Positive Commutators and Spectrum of Nonrelativistic QED Volker Bach FB Mathematik MA 7-2; TU Berlin; Str. d. 17 Juni 136 D-10623 Berlin, Germany
[email protected] Jurg Frohlich Inst. f. Theor. Physik; ETH Honggerberg; CH-8093 Zurich, Switzerland
[email protected] Israel Michael Sigal Dept. of Math.; Univ. of Toronto; Toronto, M5S 3G3, Canada
[email protected] Avy So er Dept. of Math.; Rutgers Univ. New Brunswick, NJ 08903; USA so
[email protected] Apr. 28, 97 Abstract
In this paper we consider the Hamiltonian of the standard model of nonrelativistic QED. In this model non-relativistic quantum particles interact with quantized electro-magnetic eld and their interaction is subjected to an ultra-violet cut-o . We prove absence of excited states and absolute continuity of the spectrum for suciently small charges under conditions on the coupling functions which milder than those of [2]. We use the method of positive commutators with the \conjugate" operator deformed appropriately in order to accomodate the interaction term.
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1 Introduction In this paper we develop the method of positive commutators for nonrelativistic quantum electrodynamics (QED). The latter is described by the Schrodinger operator (in the Coulomb gauge and in the units in which ~ = 1 and c = 1 ) 2 N 1 X H (e) = 2m pj ? ej A(xj ) + V (x) 1f + 1part Hf ; (1.1) j j =1 corresponding to a system of N quantum particles with masses m1; : : :; mN , charges e1; : : :; eN , coordinates x1; : : : ; xN and (quantum) momenta p1; : : :; pN , interacting with quantized electromagnetic eld. The latter is described by the (transverse) vector potential A(y), the scalar (longitudinal) potential V (x) , x = (x1; : : : ; xN ) , and the eld Hamiltonian Hf . Here, e = (e1; : : :; eN ) and H (e) and A(y) act on Hpart Hf , where Hpart and Hf are the particle and radiation Hilbert spaces, respectively. Usually, Hpart is L2(X ) or a subspace thereof, where X is the particle con guration space, and Hf is a Bosonic Fock space. Note that p 1 , the ne in our units the charge of the electron, ?e , is ?2 , where = 137 structure constant. For jej j 's suciently small we construct a deformation of the second quantized generator of dilations for photons so that the resulting operator has a positive commutator with H (e) locally in small neighbourhoods of the excited eigenvalues of the particle Hamiltonian N 1 X Hpart = 2m p2j + V (x) ; (1.2) j j =1 acting on Hpart . This implies that in those neighbourhoods (i) H (e) has no eigenvalues, (ii) H (e) has purely absolutely continuous spectrum and (iii) H (e) satis es the limiting absorption principle. Conclusion (i) is derived via a virial theorem and (ii) and (iii), with a help of the Kato-Mourre theory extended further in this paper. Note that the limiting absorption principle constitutes the rst step in the analysis of the time behaviour of a system in question. A result similar to our spectral result was obtained rst under more stringent conditions in [1, 2, 3] (see the paragraph after Theorem 3.1). For massive quan-
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tum elds such results were given in [16, 6, 8, 7, 10, 11]. Also, [8] introduced commutator estimates which inspired our method. The commutator methods were initiated in [18, 12], developed in [13] and turned into a deep theory in [14]. They were shown in [14, 15, 17, 20] to be a powerful tool in understanding spectral properties of quantum Hamiltonians and in controlling the time evolution generated by these Hamiltonians. We see the present paper as a step toward these goals, more precisely, toward a dynamical theory of photons interacting with matter.
2 Nonrelativistic QED Hamiltonian Consider a particle system interacting with the electromagnetic eld. Both the particles and transverse part of the eld are quantized. The corresponding Hamiltonian is given by (1.1). We assume that V (x) is the classical potential of interaction between the particles (in particular, it absorbs the longitudinal electric potential). The radiation Hamiltonian, Hf , can be expressed in terms of the (photon) creation and annihilation operators, a(k) and a(k) as Z Hf = !(k) a(k) a(k) dk ; (2.1) where ! = !(k) = jkj is the photon energy. Here a(k) and a(k) are operatorvalued transverse vectors, i.e., they satisfy k a(k) = 0 and k a(k) = 0 . For each x 2 R 3 , the vector potential A(x) acts on Hf as an operator-valued distribution and is given by the following expression. Z (2.2) A(x) = e?ikx a(k) + eikx a(k) q(k) dk ; !(k) where is an ultraviolet cut-o , i.e., a real function on R 3 vanishing suciently fast as jkj ! 1 . In order to keep the infrared features of the original system, one can take 1 in, say, the unit ball, fjkj 1g . However, the particular form of will be of no importance in our analysis. Only certain estimates on will be needed. Thus we can, as well, consider the vector potentials of a more general form Z A(x) = Gx(k) a(k) + Gx(k) a(k) dk ; (2.3)
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where the function Gx(k) will be assumed to satisfy several conditions (depending on the problem we tackle), the most important of which is ! ) (Z 1 2 (2.4) 1 + !(k) jGx (k)j dk < 1 ; sup x as this condition ensures semiboundedness of H (e) and essential self-adjointness on the domain of H (e = 0) . We remark that we have ignored the Zeeman term, X ? 2emi i B (xi) ; (2.5) i which is due to the interaction of the particle spins, i , with the magnetic eld B (x) = curl A(x) . In order to simplify notation and exposition, we demonstrate our approach on the model of a particle system interacting with a massless scalar eld instead of the vector one. The Hamiltonian for such a model is given by
H = Hpart 1f + 1part Hf + I ;
(2.6)
acting on Hpart F , where Hpart is a particle (atomic) Hamiltonian, X 1 2 Hpart = 2m pi + V (x) (2.7) i (with the same notation as above), acting on a dense domain in Hpart . Hf is a scalar eld Hamiltonian, Z Hf = !(k) a(k) a(k) dk ; (2.8) with ! = !(k) = jkj , as above, acting on F , and I is the interaction given by Z I := a(Gx) + a(Gx) = Gx(k) a(k) + Gx(k) a(k) dk ; (2.9) with Gx(k) satisfying (2.4) (for referencing convenience we use here the same notation for coupling functions as in the vector case). Here a(k) and a(k) are creation and annihilation operators of a quantum scalar eld acting on F . Hpart is either L2(X ) of the particle con guration space X , which typically is R 3N ,
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or a fermionic subspace of L2(X ) . Finally, F is the Fock space for a scalar eld, i.e., the Fock space built upon the one-photon space L2(R 3) . (We will still refer to the scalar eld in question as photon eld.) Note that for the scalar eld the coupling cannot be minimal, i.e., through replacing the momentum operator by the covariant one. This model contains all the diculties of the vector one, but the infrared problem comes out in its pure form unencumbered by vector notation and inessential particulars of the model in question. It is straightforward to include in (2.9) also terms quadratic in a and a . We do not pursue this in order not to muddle the simple ideas involved. Finally we mention that the scalar model is a generalization of the so-called electric dipole approximation in QED. Throughout the paper we assume that: V (x) is a real function of the form P V , where are linear maps from X ( = the particle con guration space) i i i n i to R and V 2 Lpi (R ni) + L1(R ni) with pi = 2 if ni 3 , pi > 2 if ni = 4 and pi = n2i if ni 5 . This is not the most general assumption compatible with our analysis, but it is simple and explicit. Under this assumption Hpart is N self-adjoint on the domain of P 2m1 j p2j . Its spectrum, (Hpart) , is of the form j =1 fEj g [ [; 1), where E0 E1 E2 . Moreover, fEj g pp(Hpart) and cont(Hpart) = [; 1) . In what follows E(H ) stands for a smoothed out cut-o function of a selfadjoint operator H , supported in . Below we make use of the exponential decay of eigenfunctions of Hpart in the sense that we use the existence of a constant, C;M , such that
M
hxi E(Hpart)
C;M ; (2.10)
p
for all M 0 and any compact set (?1; ) . Here, hxi := 1 + x2 .
3 Results First, we consider the Hamiltonian H , as de ned in (2.6){(2.9). We show in the next section that H is self-adjoint, provided (2.4) holds. For our main result we
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impose a somewhat stronger condition on Gx = Gx(k) . 8 2 9 1=2 ! > > 2 Z = < 1 : g := sup 1 + x (k )j dk + hxi > !(k) !(k) x : ; (3.1) Note that the last term in (3.1) can be replaced by k!?1=2(k rk Gx ? x rxGx )k2 . If we use this alternative, then we have to make additional assumptions on the classical potential V (x) typical for the Mourre theory (see e.g. [4, 9]). is , s = 1; : : : ; mi , be the eigenvalues and corresponding Now let E i and part eigenfunctions of Hpart , where i = 0; 1; : : : , and E 0 < E 1 < : : : . For i; j 0 , we R assume that jkj=! (Aij )Aij dS! is continuous in ! and vanishes at ! = 0 . Here i` ; Gx jr i , ` = 1; : : :; mi and Aij are the mi mj matrices with the entries h part part r = 1; : : : ; mj , and dS! is the area element on the sphere fk 2 R 3 j jkj = !g . js ) we de ne the self-adjoint matrix ? For j 1 (i.e., for an excited states part j by X Z (Aij )Aij (! ? E j + E i ) d3k : (3.2) ?j = i:E i E j
The eigenvalues of this matrix are the resonance widths at the second order associated with the eigenvalue E j , as in Fermi's Golden Rule. We assume that we are given a family of coupling functions Gx (labeled by Gx ) satisfying
j = lim inf g?2?j > 0 : (3.3) g!0 Now we consider the Hamiltonian H (e) , de ned in (1.1). Using Kato's inequality one can easily show that H (e) is self-adjoint under condition (2.4) (see [2]). We impose on Gx the same additional conditions as in the case of H , i.e. (3.1) and the condition formulated in the paragraph preceding Eqn. (3.2). Moreover, we de ne the matrix X Z ?j (e) = (Bij )Bij (! ? E j + E i) d3 k ; (3.4) i: E i 0 jej!0 where jej = max jeij . The main result of this paper is i
(3.5)
Theorem 3.1 Assume (3.1) and, for a given j 1 , (3.5) (resp. (3.3)). Then for jej (resp. g ) suciently small, the spectrum of H (e) (resp. H ) in any
interval, containing E j but containing no other parts of the spectrum of Hpart and whose distance to spec Hpart \ (?1; E j ) is jej (resp. g ), is purely absolutely continuous. Moreover, H (e) (resp. H ) has the local decay property (formulated below) in such an interval.
The rst statement of the theorem was proved in [1, 2, 3] under additional asjGx j2 < 1 for some > 0 , which is sumptions of analyticity for Gx and R sup 1+ x ! a stronger condition in the infrared region, k ! 0 . Now we formulate the local decay property mentioned in Theorem 3.1. To this end, we introduce the anti-selfadjoint operator Z 1 A = 1part 2 a(k) k rk + rk k a(k) d3k : (3.6) This operator is a second quantization of the generator of dilations in the onep photon momentum space times ?1 , i.e., 12 (k rk + rk k) . In what follows, whenever no danger of confusion arises, we omit the trivial factors 1part and
1f . We say that H has the local decay property in a spectral interval (with respect to an operator A ), if the following estimate holds Z 1
2
hAi? e?iHt
dt C k k2 ; (3.7) ?1
for any > 1=2 and any 2 Ran E (H ) . (In fact, a slightly stronger property, the limiting absorption principle with the Holder constant < ? 21 , holds in our case.) Theorem 3.1 follows from a positive commutator estimate derived below (Theorem 5.1) and from the Kato-Mourre theory mentioned in the introduction and expounded upon in Section 5. We prove only the part of Theorem 3.1 concerning
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the operator H . The corresponding part for the operator H (e) , given in (1.1), is proven in exactly the same way but with a couple more simple estimates related 2 to the quadratic part P 2emj j A(xj )2 of the perturbation H (e) ? H (0) . Observe that absolute continuity of the spectrum and the local decay property outside of O(g2 ) (resp. O(jej2) ) neighbourhood of the eigenvalues and thresholds of Hpart was proven in [1, 2].
4 Relative Bounds on the Interaction In this section we collect some elementary bounds needed for the proof of Theorem 3.1. In what follows by Hf?1=2 , we always understand Hf?1=2P , where P is the projection onto the orthogonal complement of the vacuum state in the Fock space.
Lemma 4.1 (Relative bounds)
Z jf j2 !1=2
(Hf )1=2 ka(f ) kFock !
Fock
(4.1)
and
Z Z jf j2 !
1 =2
2 +
( H ) (4.2) jf j2 k k2Fock : f Fock ! Proof: We drop the subindex \Fock" in the proof. By Schwarz' inequality we have 1=2 Z Z jf j2 !1=2 Z 2 !(k) ka(k) k ka(f ) k jf (k)j ka(k) k : (4.3) ! This, due to Z !(k) ka(k) k2 = h ; Hf i ; (4.4)
ka(f )
k2Fock
implies (4.1). Inequality (4.2) follows from
a(f )a(f ) = a(f )a(f ) + hf; f i1 ;
h ; a(f )a(f ) i = ka(f ) k2 and (4.1).
(4.5)
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We rewrite bound (4.1) as
a(f ) Hf?1=2
Fock
?1=2
Hf a (f ) Fock
Z jf j2 !1=2 ; (4.6) ! Z jf j2 !1=2 : (4.7) ! These two bounds are equivalent since the expressions under the norm signs are adjoint to each other. Moreover, (4.1) implies Z jf j2 !1=2
1=2
(4.8)
Hf k k ; ha (f ) + a(f )i 2 ! which yields, for any > 0 , Z jf j2 1 a (f ) + a(f ) Hf + ! : (4.9) Furthermore, inequalities (4.1) and (4.2) imply Z jf j2 !1=2
1=2
Z 21=2
Hf : (4.10) jf j k k + 2
a (f ) + a(f ) ! Eqn. (4.9) implies that I is Hf -form bounded with relative bound zero, provided (2.4) holds, while Eqn. (4.10) implies that I is Hf1=2 -bounded with relR 2 1=2 ative bound 2 supx jG!x j , provided (2.4) holds. The latter of these two statements implies that, if (2.4) is satis ed, then H is self-adjoint on the domain of Hf .
5 Positive Commutators In this section we formulate our key technical result. When we talk below about a communtator of two, in general unbounded, operators, say H and A , we understand that D(H ) \ D(A) is dense (if necessary, in D(H ) , in the graph norm) and [H; A] is de ned rst as a form on D(H ) \ D(A) and then extended to a bounded or relatively bounded operator. j be the projection operator on We x j 1 once and for all. Let Ppart = Ppart the eigenspace of Hpart corresponding to the eigenvalue E j . For a xed energy scale we de ne the projection operator
P = Ppart Hf
(5.1)
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and, as usual, P = 1 ? P . Now we de ne a family of operators
AV = A + PV P ? PV P ;
(5.2)
where A0 is the second quantized dilatation generator de ned in (3.6), and
V = R2" I ;
R " = R" P ;
(5.3)
with
i?1=2 h R" = (H0 ? E j )2 + "2 : (5.4) Note that "R2" ! (H0 ? E j ) , as " ! 0 . Also, we remark that AV depends on four parameters, g , " , and . Let be an energy interval containing E j but no other parts of the spectrum of Hpart , and let n o 1 = inf ? sup (Hpart) \ (?1; inf ) > 0 ; (5.5) i.e., the distance, 1 , of inf to the part of the spectrum of Hpart on the left from it is assumed to be positive. The key technical result of this paper is
Theorem 5.1 Assume conditions (3.1) and (3.3) hold and let the parameters
involved satisfy
" minf; 1g 21 ; g "3=4 1=2 :
(5.6) (5.7)
Then
E(H ) [H; AV ] E (H ) (2 ?" ) j E (H )2 ; (5.8) 1=2 ?1=2 where j is the smallest eigenvalue of ?j and = O " + "?1 + o" (1) and o" (1) ! 0 , as " ! 0 . This theorem is proven in Section 7. Proof of Theorem 3.1: The absence of eigenvalues in a vicinity of the E j described in Theorem 3.1 follows directly from Theorem 5.1. Indeed, if such an eigenvalue existed, then (5.8) on the corresponding eigenfunction would imply h ; [H; AV ] i (2 ?" ) j k k2 (5.9)
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(here we used that E (H ) = ). But this inequality would contradict the virial relation h ; [H; AV ] i = 0 : (5.10) To convince ourselves that the latter holds we observe that (i) D(A)\D(H ) is dense in D(H ) and D(A)\D(H0 A) is dense in D(H0 ) , both in the graph norms, (ii) D(H0 ) = D(H ) and [H0; AV ] is H -bounded and (iii) [H; AV ] (H ) extends R jGxj2 , to a bounded opeator, provided 2 C01 and supp ? 1; ? sup ! x where, recall, = inf cont(Hpart) . The rst two properties are straightforward if one writes AV = A + Q1 , where
Q1 := (PV P ? PV P ) and notices that [Hf ; A] = Hf and Q1 is a bounded operator. To demonstrate (iii) we compute [H; A] = Hf + a((k rk + 23 )Gx) + a((k rk + 32 )Gx ) : Next, we use the following estimate proven in [2]: if 2 C01 with supp ? 1; ? sup R jG!x j2 , then x
ke jxj (H )k C
(5.11)
R jGx j2 for suciently small < ? sup supp ? sup ! . The last two relations x together with estimates (4.1) and (4.2) for f := hxi?M (k rk + 32 )Gx and Condition (3.1), which assures that sup kf k < 1 , imply statement (iii). Now condition x (iii) is weaker than the one (the H -boundedness of [H; AV ] ) under which the virial relation is proven in [17, 6]. However, replacing AV by (H )AV (H ) shows that the proof of [17, 6] goes through under conditions (i)-(iii). To prove the statements about absolute continuity and local decay we use abstract Kato-Mourre theory. A standard variant of this theory (see e.g. h i [4, 9]) requires H -boundedness of the commutators [AV ; H ] and AV ; [AV ; H ] . In our case these commutators are not H -bounded for two reasons. First, [A; H ] and h i A; [A; H ] are H -bounded under Condition (3.1) only for M = 0 . This follows
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from the straightforward computation adnA(H ) = Hf + a((k rk + 32 )nGx ) + a((k rk + 32 )nGx) ; (5.12) where we used the standard notation adA(H ) = [H; A]. The second reason is that the second part of the operator AV (see Eqns (5.2){(5.4)) contains the cut-o function Hf , entering into P , and this cut-o function, not being di erentiable in Hf , has a very singular commutator with the Hamiltonian H (or any other operator not commuting with Hf ). To remedy the rst problem we weaken the conditions in the Mourre theory (see Lemma 5.4 below) and use (5.11) (see Lemma 5.3 below). We go around the second problem by replacing AV by its smooth version as follows. In de nition (5.2){(5.4) of the operator AV we replace the projection P by the projection Ps , where
Ps = Ppart Hf s :
(5.13)
Thus we just vary a bit the photon energy scale. Denote the resulting operator by AV;s . Let now be a non-negative function supported in the interval [1; 2] and satisfying R = 1 . De ne Z AV(av) := (s) AV;s ds : (5.14) The next two lemmas establish the desired properties of AV(av) .
Lemma 5.2 Theorem 5.1 still holds if we replace AV by AV(av) . Proof: Inequalities (5.6){(5.7) are still true if we replace by s with 1 s 2 . Hence (5.8) holds after AV is replaced by AV;s , for 1 s 2 . Since 0 R and = 1 , this implies (5.8) with AV replaced by AV(av) .
Lemma 5.3 Let be as in (5.11). Then the operators [AV(av); H ] (H ) and [AV(av); [AV(av); H ]] (H ) are bounded.
Proof: We write AV(av) = A + Q , where
Z Q := (P s V Ps ? PsV P s)(s)ds :
(5.15)
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Now we can present the rst commutator as [H; AV(av)] = [H; A] + [H; Q] :
(5.16)
Applying Eqn. (4.10) with f := hxi?M (k rk + 32 )nGx to (5.12) and observing that Condition (3.1) guarantees that supx kf k is nite, we conclude that adnA (H )hxi?M are H -bounded for n = 1; 2 . Hence, due to Eqn. (5.11), adnA (H ) (H ) are bounded for n = 1; 2 . Furthermore, Q is a bounded operator and Eqns (4.3) and (4.4) show that so are the operators H Q and Q H . Hence [AV(av); H ] (H ) is bounded. Now we write i h i h i h [H; AV(av)]; AV(av) = [H; A]; A + [H; A]; Q h i h i + [H; Q]; Q + [H; Q]; Q : (5.17) By the conclusions at the end of the previous paragraph, the rst three terms on the r.h.s. of (5.17), multipliedh by (H )i , are bounded. It remains to show that A; [Q; H ] , the fourth term on the r.h.s. of (5.14), times (H ) , is bounded. To this end, we use the Jacobi identity and rewrite this term as h i h i h i A ; [Q ; H ] = [A ; Q] ; H + Q ; [A ; H ] : (5.18) By Eqn. (5.12) with n = 1 it suces to demonstrate that [A ; Q] and H [A; Q] are is bounded. We write [A ; Q] = (S + S ) ; where
S = A;
Z
(s) P s R2IPs "
(5.19)
:
Now we present S in the form Z d 2 ? A A S = d (s) e P s R" I Ps e ds =0 Z d 2 = d (s) P se? R"; I Pse? ds ; =0
(5.20)
(5.21)
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where R"; = eAR" e?A , etc., and where we have used that eAHf e?A = e Hf and therefore (5.22) eA Ps e?A = Pse? : Next using Leibnitz' rule, we transform this relation as Z Z S = (s) P s [A; R2" ] P s I Ps ds + (s) P s R2" [A; I ] Ps ds Z d P ? R2 I P ? ds : (5.23) + (s) d se se =0 " Since [A; H0] and [A; I ]hxi?M are Hf -bounded, the rst two terms on the r.h.s. are bounded. The last term on the r.h.s. can be rewritten as Z Z (s)) ? s (s) dsd P s R2" I Ps ds = d(sds P s R2" I Ps ds ; (5.24) which shows that it is bounded, as well. Using the above analysis and Eqns. (4.1) and (4.2) one shows that H [A; Q] is bounded as well. In the next lemma we weaken slightly standard hypotheses of the Mourre theory (see e.g. [6]) is order to accomodate our situation (see Lemma 5.3).
Lemma 5.4 Assume a self-adjoint operator H and an anti-self-adjoint operator A , de ned on the same Hilbert space, satisfy for some interval R the
following conditions ( ) [H; A] (H ) and (H )[[H; A]; A] (H ) are bounded for some 2 C01 s.t. 1 on and ( ) E(H )[H; A]E(H ) E (H )2 for some > 0 . Then the spectrum of H in is absolutely continuous and H has the local decay property in with respect the operator A and therefore the operator A as well. Proof: De ne A = (H )A (H ) and let f be a function on R satisfying f 1 on and suppf fj () = 1g . Observe that ( 0) [H; A ] and [f (H )[H; A ]f (H ); A ] are bounded. The boundedness of the rst term follows readily from the rst part of condition ( ) . To prove the boundedness of the second term we write
[f [H; A ]f; A ] = fB [f; A] ? [A; f ]Bf + f [B; A]f ;
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where f = f (H ) , = (H ) and B = [H; A] . The rst two terms on the r.h.s. are bounded due to the boundedness of fB , Bf and [f; A] and the third term is bounded by the second part of ( ) . Next, ( 0) condition ( ) still holds with A replaced by A . Now, keeping in mind the observations made above we can follow a simpli ed version the arguments of [17, 6] (see the proof Theorem 4.9 of [6] with Lemmas 4.12 and 4.13 omitted or the proof of Theorem 7.1 of [17] with Lemmas 7.4 and 7.5 dropped (those lemmas are replaced by the second part of ( 0) above) and Theorem 7.8 of [17] and use a general result of Kato (see Theorem XIII.25 of [19]) in order to derive the statements of Lemma 5.4. The fact that the local decay property w.r. to A implies that w.r. to A follows from the observation that, by an operator calculus of [9] and property ( ) , hAi?1A is a bounded operator. Now we can complete the proof of Theorem 3.1. The absolute continuity and local decay property, with A replaced by the operator AV(av) , follow from Lemmata 5.2-5.4. To pass to the local decay property w.r. to the operator A , it suces to observe that due to (4.10), Q is a bounded operator and, therefore, hAi? hAV(av)i const , for > 0 .
6 Positivity of Truncated Commutator Before tackling the proof of Theorem 5.1 head on we go a part way by proving a positivity of a simpler commutator. Namely, let where and
B 0; = P B0 P ;
(6.1)
B0 = [ H ; A 0 ]
(6.2)
P = P E (H0) : The main result of this section is Lemma 6.1 Assume g2 . Then B0; 21 Hf P 12 P ;
(6.3)
(6.4)
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and, if in addition 3jzj , then
p
jB0; ? zj?1=2 P
2
Hf?1=2 P
;
(6.5)
where jAj := AA , for a closed operator A . Proof: We begin with a computation:
B0 = Hf + Ie ;
(6.6)
where Ie = a(Ge x ) + a(Ge x) , and
Ge x(k) := k rkGx (k) + 32 Gx (k) : Inequality (4.9) with = 1=4 and f = Ge x yields Z e 2 Ie 41 Hf + 4 jG!xj which implies Z jGe x j2 3 B0 4 Hf ? 4 ! : Since Hf 0 , Inequality (2.10) implies that
M
hxi E(H0)
CM
(6.7) (6.8) (6.9) (6.10)
for any M < 1 , provided sup < inf cont spec Hpart . The last two inequalities and the de nition (3.1) of g imply that 3 2 B0; 4 Hf ? Cg P : (6.11) Next, De nition (5.1) yields that
P = P part 1 + Ppart Hf : Since, by the energy conservation,
P part E (H0) = we have
X i: E ij ) (j ) (j ) := X P i (7.46) Ppart part and Ppart := 1part ? Ppart ;
C g2
i: E i E j
(>j ) P (>j ) , for some > 0 , we estimate and noting that (Hpart ? E j ) Ppart part
Z
(j ) R2 2
PMP ? P Gx (k) Ppart (7.47) ";!(k) Gx (k ) dk P
Cg :
Thus, remembering the de nition of the matrices Aij (see the paragraph preceeding (3.2)), we obtain h i?1 X Z PMP = (Aij )Aij (Hf + ! ? E ji )2 + "2 dkP + O(g2 ) (7.48) i: E i E j
where E ji := E j ? E i . Next, using the identity
! Hf2 1 2 H 1 f (Hf ? ) (Hf ? )2 + "2 = 2 + "2 1 + (Hf ? )2 + "2 ? (Hf ? )2 + "2 ; (7.49) we obtain on Ran P 0 Z Aij Aij dk 1 h i X ?1 )+ O("?2 2) + O(g 2 ) ; (7.50) A 1+ O ( " M =@ ji 2 2 i:E i E j (! ? E ) + "
BFSS; Apr. 28, 97
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Since !2 RS2 Aij Aij dS (k) , where dS (k) is the area element on S 2 , is continuous in ! and vanishing at ! = 0 and since j > const (see Eqn. (3.3)), we have on Ran P 1 0 Z h i X 1 jGij (k)j2 !(k) ? E ji dkA 1 + o" (1) + O("?1 ) + O(g2) ; M = "@ i: E i
