**Hitoshi Kitada** (*hitoshi@kitada.com*)

*Sat, 25 Sep 1999 20:03:51 +0900*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 805] Re: [time 804] Re: [time 803] Re: [time 801] Re: [time 799] Still about construction of U"**Previous message:**Matti Pitkanen: "[time 803] Re: [time 801] Re: [time 799] Still about construction of U"**In reply to:**Hitoshi Kitada: "[time 801] Re: [time 799] Still about construction of U"**Next in thread:**Matti Pitkanen: "[time 805] Re: [time 804] Re: [time 803] Re: [time 801] Re: [time 799] Still about construction of U"

Dear Matti,

Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:

Subject: [time 803] Re: [time 801] Re: [time 799] Still about construction of

U

*>
*

*>
*

*> On Sat, 25 Sep 1999, Hitoshi Kitada wrote:
*

*>
*

*> > Dear Matti,
*

*> >
*

*> > I have several questions on your construction of S-matrix.
*

*> >
*

*> > 1.
*

*> >
*

*> > > Contrary to earlier expectations, it seems that one cannot assign
*

*> > >explicit Schr\"odinger equation with S-matrix although the
*

*> > >general structure of the solutions of the Virasoro conditions
*

*> > >is same as that associated with time dependent perturbation theory
*

*> > >and S-matrix is completely analogous to that obtained as
*

*> > >time evolution operator $U(-t,t)$, $t\rightarrow \infty$ in
*

*> > >the perturbation theory for Schr\"odinger equation.
*

*> >
*

*> > Does this mean that your former equation
*

*> >
*

*> > >\begin{eqnarray}
*

*> > >i\frac{d}{dt}\Psi&=& H\Psi\per , \nonumber\\
*

*> > >H &\equiv& k\left[-P_T^2 - L_0(vib)-L_0(int)\right]\Psi\per .
*

*> > >\end{eqnarray}
*

*> >
*

*> > is wrong or cannot be derived by the former note?
*

*>
*

*> As such the equation is probably wrong. The point is
*

*> that I was forced to make *ad hoc* replacement
*

*> of Diff^4 invariant translation generator p_+ with id/dt
*

*>
*

*> p_+--> id/dt
*

*>
*

*>
*

*> in order to obtain Schrodinger equation. The introduction
*

*> of time t leads to potential problems with Poincare invariance,
*

*> which however could be avoided. But this is the main
*

*> ad hoc element of construction.
*

*>
*

*>
*

*> Starting directly from Super Virasoro conditions and just
*

*> writing "scattering solution" for them one avoids
*

*> all ad hoc hypothesis and manifest Poincare invariance is achieved.
*

*> One however loses Schrodinger equation but this is not needed
*

*> since informational "time evolution" is
*

*> totally characterized by S-matrix. Thus I am tending to believe that
*

*> Heisenberg was right: S-matrix has nothing to do with time evolution with
*

*> respect to geometric time.
*

I will show at the bottom that there is a relation between S-matrix

formulation and the time-dependent formulation. Of course, this is possible

when there are scattering states associated with the Hamiltonian H.

On the point that the total energy is zero is equivalent to non-existence of

time, consider the case:

H\Psi = E\Psi with E not = 0,

where \Psi belongs to the total Hilbert space. That is, \Psi is an eigenstate

of H with non-zero energy.

Then time evolution is

exp(-itH)\Psi = exp(-itE)\Psi.

This means there is no QM motion, i.e. there is no time. That is, even if

there is non zero energy state, it happens that the universe has no time.

*>
*

*>
*

*>
*

*> >
*

*> > 2.
*

*> >
*

*> > >\begin{eqnarray}
*

*> > >\Psi&=&\Psi_0 + \frac{V}{E-H_0-V+i\epsilon} \Psi \per .
*

*> > >\end{eqnarray}
*

*> >
*

*> > (This is equation (1) of your note.)
*

*> >
*

*> Sorry: this equation is mistyped:
*

*>
*

*> \Psi&=&\Psi_0 + \frac{V}{E-H_0+i\epsilon} \Psi \per .
*

*>
*

*> The presence of V in denominator would make it to diverge.
*

*> The ordering is also important: V is to the right.
*

Either equation leads to a correct time dependent formulation. Just by

exchanging roles of some factors.

*>
*

*>
*

*> >
*

*> > >Since ordinary Schr\"odinger equation is consistent with the scattering
*

*> > >matrix formalism avoiding elegantly the difficulties with the
*

*> > >definition of the limit $U(-t,t)$, $t\rightarrow \infty$, it
*

*> > >is natural to take this form of Schr\"odinger equation as starting
*

*> > >point when trying to find Schr\"odinger equation for the 'time'
*

evolution

*> > >operator $U$. One can even forget the assumption
*

*> > >about time evolution and require only
*

*> > >that the basic algebraic information guaranteing
*

*> > >unitarity is preserved. This information boils down to the Hermiticity
*

*> > > of free and interacting Hamiltonians and
*

*> > >to the assumption that the spectra
*

*> > >non-bound states for free and interacting Hamiltonians
*

*> > >are identical.
*

*> >
*

*> >
*

*> > It is known that to consider the limit as \epsilon -> 0 in the
*

Schrodinger

*> > equation (1) of your note is equivalent to considering the time limit as
*

t ->

*> > \infty of exp(-itH). So you cannot avoid the difficulty: Below I will try
*

to

*> > show this.
*

*>
*

*> In TGD framework single particle Virasoro generators L_0(n) define
*

*> propagators
*

*>
*

*> 1/(p^2-L_0(vib)+i*epsilon)
*

*>
*

*> appearing in stringy diagrams. L_0(vib) is integer valued and gives rise
*

*> to the universal non-negative integer valued mass squared spectrum of
*

*> string models (in suitable units).
*

*>
*

*> In present case i*epsilon is completely equivalent with
*

*> the presence of i*epsilon in the propagators of relativistic quantum field
*

*> theory: epsilon term guarantees that momentum spacetime integration
*

*> over virtual momenta is performed correctly in case that one
*

*> is forced to integrate over pole of propagator.
*

*>
*

*> As far as I can understand this has nothing to do with time but I
*

*> could of course be wrong.
*

That time does not exist follows from the eigenequation

H\Psi = E\Psi.

But it does not follow from the form of propagators. If H has a continuous

spectral subspace (this space is sometimes called scattering space of H),

then H can have time.

In fact in my formulation,

H \Psi = 0

implies non-existence of time of the universe. But if we want to consider a

scattering state \Phi as the total satte that is orthogonal to eigenspace,

hence to \Psi, then one can recover time.

The local time of a local system arises in the same way. A state \psi of a

local system L is considered a kind of a part of the total state \Psi. Then

it can be shown that \psi can be a scattering state of the local Hamiltonian

H_L. This gives the local time t_L of the system L, but at the same time

there is no time of the total universe.

*>
*

*>
*

*> [In string models similar expansion is derived from
*

*> stringy perturbation theory in which time parameter corresponds
*

*> to string coordinate. The problem in string models is
*

*> that this gives only first quantized theory and one should
*

*> second quantize. In TGD this problem is not encountered. Super
*

*> Virasoro generators are of second quantized form from
*

*> beginning. Also topological description of particle reactions
*

*> emerges automatically via the decomposition of L_0(tot) to
*

*> a sum of single particle Virasoro generators L_0(n). This
*

*> decomposition was described in later posting.]
*

*>
*

*>
*

*>
*

*>
*

*>
*

*> >
*

*> > The Schrodinger equation (1) in your note should be written as:
*

*> >
*

*> > \Psi = \Psi_0 - V ((E+i \epsilon)-H)^{-1} \Psi, (1)
*

*>
*

*> Yes with H replaced with free Hamiltonian: I am sorry for my typo.
*

*> I hope that it does not affect you conclusions.
*

No, it does not affect my conclusion at all.

*>
*

*>
*

*> >
*

*> > or
*

*> >
*

*> > \Psi = \Psi_0 - ((E+i \epsilon)-H)^{-1} V \Psi. (2)
*

*> >
*

*> > The reason we consider two equations is that the fractional expression in
*

*> > your note has two interpretations, since the relevant factors are
*

operators,

*> > hence they are noncommutative in general.
*

*>
*

*> Yes. In my equation 1/(E-H_0 +iepsilon)*V is meant to be the
*

*> correct ordering. This is easy to see: by operating with E-H_0 to both
*

*> sides of the equation one obtains (E-H_0) Psi= (E-H_0)Psi+ VPsi =VPsi
*

*> which is just Schrodinger equation. Provided that order is
*

*> correct.
*

*>
*

*>
*

*> > Also the sign on the RHS (right
*

*> > hand side) should be minus if the definition of H is H = H_0 + V.
*

*> >
*

*>
*

*>
*

*> > We write z = E+i\ep, where \ep = \epsilon > 0 and E is any real number.
*

*> >
*

*> > In case (1), the equation is equivalent to
*

*> >
*

*> > (I - V (H-z)^{-1} ) \Psi = \Psi_0.
*

*> >
*

*> > This is rewritten as
*

*> >
*

*> > (H - z - V) (H - z)^{-1} \Psi = \Psi_0,
*

*> >
*

*> > which is equivalent to
*

*> >
*

*> > (H_0-z) (H-z)^{-1}\Psi = \Psi_0 (1)'
*

*> >
*

*> > or
*

*> >
*

*> > (H-z)^{-1}\Psi = (H_0-z)^{-1}\Psi_0 (1)"
*

*> >
*

*> > or
*

*> >
*

*> > \Psi = (H-z)(H_0-z)^{-1}\Psi_0
*

*> >
*

*> > = \Psi_0 + V(H_0-z)^{-1}\Psi_0 (1)'''
*

*> >
*

*> > by H = H_0 + V.
*

*> >
*

*> > Thus if \Psi_0 is a given fixed state function, \Psi depends on z =
*

E+i\ep,

*> > and it should be written as a function of z:
*

*> >
*

*> > \Psi = \Psi(z).
*

*> >
*

*> > Note that these hold only when \ep > 0, NOT for an infinitesimal number
*

\ep

*> > because the inverse (H-(E+i\ep))^{-1} does not exist for \ep = 0 in
*

general.

*> >
*

*> > In case (2), the equation is equivalent to
*

*> >
*

*> > \Psi_0
*

*> >
*

*> > = (I - (H-z)^{-1} V)\Psi (2)'
*

*> >
*

*> > = (H-z)^{-1} (H-z-V)\Psi
*

*> >
*

*> > = (H-z)^{-1} (H_0 - z)\Psi.
*

*> >
*

*> > This is rewritten:
*

*> >
*

*> > \Psi(z) = (H_0-z)^{-1} (H-z) \Psi_0. (2)"
*

*> >
*

*> > If we transform \Psi_0 and \Psi to
*

*> >
*

*> > \Phi_0 = (H-z) \Psi_0,
*

*> >
*

*> > \Phi = (H-z) \Psi,
*

*> >
*

*> > the equation (2)" becomes
*

*> >
*

*> > \Phi(z) = (H-z) (H_0-z)^{-1} \Phi_0. (2)'''
*

*> >
*

*>
*

*>
*

*>
*

*> Certainly one cannot get rid of time in ordinary wave mechanics.
*

As I stated, it is possible to have time locally without the total time.

*> If you have energy, you have also time!
*

Even when an energy is not zero, if that energy is eigenenergy, then there is

no time.

*>
*

*>
*

*> In TGD approach one has
*

*>
*

*> L_0(tot) Psi=0 rather than HPsi = EPsi! No energy, no time!!
*

*>
*

*> By the way, this condition is analogous to your condition
*

*> that entire universe has vanishing energy
*

*>
*

*> HPsi=0
*

*>
*

*> Thus there is something common between our approaches!
*

Then you agree with that there is no time for the total universe?

*>
*

*>
*

*> Also in general Relativity Hamiltonian vanishes as a constraint.
*

*> In TGD However General Coordinate invariant
*

*> as gauge invariance is replaced by Super Virasoro invariance
*

*> which operates at lightcone boundary and H is replaced with L_0.
*

*>
*

*>
*

*> Best,
*

*>
*

*> MP
*

*>
*

*>
*

Note:

This note is to show a relation between time independent method and time

dependent method. I attached a LaTeX file for your convenience:

Consider the scattering operator:

S f = lim_{t->\infty} exp(itH_0)exp(-2itH) exp(itH_0) f

= (W_+)^* (W_-) f (1)

and compute the inner product:

(S f - f, g)

= ((W_+)^* ((W_-) - (W_+))f, g)

= (((W_-) - (W_+))f, (W_+) g)

= - (lim_{t->\infty} \int_{-t}^t (d/ds) [exp(isH) exp(-isH_0)f], (W_+) g) ds

= - lim_{t->\infty} \int_{-t}^t i (exp(isH) V exp(-isH_0)f, (W_+) g) ds. (2)

We note the intertwining property of wave operators W = W_+ or W_-:

exp(-isH) W = W exp(-isH_0).

By this the RHS of (2) is equal to

= -i lim_{t->\infty} \int_{-t}^t (V exp(-isH_0)f, (W_+) exp(-isH_0)g) ds. (3)

Similarly, we have

(W_+) exp(-isH_0)g

= exp(-isH_0)g + lim_{s->\infty}i\int_0^s exp(iuH)Vexp(-i(s+u)H_0)g du.

Inserting this into (3) we have

(S f - f, g)

= -i lim_{t->\infty}\int_{-t}^t (V exp(-isH_0)f, exp(-isH_0)g) ds

+ i lim_{t->\infty} i \int_{-t}^t \int_0^s

(V exp(-isH_0)f, exp(iuH)Vexp(-i(s+u)H_0)g) du ds. (4)

This time limit is equal to the Abelian limit

(S f - f, g)

= lim_{\ep,\ep'->+0}

[ i \int_0^\infty exp(-\ep u) i \int_{-\infty}^\infty

exp(-\ep'|s|)(exp(i(s+u)H_0)Vexp(-iuH)Vexp(-isH_0)f, g) ds du

-i \int_{-\infty}^\infty exp(-\ep'|s|)(exp(isH_0)V exp(-isH_0)f,g) ds ]. (5)

Letting E_0(\lambda) be the spectral measure for the self-adjoint operator

H_0, we rewrite the inner products (with \lam = \lambda):

(exp(i(s+u)H_0)V exp(-iuH)V exp(-isH_0)f, g)

= \int_{-\infty}^\infty

exp(i(s+u)\lam) (dE_0(\lam)V exp(-iuH)V exp(-isH_0)f, g),

and

(exp(isH_0)V exp(-isH_0)f,g)

= \int_{-\infty}^\infty exp(is\lam) (dE_0(\lam)V exp(-isH_0)f,g).

Then (5) becomes

(S f - f, g)

= lim_{\ep,\ep'->+0}

[ i\int_0^\infty i \int_{-\infty}^\infty

\int_{-\infty}^\infty (dE_0(\lam)V exp(-iu(H-\lam-i\ep))

V exp(-is(H_0-\lam)-|s|\ep')f, g) ds du

-i \int_{-\infty}^\infty \int_{-\infty}^\infty

(dE_0(\lam)V exp(-is(H_0-\lam)-|s|\ep')f, g) ds ]. (6)

Here

\int_0^\infty exp(-iu(H-\lam-i\ep)) du = (1/i) R(\lam+i\ep),

where R(z) = (H-z)^{-1} is the resovent of H, and

\int_{-\infty}^\infty exp(-is(H_0-\lam)-|s|\ep') ds

= (1/i)[R_0(\lam+i\ep') - R_0(\lam-i\ep')].

Thus (6) becomes

(S f - f, g)

= lim_{\ep,\ep'->0} \int_{-\infty}^\infty

(dE_0(\lam)V R(\lam+i\ep)V [R_0(\lam+i\ep')-R_0(\lam-i\ep')]f, g)

- lim_{\ep'->0} \int_{-\infty}^\infty

(dE_0(\lam)V [R_0(\lam+i\ep')-R_0(\lam-i\ep')]f,g). (7)

If H_0 is simple enough such that H_0 has only absolutely continuous

spectrum, then one has

E'_0(\lam)=dE_0/d\lam(\lam) = (2i\pi)^{-1}[R_0(\lam+i0)-R_0(\lam-i0)],

where

R_0(\lam+i0) = lim_{\ep'->0} R_0(\lam+i\ep'), etc.

as an operator between Hilbert spaces with suitable topologies. Thus

(Sf - f, g)

= 2i\pi \int_{-\infty}^\infty (E'_0(\lam)V R(\lam+i0) VE'_0(\lam)f,g)d\lam

-2i\pi \int_{-\infty}^\infty (E'_0(\lam)VE'_0(\lam)f,g)d\lam. (8)

This shows that if f is restricted to the spectrum \lam wrt H_0, then the

image is also restricted to \lam wrt H_0. In particular, S is decomposable

with respect to the spectrum of H_0, or in other words, S is decomposable wrt

the direct integral expression wrt H_0 of the Hilbert space on which H_0 is

defined.Thus S is expressed as wrt to this direct integral decomposition

S = \int S(\lam) d\lam, (9)

where each S(\lam) is an operator in the fiber H(\lam) of the direct integral

expression of the Hilbert space H:

H = \int H(\lam) d\lam.

(H here is not the operator H above.)

The family {S(\lam)} of S(\lam) in (9) is called S-matrix.

Summary: We started with the time dependent definition of the scattering

operator S, and ariived at its stationary definiton.

\beq

S f&=& (W_+)^* (W_-) f= \lim_{t\to\infty} \exp(itH_0)\exp(-2itH) \exp(itH_0)

f\nonumber\\

&=& I+2i\pi \int_{-\infty}^\infty E'_0(\lam)V R(\lam+i0) VE'_0(\lam)d\lam

-2i\pi \int_{-\infty}^\infty E'_0(\lam)VE'_0(\lam)d\lam.

\ene

********LaTeX file*********

\documentstyle[12pt]{article}

\oddsidemargin 0pt

\evensidemargin 0pt

\topmargin 0pt

\textwidth 16cm

\textheight 23cm

\newcommand{\F}{\noindent}

\newcommand{\qqq}{\qquad\qquad}

\newcommand{\q}{\qquad}

\newcommand{\SP}{\smallskip}

\newcommand{\MP}{\medskip}

\newcommand{\BP}{\bigskip}

\newcommand{\beq}{\begin{eqnarray}}

\newcommand{\ene}{\end{eqnarray}}

\newcommand{\HH}{{\cal H}}

\newcommand{\UU}{{\cal U}}

\newcommand{\OO}{{\cal O}}

\newcommand{\SS}{{\cal S}}

\newcommand{\ep}{{\epsilon}}

\newcommand{\lam}{{\lambda}}

\newcommand{\Ltn}{{L^2(R^\nu)}}

\newcommand{\Ltnn}{{L^2(R^{3n})}}

\newcommand{\tX}{{\tilde X}}

\newcommand{\tP}{{\tilde P}}

\newcommand{\ve}{\vert}

\newcommand{\V}{\Vert}

\newcommand{\tx}{(x_0,x_1,x_2,x_3,x_4)}

\newcommand{\txp}{(x_0^\prime,x_1^\prime,x_2^\prime,x_3^\prime,x_4^\prime)}

\begin{document}

{\bf A Relation between time dependent and stationary representations}

\normalsize

\BP

\F

Consider the scattering operator:

\beq

S f &=& \lim_{t\to\infty} \exp(itH_0)\exp(-2itH) \exp(itH_0) f\nonumber\\

&=& (W_+)^* (W_-) f

\ene

and compute the inner product:

\beq

(S f - f, g)&

=& (W_+^* (W_- - W_+)f, g)\nonumber\\

&=& ((W_- - W_+)f, W_+ g)\nonumber\\

&=& - \left(\lim_{t\to\infty} \int_{-t}^t (d/ds) [\exp(isH) \exp(-isH_0)f],

W_+ g\right) ds \nonumber\\

&=& - \lim_{t\to\infty} \int_{-t}^t i (\exp(isH) V \exp(-isH_0)f, W_+ g) ds.

\ene

We note the intertwining property of wave operators $W = W_+$ or $W_-$:

\beq

\exp(-isH) W = W \exp(-isH_0).\nonumber

\ene

By this the RHS of (2) is equal to

\beq

= -i \lim_{t\to\infty} \int_{-t}^t (V \exp(-isH_0)f, W_+ \exp(-isH_0)g) ds.

\ene

Similarly, we have

\beq

(W_+) \exp(-isH_0)g

= \exp(-isH_0)g + \lim_{s\to\infty}i\int_0^s \exp(iuH)V\exp(-i(s+u)H_0)g

du.\nonumber

\ene

Inserting this into (3) we have

\beq

&&(S f - f, g)\nonumber\\

&=& -i \lim_{t\to\infty}\int_{-t}^t (V \exp(-isH_0)f, \exp(-isH_0)g)

ds\nonumber\\

&&+ i \lim_{t\to\infty} i \int_{-t}^t \int_0^s

(V \exp(-isH_0)f, \exp(iuH)V\exp(-i(s+u)H_0)g) du ds.

\ene

This time limit is equal to the Abelian limit

\beq

&&(S f - f, g)\nonumber\\

&=& \lim_{\ep,\ep'\to+0}

[ i \int_0^\infty \exp(-\ep u) i \nonumber\\

&&\times\int_{-\infty}^\infty

\exp(-\ep'|s|)(\exp(i(s+u)H_0)V\exp(-iuH)V\exp(-isH_0)f, g) ds du\nonumber\\

&&-i \int_{-\infty}^\infty \exp(-\ep'|s|)(\exp(isH_0)V \exp(-isH_0)f,g) ds ].

\ene

Letting $E_0(\lambda)$ be the spectral measure for the self-adjoint operator

$H_0$, we rewrite the inner products:

\beq

&&(\exp(i(s+u)H_0)V \exp(-iuH)V \exp(-isH_0)f, g)\nonumber\\

&=& \int_{-\infty}^\infty

\exp(i(s+u)\lam) (dE_0(\lam)V \exp(-iuH)V\exp(-isH_0)f, g),\nonumber

\ene

and

\beq

(\exp(isH_0)V \exp(-isH_0)f,g)

= \int_{-\infty}^\infty \exp(is\lam) (dE_0(\lam)V \exp(-isH_0)f,g).\nonumber

\ene

Then (5) becomes

\beq

&&(S f - f, g)\nonumber\\

&&= \lim_{\ep,\ep'\to+0} \nonumber\\

&&[ i\int_0^\infty i \int_{-\infty}^\infty

\int_{-\infty}^\infty (dE_0(\lam)V \exp(-iu(H-\lam-i\ep))

V \exp(-is(H_0-\lam)-|s|\ep')f, g) ds du\nonumber\\

&&-i \int_{-\infty}^\infty \int_{-\infty}^\infty

(dE_0(\lam)V \exp(-is(H_0-\lam)-|s|\ep')f, g) ds ].

\ene

Here

\beq

\int_0^\infty \exp(-iu(H-\lam-i\ep)) du = (1/i) R(\lam+i\ep),\nonumber

\ene

where $R(z) = (H-z)^{-1}$ is the resolvent of $H$, and

\beq

\int_{-\infty}^\infty \exp(-is(H_0-\lam)-|s|\ep') ds

= (1/i)[R_0(\lam+i\ep') - R_0(\lam-i\ep')].\nonumber

\ene

Thus (6) becomes

\beq

(S f - f, g)

&&= \lim_{\ep,\ep'\to0} \int_{-\infty}^\infty

(dE_0(\lam)V R(\lam+i\ep)V [R_0(\lam+i\ep')-R_0(\lam-i\ep')]f, g)\nonumber\\

&&- \lim_{\ep'\to0} \int_{-\infty}^\infty

(dE_0(\lam)V [R_0(\lam+i\ep')-R_0(\lam-i\ep')]f,g).

\ene

If $H_0$ is simple enough such that $H_0$ has only absolutely continuous

spectrum, then one has

\beq

E'_0(\lam)=dE_0/d\lam(\lam) =

(2i\pi)^{-1}[R_0(\lam+i0)-R_0(\lam-i0)],\nonumber

\ene

where

\beq

R_0(\lam+i0) = \lim_{\ep'\to0} R_0(\lam+i\ep'), \q \mbox{etc.}\nonumber

\ene

as an operator between Hilbert spaces with suitable topologies. Thus

\beq

(Sf - f, g)

&=& 2i\pi \int_{-\infty}^\infty (E'_0(\lam)V R(\lam+i0)

VE'_0(\lam)f,g)d\lam\nonumber\\

&&-2i\pi \int_{-\infty}^\infty (E'_0(\lam)VE'_0(\lam)f,g)d\lam.

\ene

This shows that if $f$ is restricted to the spectrum $\lam$ wrt $H_0$, then

the image is also restricted to $\lam$ wrt $H_0$. In particular, $S$ is

decomposable with respect to the spectrum of $H_0$, or in other words, $S$ is

decomposable wrt the direct integral expression wrt $H_0$ of the Hilbert

space $\HH$ on which $H_0$ is defined. Thus $S$ is expressed wrt to this

direct integral decomposition as follows:

\beq

S = \int^\oplus \SS(\lam) d\lam,

\ene

where each $\SS(\lam)$ is an operator in the fiber $\HH(\lam)$ of the direct

integral expression of the Hilbert space $\HH$:

\beq

\HH = \int^\oplus \HH(\lam) d\lam.\nonumber

\ene

The family $\{\SS(\lam)\}$ of $\SS(\lam)$ in (9) is called S-matrix.

\BP

\F

{\bf Summary}: We started with the time dependent definition of the

scattering operator $S$, and arrived at its stationary definition:

\beq

S f&=& (W_+)^* (W_-) f= \lim_{t\to\infty} \exp(itH_0)\exp(-2itH) \exp(itH_0)

f\nonumber\\

&=& I+2i\pi \int_{-\infty}^\infty E'_0(\lam)V R(\lam+i0) VE'_0(\lam)d\lam

-2i\pi \int_{-\infty}^\infty E'_0(\lam)VE'_0(\lam)d\lam.

\ene

\end{document}

**Next message:**Matti Pitkanen: "[time 805] Re: [time 804] Re: [time 803] Re: [time 801] Re: [time 799] Still about construction of U"**Previous message:**Matti Pitkanen: "[time 803] Re: [time 801] Re: [time 799] Still about construction of U"**In reply to:**Hitoshi Kitada: "[time 801] Re: [time 799] Still about construction of U"**Next in thread:**Matti Pitkanen: "[time 805] Re: [time 804] Re: [time 803] Re: [time 801] Re: [time 799] Still about construction of U"

*
This archive was generated by hypermail 2.0b3
on Sat Oct 16 1999 - 00:36:41 JST
*