Physics: Commun. Math. Phys. 131, 333-346 (1990) Mathematical
Physics: Commun. Math. Phys. 131, 333-346 (1990) Mathematical
Physics: Commun. Math. Phys. 131, 333-346 (1990) Mathematical
1. Introduction
Let A and B be linear operators in a separable complex Hubert space Jjf. Then,
under suitable conditions concerning {A, B} the strong limit
/ t/ \ ( t \V
s-lim I expl --4 expl B I =exp(-ίC) (1.1)
«->oo \ V n ) \ n JJ
exists for t ^ 0, where the operator C can be constructed by means of A and B. This
is the well-known Trotter-Lie product formula for strongly continuous (C0-)
semigroups [1], (For finite matrices it has been established by Sophus Lie about
1875.) Since the discovery of the product formula, it has permeated through
mathematics and mathematical physics, challenging the problem of relaxation and
generalization of the hypotheses under which the formula holds, see [2-10].
A solution of this problem implies that one has to do the following:
(i) to find the set of pairs {A,B} for which the limit (1.1) exists;
(ii) to identify the operator C and to describe the mapping {A,B}: ->C;
(iii) to generalize (if possible) the exponential functions involved in the left-hand
side of (1.1) to a class of real-valued, Borel measurable functions /( ), g( ) such
that in some operator topology τ,
ί it \ it \\n
τ-lim / ( - A g ( - B ] } =exp(-ίCμi (1.2)
n->«> V \n J \n JJ
(1.6)
2. Technical Preliminaries
In the following, we prove some kind of bounded convergence theorem for trace
ideals which will be necessary in the sequel.
For X,LeJ^f) we write K<ζL if \\Ku\\ ^ \\Lu\\, uetf. This condition is
equivalent to K*K^L*L and also to |JC|^|L|. It is obvious that K<ζL with
LeVpW) implies that Xe<ί?p(Jf) and \\K\\p£ ||L||p, 1 ^p< + oo.
Proposition2.1. Let Kn = K* e Λ(Jtf), Jw = J*e^pf), n = l,2,.... Assume that
s-limJn = I and that Kn<ζLJn, n = l,2,..., where L=L*e^p(^f) for some
(so that KnE^p(J^)). If w-lim Kn = K, then KE^P(^) and
|| 11,-UmK^K.
Proof. By w-lim Kn = K we find \\Ku\\ ^ lim inf \\Knu\\ ^ lim inf ||LJMw||
«—»• oo n~* oo W-*QO
= \\Lu\\, uEJf. Hence K^L and Ke^p(J^).
Since Le^pf) there is a sequence {P/Jjii of finite-dimensional orthogonal
projections such that LPj=PjLJ = \,2,..., and ||Lβ;||p= ||β/L||p< τ, J = l,2,...,
where Qj = I-Pj. By w-lim Kn = K we have || - ||p- lim PjKnPj = PjKPj for
w~* oo w -~* oo
every j=l,2,..., and every l ^ p < + oo. Given ε>0, therefore, we have
|| Pj{Kn — K)Pj\\p<ε for sufficiently large n depending on ε and j. On the other hand,
we have \\(Kn-K)Qj\\p£ \\KHQj\\p+ \\KQj\\p£ \\QjLJnQj\\p+ \\LPjJnQj\\p
+ ||Lβj||p. Since 5-lim Jn = I, we have sup ||Jn|| =c< + 00. Consequently, we
obtain the estimate
1 +r ' . II T II I I ./"Λ T T^ II /^ /< \
' J
; = 1,2,.... Since Pj is a finite-dimensional projection, we get
|| - Up— lim QjJnPj=Q, 1 ^p< + oo. Therefore, choosing; and n sufficiently large,
we can achieve that \\Kn — K\\p<s. This proves the proposition. Π
Corollary2.2. Let Kn( }'.tf-*@(3?\ Jn( }\tf^>@(3?\ n=l,2,..., and K(-):tf
-+&(-) operator-valued functions such that the assumptions of Proposition 2.1 are
poίntwise satisfied. If in addition \\Jn(t)\\ ^c< H-oo, s-lim Jn(t) = I and
«->oo
w-lim Kn(t) = K(t) hold uniformly intεJf, then \\ - L-lim Kn(t) = K(t) uniformly in
Jn by Kn(t), K(f) and Jn(t\ respectively, the estimate (2.1) remains true. But the
uniformity of s-lim Jn(i) = I in ίeJf implies the uniformity of
«-*oo
Proposition 2.3. Let Kn e <ίί2ί,( Jf ) and £„ e ^2p(^l n = 1,2,..., for some p e [1 , oo).
Assume s-lim Kn = KeV2p(J*?) and \\ ||2p-lίm &n = K. If \\Kn\\2p^ \\Kn\\2p,
n-*oo
Since K(t)<ϋL we find that \\K(t)Qj\\2p^ \\LQj\\2p< τ,j=l,2, ... . Hence, we have
1 ε
« = 1,2, ..., ίejf. Given ε>0 we fix a j such that τ < -. Since Pj is a finite-
Since lim \\Kn(t)\\2P = ||*(f)||2j,and || K(t) \\2p ^ \\L\\2p uniformly in ίeJf, there
is a n 0 ^Γsuch that the sequences {\\Kn(t)\\2P}n=no and {II^W + ^WM^no are
uniformly bounded in ίeJf. Hence, it is not hard to see that
and H^W + ^ W I I i S - i ^ l l X W l l i J uniformly in
ί e Jf. Taking into account (2.2) we get || Kn(t) - K(t) \\22PP -^> 0 uniformly in t e JΓ
which immediately completes the proof. Π
3. Product Formula
Let ^4^0 and 5^0 be self-adjoint operators in a separable Hubert space ffl.
Denoting by Q = @(A1/2)n@(B1/2) we do not assume that Q is dense in Jf . By 3?'
we denote the closure of β, i.e. J"f ' = β " . In general, J f ' is a proper subspace of $? ,
i.e. J f Φ jf '. The orthogonal projection from Jf onto ^fr is indicated by Π. We
recall that C is the self-adjoint operator in 3? ' associated with the non-negative
closed quadratic form f ^\\Aί/2f\\2 + \ \ B 1 / 2 f \ \ 2 , /eβ, i.e. C = A + B.
Further, we introduce a class of Borel functions / and g defined on
Ri = {t e R 1 : t ^ 0} characterized by
/(o)=ι, /'(o)=-
g(0)=l, g'(0)=-l.
Notice that f(tA)a^->I and g(tB)a-^I as ί-> +0 for any α^O. At the beginning
we assume that
0</(ί), ίeRi, (3.2)
and that
and v(ί)={(l-gW)
(3.3)
are monotonously nonincreasing functions .
Trotter-Kato Product Formula for Gibbs Semigroups 339
- (3.4)
Lemma 3.1. // the conditions (3.1)-(3.3) and (3.5) are satisfied, then
(λ + MMΓ^eVJίJr) (3.6)
for λ>0 and t>0.
Proof. On account of the identity
(3.7)
λ, t > 0, the result follows if one proves that the operator in the curved brackets is
boundedly invertible for λ > 0 and t > 0. For λt ^ 1 we get
Lemma 3.2. Let A and B be two non-negative self-adjoint operators defined on the
separable Hilbert ffl ana let f and g be two Borel functions obeying (3.1)-(3.3). Let
F(t) = g(tB)l/2 f(tA)g(tB)112, t e Ri . // (3.5) is satisfied, then e~tce ^2p(^f} fort>0
and
e-'cΘO (3.8)
On the other hand, starting from the identity (3.7) one gets
1
g(tS)1/2 = (λ + S(ί)) -1 F(t)
+ λ(λ + S(t)Γ 1 g(ίB)1/2 [/-/(ίX)] (λ + M(t)Γ 1 g(ίβ)1/2 ,
λ,t>0. Setting t=( and A = l/ί as well as using ||g(ί'B)1/2|| gl, \\λ(λ + S(t')Γl\\
and |1 /-/(L4) H ^ l we find
/2
1/2
1/3/1 V
ί6[if,τ] Here 2l/- ί-+M(τ)J e^2p(^) by (3.6). Since
/ t V+1 / ί \
5-limF -- =e~tC®Q (see [6,7]), and s-limF -- =/ we obtain
n-»oo \n-\-\J n^oo \Π+1/
Setting
1/2 1/2
1/3/1 V / t \
L = 2 / - - + M(τ) , JM(ί) = g -- B and applying Proposition 2. 1 we
I/ n \τ / \n+l /
(3.10)
Trotter-Kato Product Formula for Gibbs Semigroups 341
uniformly in t e [0, τ]. For instance, denoting by EB( -) the spectral measure of B we
get the representation
. Assuming uε@(B) and taking into account (3.1) we obtain the estimate
t
^ const. to ||^ const. || Bu \\ ,
t
ίe[0,τ], which yields the uniformity of lim g w = u for every u
tt + 1
Since 3>(B) is dense in ffl and I — is uniformly bounded we obtain
(3.10).
The estimate yields the uni-
/2
formity of s-lim g
Y
B] =1 in ί e [0, τ]. Since s-lim / ^— B]=I and
/ H-00
/ ί t
5- lim g B I =/ uniformly in ίe[0,τ], we get 5- lim F =/ uni-
/ t Y
ίe[>,τ] (cf. [6]) we find the same for 5-lim F f -) =e~tc®0. Taking into
= inf
0<s<ί
= - lim
ί>0
.
Since φ0(ί)^φ(ί)? ί>0, we get 0^g(ί) = l-ίtp(ί)^l -ίtp0(ί) = g0(ί), ί>0. Fur-
thermore, the representation
= \\F(t)m\\pp (3.15)
and, analogously,
\\F0(t)m\\pp=\\l (3.16)
On account of (3.14), (3.15), (3.16) and f(t)^fΌ(t) we find
Theorem 3.4. Let A and B be two non-negative self-adjoint operators defined in the
separable Hilbert space ffl and let f and g be two Borel functions obeying (3.1). //
(3.12) is satisfied, then e~ιc^γ(^f'\ ί>0, and
f f t \ f t \V
IHIi-lim / (-A )g (~B\) =e-*c®Q (3.17)
π
/ι>°°)
holds uniformly in t ε [//, τ], 0 < η < τ < -f oo.
ί>0
Proof. By Lemma 3.2 we have e~(tl2p)Ce<e2pW)> Since (e-^^c}^ =e-tc
(3.18)
2p
π-1
(3.19)
2p
tc
Furthermore, we note that s-lim = e~ ®0 and
n /~\n ''
\n-l
B
-'™ C e o (3 20
'
344 H. Neidhardt and V. A. Zagrebnov
\kn + l
k-l
f A B
^ Σ
V kn + l
B
^ι )4knτϊ
_{ e -< /*>cΘO} ||β-<"*>cΘO||ίfc-m-1
IHIrlim
' \
uniformly in ί e [f?, τ] and / = 0, 1 , 2, . . . , k — 1 . But the last assertion coincides with
(3.17). Π
_ V I
— L \ — 2m *j= αΣ= l
with the domain &(TN) = C$(ΛN). The domain &(Tσ) is specified by a boundary
condition σ e C(dA). Then, one can check that Tσ is a p-generator for the self-
Nv
adjoint Gibbs semigroup GTσ(t) [12], i.e., for — ^p< -f oo we have
(4.1)
Trotter-Kato Product Formula for Gibbs Semigroups 345
uniformly in β>0 on any compact interval bounded away from zero, where
C = A + B = H + NuI. Cancelling both parts of (4.3) by exp(-βwJV) one gets
(4.2). D
Remarks 4.2. Our results are not applicable to interactions 17 which are not
semibounded from below (e.g. for Coulomb systems) in spite of the semibounded-
ness of the operator H.
Acknowledgement. The authors wish to thank the referee for his hints which have allowed them to
essentially simplify the proofs of their earlier work.
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Communicated by H. Araki