ASYMPTOTIC COMPLETENESS OF N-BODY SCATTERING JAN DEREZI NSKI - PowerPoint PPT Presentation

ASYMPTOTIC COMPLETENESS OF N-BODY SCATTERING JAN DEREZI´ NSKI Dept. of Math. Methods in Phys., Faculty of Physics, University of Warsaw

In my opinion, scattering theory for N -body Schr¨ odinger opera- tors is one of the greatest successes of 20th century mathematical physics. On the physical side, we have a rigorous framework that explaines why nonrelativistic matter is built out of well defined clusters of nuclei and electrons, such as atoms, ions, molecules. On the mathematical side, we have a deep analysis of a large family of nontrivial operators with continuous spectrum, combining ideas from classical and quantum mechanics.

A single quantum particle in an external potential is described by the Hilbert space L 2 ( R d ) and the Schr¨ odinger Hamiltonian H = H 0 + V ( x ) , where H 0 = p 2 2 m, p = 1 i ∂ x . A typical example of a potential is V ( x ) = c | x | .

THEOREM. Assume that V ( x ) is short range, that is, | V ( x ) | ≤ c � x � − µ s , µ s > 1 . Then there exist wave (Møller) operators Ω ± := s − lim t →±∞ e i tH e − i tH 0 , they are isometric, they intertwine the free and full Hamiltonian: Ω ± H 0 = H Ω ± , and they are complete: Ω ± Ω ±∗ = 1 l c ( H ) .

THEOREM. Assume that V ( x ) is long range, that is, V ( x ) = V l ( x ) + V s ( x ) , where V s ( x ) is short range and | ∂ α x V l ( x ) | ≤ c α � x � −| α |− µ l , µ l > 0 , α ∈ N d . Then there exists a function ( t, ξ ) �→ S t ( ξ ) and modified Møller operators Ω ± := s − lim t →±∞ e i tH e − i S t ( p ) , which satisfy the same properties as those stated for the short-range case.

Thus the Hilbert space is the direct sum of bound states and of scattering states – states which evolve for large times as free waves. One can define the scattering operator, S := Ω + Ω −∗ , which is unitary. The integral kernel of S defines scattering ampli- tudes. The square of the absolute value of a scattering amplitude is the scattering cross-section describing the probability of a scatteting process. The most difficult part of the above theorems is to prove that the range of (modified) wave operators fills the whole continuous spectral space of H . This is called asymptotic completeness (AC).

2 interacting quantum particles are described by the Hilbert space L 2 ( R d ) ⊗ L 2 ( R d ) ≃ L 2 ( R 2 d ) and the Hamiltonian H = p 2 + p 2 1 2 + V ( x 1 − x 2 ) . 2 m 1 2 m 2 Introduce the center-of-mass coordinate x 12 := m 1 x 1 + m 2 x 2 and the m 1 + m 2 relative coordinate x 12 := x 2 − x 1 . The Hilbert space factorizes L 2 ( R 2 d ) = L 2 ( X 12 ) ⊗ L 2 ( X 12 ) .

Let m 12 := m 1 + m 2 be the total mass and m 12 := ( m − 1 1 + m − 1 2 ) − 1 be the reduced mass. Then we can write H = p 2 12 + H 12 , 2 m 12 where H 12 := ( p 12 ) 2 2 m 12 + V ( x 12 ) . Thus the problem of two interacting particles is reduced to a single particle in an external potential.

N interacting quantum particles are described by the Hilbert space N i =1 L 2 ( R d ) ≃ L 2 ( X ) , ⊗ where X := R Nd , and the Hamiltonian is N p 2 j � � H := + V ij ( x i − x j ) . 2 m j j =1 1 ≤ i<j ≤ N A typical potential is Z i Z j e 2 V ij ( x i − x j ) = 4 π | x i − x j | .

A cluster decomposition is a partition of { 1 , . . . , N } into clusters: a = { c 1 , . . . , c k } . The Hamiltonian of a cluster c is p 2 j � � H c := + V ij ( x i − x j ) . 2 m j j ∈ c i,j ∈ c The Hamiltonian of a cluster decomposition a is H a = H c 1 + · · · + H c k .

Note that cluster decompositions have a natural order. In partic- ular, there is a minimal cluster decompostion, where all clusters are 1-element. Every pair determines a cluster decomposition. Define the collision plane of a as X a := { ( x 1 , . . . , x N ) ∈ R Nd : ( ij ) ≤ a ⇒ x i = x j } . Consider the quadratic form on X � m i 2 x 2 i . Let X a denote the internal plane of a , defined as the orthogonal complement of X a wrt this form. We will write x �→ x a and x �→ x a for the orthogonal projections onto X a and X a .

We have X = X a ⊕ X a , X a = X c 1 ⊕ · · · ⊕ X c k . Therefore, L 2 ( X ) = L 2 ( X a ) ⊗ L 2 ( X a ) , L 2 ( X a ) = L 2 ( X c 1 ) ⊗ · · · ⊗ L 2 ( X c k ) , ∆ a = ∆ c 1 + · · · + ∆ c k . ∆ = ∆ a + ∆ a ,

For a cluster decomposition a = { c 1 , . . . , c k } set � � � V a ( x ) = V ij ( x i − x j ) = V ij ( x i − x j )+ · · · + V ij ( x i − x j ) . i,j ∈ c 1 i,j ∈ c k ( ij ) ≤ a The cluster Hamiltonian decomposes: H a = ∆ a + H a , H a = ∆ a + V a ( x a ) , H a = H c 1 + · · · + H c k .

Introduce H a := Ran1 l p ( H a ) ≃ Ran1 l p ( H c 1 ) ⊗ · · · ⊗ Ran1 l p ( H c k ) . Let E a := H a � � � H a = H c 1 H c 1 + · · · + H c k � � � H ck � � � be the operator describing the bound state energies of clusters. Let J a : L 2 ( X a ) ⊗ H a → L 2 ( X ) be the embedding of bound states of clusters into the full Hilbert space.

THEOREM. Assume that the potentials V ij are short range. Then for any cluster decompostion a there exists the corresponding partial wave operator t →±∞ e i tH J a e − i t (∆ a + E a ) . Ω ± a := s − lim Ω ± a are isometric, they intertwine the cluster and the full Hamilto- nian: Ω ± a (∆ a + E a ) = H Ω ± a and are complete: a RanΩ ± a = L 2 ( X ) . ⊕

THEOREM. Assume that the potentials V ij are long range with √ µ l > 3 − 1 . Then for any cluster decompostion a there exists a function ( t, ξ a ) �→ S a,t ( ξ a ) , the corresponding partial modified wave operator t →±∞ e i tH J a e − i( S a,t ( p a )+ tE a ) , Ω ± a := s − lim which satisfy the same properties as those stated in the short range case.

AC means that all states in L 2 ( X ) can be decomposed into states with a clear physical/chemical interpretation such as atoms, ions and molecules. We can introduce partial scattering operators S ab := Ω + ∗ a Ω − b describing various processes, such as elastic and inelastic scattering, ionization, capture of an electron, chemical reactions. The partial wave operators Ω ± a can be organized into a L 2 ( X a ) ⊗ H a ∋ ( ψ a ) �→ � Ω ± a ψ a ∈ L 2 ( X ) , ⊕ a which is unitary. The partial scattering operators S ab arranged in the matrix [ S ab ] also describe a unitary operator.

2 -body scattering theory, including AC in both short- and long- range case, was understood already in the 60’s. Existence of N -body wave operators and the orthogonality of their ranges was established about the same time. What was missing for a long time was Asymptotic Completeness – the fact that the ranges of wave operators span the whole Hilbert space. Below I review the various methods that were used, more or less successfully, to prove this.

The stationary approach to scattering theory is based on resolvent identities. For example, if H = H 0 + V , then the identity ( z − H ) − 1 = ( z − H 0 ) − 1 1 − | V | 1 / 2 ( z − H 0 ) − 1 V 1 / 2 � − 1 +( z − H 0 ) − 1 V 1 / 2 � | V | 1 / 2 ( z − H 0 ) − 1 can be used to prove AC in the 2-body case.

L.Faddeev found a resolvent identity that can be used to study 3-body scattering. A number of other resolvent identities were used (eg. G.Hagedorn’s for 4 bodies). The results about AC with N ≥ 3 proven using the stationary approach involve implicit assumptions on invertibility of certain complicated operators and on properties of bound and almost-bound states. They also require a very fast decay of potentials and d ≥ 3 . However, in principle, the stationary approach leads to explicit formulas for scattering amplitudes.

V.Enss introduced time-dependent methods into proofs of AC. In his approach an important tool was the RAGE Theorem saying that l c ( H ) for K compact and ψ ∈ Ran1 � T 1 � K e i tH ψ � 2 d t = 0 . lim T T →∞ 0 Enss started with proving the 2-body AC (late 70’s), and managed √ to prove 3-body AC including the long-range case with µ l > 3 − 1 (late 80’s).

Let us describe an idea that turned out to be important: One needs to look for observables A such that i[ H, A ] is in some sense positive. Here is an important example of this idea: E.Mourre (1981). Suppose that E is not a threshold (it is not an eigenvalue of H a for any a ). Then there exists an interval I around E and c 0 > 0 such that 1 l I ( H )i[ H, A ]1 l I ( H ) ≥ c 0 1 l I ( H ) , 1 where A = � 2 ( p i x i + x i p i ) is the generator of dilations. i The Mourre estimate has important implications both in the sta- tionary and time-dependent approach.

I.M.Sigal devoted a large part of his research carreer to N -body AC. After working with the stationary approach he switched to the time-dependent approach. Together with A.Soffer he obtained the first proof of the N -body AC in the short range case (announced 1985, published 1987). They first used heavily propagation esti- mates. Below we summarize abstractly the time-dependent version of this technique: If Φ( t ) is a uniformly bounded observable on a Hilbert space H and d d t Φ( t ) + i[ H, Φ( t )] ≥ Ψ ∗ ( t )Ψ( t ) , then � ∞ � Ψ( t )e − i tH v � 2 d t < ∞ , v ∈ H . 1