b,t)=Sum_j
A_j(q_a,q_b,t)*exp(iR_j(q_a,q_b,t)/hbar), where j labels classical
trajectories going from q_a to q_b in time t.

Such a form is called WKB ansatz or BKW ansatz or Van Vleck ansatz
depending on the author. It is a generalization of the 1D case, and the
phase functions R_j are Hamilton's principal function from classical
dynamics (the correct expression of K is a bit more complicated, some
crucial topological phases arise).

For a mathematical point of view on this, see the introduction chapter
of the old book of Guillemin & Sternberg available at
http://www.ams.org/online_bks/surv14/

There are also very recent and clear lecture notes by Zworski and Evans
at http://math.berkeley.edu/~zworski/semiclassical.pdf

A physics approach of all this can be found in chapter 28 of the book
of Cvitanovic-et-al at
http://www.nbi.dk/ChaosBook/stable/chapters/Webbook.pdf


> Also in which kind of physical situation is this approximation
> relevant??

The semiclassical limit and this kind of propagator is by definition
relevant when hbar is small compared to the characteristic actions
involved in the system. Bluntly this means considering highly excited
states. In practise (i.e. when forgeting theorems and doing actual
physics computations) one can sometimes obtain fairly good
approximations even down to the ground state. It depends a lot on the
detailed structure of the classical system (its geometry and dynamics,
in particular the actual values of the actions of short periodic orbits
-- something which can only be found numerically in non-trivial system)