数理解析研究所講究録 1208 巻 2001 年 128-134

128
ON THE NORM CONVERGENCE OF THE TROTTER-KATO
PRODUCT FORMULA WITH ERROR BOUND

金沢大理

岡山大理

孝 (Takashi Ichinose *)田村 英男 (Hideo Tamura **)一瀬

Department of Mathematics, Faculty of Science, Kanazawa Univ.
** Department of Mathematics, Faculty of Science, Okayama Univ.
*

Abstract. The norm convergence of the Trotter-Kato product formula with error
bound is shown for the semigroup generated by that operator sum of two nonnegative
selfadjoint operators and which is selfadjoint.
$B$

$A$

1. Introduction and Result
It is well-known ([23], [15]; [19]) that the Trotter-Kato product formula for theareselfadjoint semigroup holds in strong operator topology. Namely, when and$D[A]$$D[B]$
,
space
and
with
domains
inaHilbert
operators
nonnegative selfadjoint
then
$A$

$B$

$\mathcal{H}$$\mathrm{s}-\lim_{\mathrm{n}arrow\infty}(e^{-tB/2n}e^{-tA/n}e^{-tB/2n})^{n}=\mathrm{s}-\lim_{r\iotaarrow\infty}(e^{-tA/n}e^{-tB/n})^{n}=e^{-tC}$

,

(1.1)

which is selfadjoint, or, in particular, if the operator sum
is the form sum$A+B$ is essentially selfadjoint on $D[A]\cap D[B]$ with $C$ its closure. The convergence is.
uniform on each compact -interval in the closed half line
The aim of this note is to briefly announce our recent results on its operator-norm
convergence with error bound. In [12] we have shown

if$A\dotplus B$

$C$

$t$

$[0, \infty)$

with domains
Theorem 1.1. If and $B$ are nonnegative selfadjoint operators in$D[A]$ and $D[B]$ and if their operator sum $C:=A+B$ is selfadjoint on $D[C]=D[A]\cap$$D[B]$ , then the prvxiuct formula in operator norm holds with error bound:$A$$\prime H$$||(e^{-tB/2n}e^{-tA/n}e^{-tB/2n})^{n}-e^{-tC}||=O(n^{-1/2})$,$||(e^{-tA/n}e^{-tB/n})^{n}-e^{-tC}||=O(n^{-1/2})$,$narrow\infty$

.

(1.1)

129
The convergence is unifom on each compact -interval in the open half line $(0, \infty)$ , andfurther, if $C$ is strictly positive, unifom on the closed half line $[T, \infty)$ for every$t$$\hslash ed$$T>0$ .

One of the typical examples of such aselfadjoint operator $C=A+B$ is the
Schr\"odinger operator$H=- \frac{1}{2}\Delta+P|x|^{-1}+D|x|^{2}+E|x|^{2000}$in$L^{2}(\mathrm{R}^{3})$, where $P$, $D$ and $E$ are nonnegative constants.

Remark 1.1 The first result of such anorm convergence of the Trotter-Kato productformula (1.1) was proved by Rogava [20] in the abstract case under an additional conis -bounded, with error bound $O(n^{-1/2}\log n)$ . The next was by Helffer
dition that
nonnegative[5] for the Schr\"odinger operators $H=H0+V \equiv-\frac{1}{2}\Delta+V(x)$ with
$O(|x|^{2})$withfor large
potentials $V(x)$ , roughly speaking, growing at most of ordererror bound $O(n^{-1})$ . Each of these two results is independent of the other.
Then under some stronger or more general conditions, several further results are
obtained. As for the abstract case, abetter error bound $O(n^{-1}\log n)$ than Rogava’s-bounded for some $0<$
isis obtained by Ichinose-Tamura [11] (cf. [9]) when$\alpha<1$ , even though the $B=B(t)$ may be-dependent, and by Neidhardt-Zagrebnov
is -bounded with relative bound less than 1. As for[16], [17] (cf. [18]) when
the Schr\"odinger operators, adifferent proof to Helffer’s result was obtained by DiaSchatzman [2]. Further, more general results were proved for continuous nonnegativewith $\rho>0$ ,for large
potentials $V(x)$ , roughly speaking, growing of order$O(n^{-2/\rho})$ ,instance,oforder
together with error bounds dependent on the power (for), by Ichinose-Takanobu [6] (cf. [7]), Doumeki-Ichinose-Tamura [3], Ichinoseif
Tamura [10], Decombes-Dia [1] and others, although the primary purpose of most ofthese papers was to prove rather anorm estimate between the Kac transfer operator
and its corresponding Schr\"odinger semigroup. The Schr\"odinger operators treated in: $H=H_{0}(A)+V\equiv$[6] and [3] may even involve bounded magnetic fields$\frac{1}{2}(-i\nabla-A(x))^{2}+V(x)$ . In [7] and [8] the relativistic Schr\"odinger operator was also
dealt with.
It should be noted (see [4], [21]) that in all these cases of the Schr\"odinger operators
the sum $H=H_{0}+V$ (resp. $H=H_{0}(A)+V$ ) is selfadjoint on the domain $D[H]=$
$B$

$A$

$C^{\infty}$

$|x|$

$B$

$A^{\alpha}$

$t$

$B$

$A$

$O(|x|^{\rho})$$|x|$$\rho$$\rho\geq 2$$\nabla\cross A(x)$(resp. $D[H]=D[H_{0}(A)]\cap D[V]$ ).
Thus the present theorem not only extends Rogava’s result, but also can extend andcontain all the results mentioned above, inclusive better error bounds in some cases.
$D[H_{0}]\cap D[V]$Remark 1.2. Unless the sum $A+B$ is selfadjoint on $D[A]\cap D[B]$ , the norm convergenceof the rather-Kato product formula does not always hold, even though the sum isessentially selfadjoint there and is A-form-bounded with relative bound less than 1.
Acounterexample is due to Hiroshi Tamura [22].
$B$

replaced by real-valued,
The theorem also holds with the exponential function
satisfying that
Borel measurable functions and on$e^{-s}$$f$$g$$0\leq f(s)\leq 1$$[0, \infty)$, $f(0)=1$ , $f’(0)=-1$ ,

(1.1)

130
that for every small$>0$ there exists apositive constant$\epsilon$,swith $1<\kappa$$\leq 2$$f(s)\leq 1-\delta(\epsilon)$

and that, foT some fixed constant$\kappa$$\geq\epsilon$$=\delta(\epsilon)<1$$\delta$such that

,

(1.4)

,

$[f]_{\kappa}:= \sup_{s>0}s^{-\kappa}|f(s)-1+s|<\infty$

,

(1.5)

and the same for g. Of course, the functions $f(s)=e^{-s}$ and $f(s)=(1+k^{-1}s)^{-k}$ withk $>0$ are examples of functions having these properties.
Theorem 1.2. If $3/2\leq\kappa$$\leq 2$, it holds in operator nor$m$ that$||[g(tB/2n)f(tA/n)g(tB/2n)]^{n}-e^{-tC}||=O(n^{-1/2})$ ,$||[f(tA/n)g(tB/n)]^{n}-e^{-tC}||=O(n^{-1/2})$ ,$narrow\infty$

(1.4)

.

2. Outline of Proof

To proving the theorem, it is crucial to show the following operator-norm version ofChernoff’s theorem with error bounds. The case without error bounds was noted by

Neidhardt-Zagrebnov [18].

Lemma. Let
be a nonnegative selfadjoint operator in a Hilbert space 7{ and let$\{F(t)\}_{t}>0$ be a familyof selfadjoint operators with $0\leq F(t)\leq 1$ . Define $S_{t}=t^{-1}(1-$$F(t))$ .
en in the following teuo assertions, for $0<\alpha\leq 1$ , (a) implies (b).
$C$$\overline{\mathrm{f}}\mathrm{f}\mathrm{l}$(a)$||(1+St)^{-1}-(1+C)^{-1}||=O(t^{\alpha})$,$t\downarrow \mathrm{O}$

.

(2.1)

(b) For any $\delta>0$ with $0<\delta\leq 1$ ,$||F(t/n)^{n}-e^{-tC}||=\delta^{-2}t^{-1+\alpha}e^{\delta t}O(n^{-}’)$,

n$arrow\infty$

,

(2.2)

for all $t>0$ .

Therefore, for $0<\alpha<1$ (resp. $\alpha=1$ ), the convergence in (2.2) is unifom on eachcompact -interval in the open half line
(resp. in the closed half line [0, ) .
$C$
$i.e$
$C\geq\eta$Moreover, if is strictly positive,.
for sorne constant $\eta>0$ , the error
bound on the right-hand side of (2.2) can also be replaced by $(1+2/\eta)^{2}t^{-1+\alpha}O(n^{-\alpha})$ ,so that, for $0<\alpha<1$ (resp. $\alpha=1$ ), the convergence in (2.2) is unifom on the closed
half line
for every fied $T>0$ (resp. on the whole closed half line [0, ) .
$(0, \infty)$$t$$\infty$$[T, \infty)$$\infty$

Sketch of Proof of Lemma.
Put$F(t/n)^{n}-e^{-tC}=(F(t/n)^{n}-e^{-tS_{t/n}})+(e^{-tS_{t/n}}-e^{-tC})$.

For the first term on the right we have by the spectral theorem$||F(t/n)^{n}-e^{-t\mathrm{S}_{t/n}}||=||F(t/n)^{n}-e^{-n(1-F(t/n))}||\leq e^{-1}n^{-1}$

,

$)$

$)$

131
,

$0\leq e^{-n(1-\lambda)}-\lambda^{n}\leq e^{-1}/n$for$0\leq\lambda\leq 1$.

For the second term, we use$(1+S_{\epsilon})^{-1}[e^{-t(\delta+S_{e})}-e^{-t(\delta+C)}](1+C)^{-1}$$= \int_{0}^{t}e^{-(t-s)(\delta+S_{e})}[(1+S_{\epsilon})^{-1}-(1+C)^{-1}]e^{-s(\delta+C)}ds$$= \int_{0}^{t/2}+\int_{t/2}^{t}$, to bound these two integrals on the right by
where $0<\delta\leq 1$ and$=t/n$, we have
Taking$\epsilon>0$$(\delta^{2}t)^{-1}e^{\delta t}O(\epsilon^{\alpha})$.

$\epsilon$$||e^{-tS_{t/n}}-e^{-tC}||\leq(\delta^{2}t)^{-1}e^{\delta t}O((t/n)^{\alpha})=\delta^{-2}t^{-1+\alpha}e^{\delta t}O(n^{-\alpha})$.

Sketch of Proof of Theorems 1.1 and 1.2.
First note that since $C=A+B$ is itself selfadjoint and so aclosed operator, by theclosed graph theorem there exists aconstant such that$a$$u\in D[C]=D[A]\cap D[B]$ .

$||(1+A)u||+||(1+B)u||\leq a||(1+C)u||$ ,The proof of the theorem is divided into two cases, (a) the symmetric product case$F(t)=e^{-tB/2}e^{-tA}e^{-B/2}$

,

(2.3)

and (b) the non-symmetric product case
$G(t)=e^{-tA}e^{-tB}$

.

(2.4)

(a) In the symmetric case we put$S_{t}=t^{-1}(1-F(t))=t^{-1}(1-e^{-tB/2}e^{-tA}e^{-tB/2})$and use Lemma to show that$||(1+S_{t})^{-1}-(1+C)^{-1}||=O(t^{1/2})$ ,$t\downarrow \mathrm{O}$.

Put$A_{t}=t^{-1}(1-e^{-tA})$,$B_{t}=t^{-1}(1-e^{-tB})$,$C_{t}=t^{-1}(1-e^{-tC})$.

We have$1+S_{t}=1+A_{t}+B_{t/2}- \frac{t}{4}B_{t/2}^{2}+\frac{t^{2}}{4}B_{t/2}A_{t}B_{t/2}-\frac{t}{2}(A_{t}B_{t/2}+B_{t/2}A_{t})$$=K_{t}^{1/2}(1+Q_{t})K_{t}^{1/2}$

,

132

$K_{6}\ovalbox{\tt\small REJECT}$$1+A_{\langle}+B_{\mathit{2}\mathit{7}2}$$\ovalbox{\tt\small REJECT} B_{\ovalbox{\tt\small REJECT} \mathit{7}2}\ovalbox{\tt\small REJECT} 1$

D$t^{\mathit{2}}K1/2\mathrm{p}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} 4$$Q_{\mathit{6}}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} K\ovalbox{\tt\small REJECT}$,${}^{t}rx^{-1/2}$ $(A_{\mathit{6}}B_{\mathit{6}\mathit{7}2}+B_{\mathit{6}\mathit{1}2}A_{\mathit{6}})K\ovalbox{\tt\small REJECT} 1/2$$\mathrm{r}_{\mathrm{Z}^{-1/2}}$$\ovalbox{\tt\small REJECT}$$B_{\langle \mathit{7}2}A_{\mathit{6}}B_{\mathit{6}\mathit{7}2}K_{\mathit{6}}$$-\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} K\ovalbox{\tt\small REJECT}$

Then we can show$||(1+Q_{t})^{-1}||\leq 2/(3-\sqrt{5})$,$||(1+S_{t})^{-1}K_{t}^{1/2}||=||K_{t}^{-1/2}(1+Q_{t})^{-1}||\leq 2/(3-\sqrt{5})$

(2.5)

.

(2.6)

Then we have$(1+S_{t})^{-1}-(1+C)^{-1}$$=(1+St)^{-1}[A+B-(At+ \mathrm{B}\mathrm{t}/3-\frac{t}{4}B_{t/2}(1-tA_{t})B_{t/2}$$- \frac{t}{2}(A_{t}B_{t/2}+B_{t/2}A_{t}))](1+C)^{-1}$(2.7)$=(1+St)^{-1}(A-A_{t})(1+C)^{-1}+(1+S_{t})^{-1}(B-B_{t/2})(1+C)^{-1}$$+(1+S_{t})^{-1}[ \frac{t}{4}B_{t/2}(1-tA_{t})B_{t/2}+\frac{t}{2}(A_{t}B_{t/2}+B_{t/2}A_{t})](1+C)^{-1}$$\equiv R_{1}(t)+R_{2}(t)+R_{3}(t)$.

We can show the bounds$||R.(t)||\leq ct^{1/2}$,$i=1,2,3$ ,with some constant $c>0$ . For instance, we can get the bound forexpression(2.8)$R_{1}(t)$, via the$R_{1}(t)=[(1+S_{t})^{-1}K_{t}^{1/2}][K_{t}^{-1/2}(1+A_{t})^{1/2}]$$\mathrm{x}[(1+A_{t})^{-1/2}-(1+A_{t})^{1/2}(1+A)^{-1}](1+A)(1+C)^{-1}$by (2.6) and the spectral theorem$||R_{1}(t)||\leq\overline{3}\nabla-5^{a||(1}2+A_{t})^{-1/2}-(1+A_{t})^{1/2}(1+A)^{-1}||\leq ct^{1/2}$(b) The non-symmetric casetator argument to observe that$\mathrm{w}\mathrm{i}\mathrm{u}$.

follow from the symmetric case. We use the commu-$||G(t/n)^{n}-F(t/n)^{n}||=||(e^{-tA/n}e^{-tB/n})^{n}-(e^{-tB/2n}e^{-tA/n}e^{-tB/2n})^{n}||$$=O(1/n)$.

133
3. The Final Result

In arecent preprint [14], we have shown that if $\kappa=2$ , then Theorem 1.2 holds with
optimal error bound $O(n^{-1})$ . Further, the convergence is uniform on each compact, and further, if $C$ is strictly positive, uniform on-interval in the closed half line.
the whole closed half line
The idea of proof is simply to iterate the resolvent equation of the first identity in(2.5) with help of its adjoint form to get$[0, \infty)$$\mathrm{t}$$[0, \infty)$$(1+S_{t})^{-1}-(1+C)^{-1}$$=((1+C)^{-1}+[(1+S_{t})^{-1}-(1+C)^{-1}])(C-S_{t})(1+C)^{-1}$$=(1+C)^{-1}(C-S_{t})(1+C)^{-1}+[(C-S_{t})(1+C)^{-1}]^{*}(1+S_{t})^{-1}(C-S_{t})(1+C)^{-1}$$\equiv R_{1}’(t)+R_{2}’(t)$.

Then by the same arguments together with (2.6) we can show the bounds$||R_{i}’(t)||=O(t)$,$i=1,2$ .

Therefore it turns out that the product formula (1.2) in Theorem 1.1 holds, now withultimate error bound $O(n^{-1})$ , properly extending and containing all the known previous
related results.
Finally, we comment about optimality of the error bound $O(n^{-1})$ . We know that
if both and are bounded operators, then we have, in the symmetric product case(2.3), $||F(t/n)^{n}-e^{-tC}||=O(n^{-2})$ , while, in the non-symmetric product case (2.4),$||G(t/n)^{n}-e^{-tC}||=O(n^{-1})$ . But also in the symmetric product case, we can give an
example of two unbounded selfadjoint operators and $B$ whose operator sum $C=A+B$
is selfadjoint on $D[A]\cap D[B]$ such that $||F(t/n)^{n}-e^{-tC}||\geq L(t)n^{-1}$ , with apositive
continuous function $L(t)$ of $t>0$ independent of .
$A$

$B$

$A$

$n$

Part of the present results also was briefly announced in [13].
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