Enss method

In 1977, V. Enss [a1] introduced a new approach to the study of spectral and scattering properties of Schrödinger operators (cf. also Schrödinger equation). It was based on a combination of time-dependent scattering theory and phase-space analysis. In his first work, Enss solved the two-body scattering problem (with short-range potentials) and a few years later extended the method to the three-body problem (both short- and long-range potentials). Previous results on this problem were based on time-independent methods, primarily due to L.D. Faddeev, who worked out the three-body case in 1963. Faddeev's work was later clarified and some generalizations were made, but it remained limited to the three-body case and required further assumptions on the spectral properties of the Hamiltonian and other restrictions on the potentials.

Enss' method, on the other hand, removed all the artificial assumptions. It also initiated the fruitful approach of phase-space analysis, later further developed by E. Mourre [a3] (1979–1981) and finally led to a general phase-space theory of $N$-body Hamiltonians by I.M. Sigal and A. Soffer [a5] (1987).

The Enss method is based on using classical intuition to the study of the large-time behaviour of a quantum system. Consider the case of two-particle scattering; this system can be reduced to the study of the large-time behaviour of a quantum particle interacting with a force field, which decays to zero at large distances. The states of such a particle, being quantal, can only be localized in some energy interval, in general. If the energy is localized near a positive number, one expects the particle to escape to infinity. The problem of scattering theory is to show that every state that escapes to infinity, moves like a free particle system, for large enough times. This idea is captured using the notion of wave operator. Say the state of the system at time zero is given by a wave function $\psi ( 0 )$. One introduces the dynamics $U ( t )$ to be an operator that moves the state of the system by a time $t$. Hence

\begin{equation*} U ( t ) \psi ( 0 ) = \psi ( t ), \end{equation*}

the state of the system at time $t$.

One can also use a different, free dynamics $U _ { 0 } ( t )$; $U _ { 0 } ( t )$ is the dynamics of a particle moving without any force acting on it.

Suppose now one constructs the following state:

\begin{equation*} \Omega ( t ) \psi ( 0 ) = U _ { 0 } ( - t ) U ( t ) \psi ( 0 ). \end{equation*}

That is, $\Omega ( t ) \psi ( 0 )$ is the state of a system moved forward in time under the true (or full) dynamics, for a time $t$, and then backward under the free dynamics.

In the limit, as $t$ goes to infinity, $\Omega ( t ) \psi ( 0 )$ should approach a new state, $\psi _ { + }$ if $t \rightarrow + \infty$ and $\psi _ { - }$ if $t \rightarrow - \infty$.

The main problem of scattering theory is to show that for any $\psi ( 0 )$ for which $U ( t ) \psi ( 0 )$ disperses to infinity as $t$ approaches infinity, the limiting states $\psi _ { \pm }$ exist. To prove such results, Enss begins with proving the following basic property of states which disperse to infinity: Assuming that the force field is regular enough (that is, its value does not jump from one point to another), the wave function decays to zero inside any finite ball in space. This decay to zero is, furthermore, uniform in the choice of states, provided they all have their energy support in a same fixed finite interval $( a , b )$ with $a > 0$.

The proof of this results essentially follows from a similar theorem of D. Ruelle [a4] (1969).

Now, note that the wave operators $\Omega _ { \pm }$ which map $\psi _ { \pm }$ to $\psi ( 0 )$ measure the "difference" between the free and full dynamics (when $U ( t ) = U _ { 0 } ( t )$, one sees that $\Omega _ { \pm } = 1$).

Hence one expects that the wave operators applied to a state which is very far from the force field act like $1$. This is a key observation in the Enss method. It reduces the problem of scattering and asymptotic completeness to showing that

\begin{equation*} ( \Omega _ { + } - 1 ) \psi ( t ) \end{equation*}

goes to zero as $t$ goes to infinity.

To prove that, one now decomposes the state in the phase space, that is, in the bigger space of position and velocities of the system/particle: $P _ { + }$ will denote the projection on the part of the state where the position vector and velocity vector are related by a sharp angle between them:

\begin{equation*} \overset{\rightharpoonup} { x } \cdot \overset{\rightharpoonup} { v } > 0, \end{equation*}

and $P_-$ will be the complement.

Then,

\begin{equation*} ( \Omega _ { + } - 1 ) \psi ( t ) = ( \Omega _ { + } - 1 ) g \psi ( t ) = \end{equation*}

\begin{equation*} = ( \Omega _ { + } - 1 ) ( g - g_{0} ) \psi ( t ) + ( \Omega _ { + } - 1 ) _{g_{0}} \psi ( t ), \end{equation*}

where $g$ stands for the projection on states with total energy in the (fixed) interval $( a , b )$ and $g_{0}$ stands for the projection on states with kinetic energy in the interval $( a , b )$.

As $t$ approaches infinity, Ruelle's theorem implies that the state moves away to infinity. Hence it does not interact with the force any more; this means that all the energy of the state is kinetic. Hence one concludes that

\begin{equation*} ( g - g_0 ) \psi ( t ) \end{equation*}

vanishes as $t \rightarrow \infty$, and so is the term

\begin{equation*} ( \Omega _ { + } - 1 ) ( g - g_0 ) \psi ( t ). \end{equation*}

There remains the term

\begin{equation*} ( \Omega _ { + } - 1 ) g _ { 0 } \psi ( t ) = \end{equation*}

\begin{equation*} = ( \Omega _ { + } - 1 ) g _ { 0 } P _ { + } \psi ( t ) + ( \Omega _ { + } - 1 ) g _ { 0 } P _ { - } \psi ( t ). \end{equation*}

It is easy to see that when a free particle moves, its velocity becomes more and more parallel to its position vector. Hence the derivative with respect to time of $\overset{ \rightharpoonup} { x } \cdot \overset{ \rightharpoonup} { v }$ is positive under the free flow. This same derivative under the full flow will then be a sum of a positive term (coming from the free part of the motion) and another term, depending on the force. Since for $t$ large the force can be neglected, by Ruelle's theorem, one sees that also under the full flow, $\overset{ \rightharpoonup} { x } \cdot \overset{ \rightharpoonup} { v }$ will have positive growth. Hence, for large enough times, the support of the state will move to the region of phase space where $\overset{\rightharpoonup}{x} \cdot \overset{\rightharpoonup}{ v } > 0$. Hence $P _ { - } \psi ( t )$, the projection on the part of the state where $\overset{\rightharpoonup} { x } . \overset{\rightharpoonup} { v } < 0$, will tend to zero as $t$ approaches infinity. To complete the proof it is then left to show that

\begin{equation*} ( \Omega _ { + } - 1 ) g _ { 0 } P _ { + } \psi ( t ) \end{equation*}

also vanishes as $t$ approaches infinity.

The proof that this last term vanishes as $t \rightarrow \infty$ is the most technical part of the Enss method. It is based on Cook's original proof of the existence of the limit defining $\Omega _ { + }$, combined with the ideas of Ruelle's argument.

It should be remarked that the above description is improved over the original argument of Enss, using $P _ { \pm }$ motivated by Mourre's work.

References

[a1]	V. Enss, "Asymptotic completeness for quantum-mechanical potential scattering I. Short range potentials" Comm. Math. Phys. , 61 (1978) pp. 285–291
[a2]	V. Enss, "Quantum scattering theory of two- and three-body systems with potentials of short and long range" S. Graffi (ed.) , Schrödinger Operators , Lecture Notes in Mathematics , 1159 , Springer (1985)
[a3]	E. Mourre, "Link between the geometrical and the spectral transformation approaches in scattering theory" Comm. Math. Phys. , 68 (1979) pp. 91–94
[a4]	D. Ruelle, "A remark on bound states in potential scattering theory" Nuovo Cimento , 61A (1969) pp. 655–662
[a5]	I.M. Sigal, A. Soffer, "The $N$-particle scattering problem: Asymptotic completeness for short range systems" Ann. of Math. , 126 (1987) pp. 35–108