Ames W.F., Harrel E.M., Herod J.V. (editors). Differential Equations with Applications to Mathematical Physics
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210
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This page intentionally left blank
Topics
in
the Spectral Methods
in
Numerical Computation
-
Product Formulas
S.
T.
Kuroda
Department
of
Mathematics
Gakushuim University
Mejiro, Tokyo
171
Japan
1
Introduction
This article is
a
brief account on some aspects of the combined use
of
product formulas
of
Lie-Trotter type and the Fast Fourier Transform
(FFT)
for
solving the Schrodinger evolution equation
(1)
a
at
i
-
~(z,t)
=
(-A
-t
V(X))U(Z,~)
and for computing eigenfunctions and eigenvalues of the operator
-A
+
V.
Here,
V(z)
is
a
real valued function. The use of the
product formula for this purpose goes back to
[4]
and the combined
use with the
FFT
is due to
121.
The idea
of
[2]
is as follows.
Let
A
=
-A
and
B
=
V.
Then
exp(-itA) and e>cp(-itB) are multiplication operators, one in the
Fourier ((-)space and the other in the configuration (z-)space, re-
spectively. Therefore, products like {exp(
-i(t/n)A)
exp(-(t/n)B)}n
can be computed easily by going back and forth between these spaces.
The transformation between
2-
and (-spaces can be implemented
Differential Equations with
Applications to Mathematical
Copyright
@
1993
by Academic Press, Inc.
All
rights
of
reproduction in
any
form
reserved.
Physics
ISBN
0-12-056740-7
2
13
214
S.
T.
Kuroda
very efficiently by means of the
FFT,
which requires only
O(N
log
N)
multiplications for
a
problem with
N
mesh points. Using this idea
for solving
(l),
[a]
develops
a
method, which may be called “numer-
ical spectroscopy,’’ for simultaneously computing all eigenvalues in
a
wide energy range. Motivated by
[2], [5]
proposed
a
method, which
may be called “numerical resonant excitation” for computing
a
par-
ticular eigenfunction and the associated eigenvalue very accurately.
[2]
and
[5]
also contain ample numerical examples.
In this article we shall focus our attention on product formulas
and shall exploit various product formulas which may be used for
the numerical procedure mentioned above. Not only
a
formula itself
but the order of error
is
of
interest. In Section
2.2
we shall list
a
few
formulas with the order of error (for bounded generators). Some
of
these formulas seem not to have been noticed in the literature.
Pos-
sibilities of applying these formulas will also be discussed in Section
2.3.
Some remarks given
in
the talk on the methods developed in
[2]
and
[5],
especially on
a
way of handling remote eigenvalues in
[2]
will
be reported elsewhere.
2
Product
Formulas
2.1
Preliminaries
In this section we consider an abstract evolution equatioii in
a
Banach
space
X.
The equation and its solution with the initial data
uo
are
written as
d
-u(t)
=
Cu(t),
t
>
0;
u(t)
=
exp(tC)uo.
dt
We assume that the generator
C
and other operators appearing later
are all bounded linear operators in
X.
The reason for assuming the
boundedness is twofold. Firstly, it makes the error estimate simpler,
and secondly, in applications, product formulas will be applied after
discretization, i.e., in
a
finite-dimensional space.
The product formulas we shall discuss are written generally as
exp(tC)
=
lim
F(t/n)n.
n-+w
(3)
Topics in the Spectral Methods
in
Numerical Computation
2
15
Here,
F(t)
is an approximation
of
exp(tC) for small t.
We
may call
it
a
unit increment
of
the product approximation. The order of
error
(for large
n)
in
(3)
is related to the order
of
error (for small
t)
in the
unit increment. Namely, it can be seen
by
a
standard argument (cf.
[6],
p.
295)
that
2.2
List
of
Formulas
We
are interested in two cases
C
=
A
t
B
and C
=
[A,B]
and shall
list several product formulas. Since
F
is related
to
A
and
B
we write
F(t;
A,
B)
instead of
F(t).
In this list we use
eA
instead of exp(A).
No.
1-No.
5
and No.
8
are main formulas.
No.
6
and No.
7
will be
used in the proof
of
No.
3,
p
in the last column is
p
of
O(P)
in
(4).
F(t;
A,
B)
P
etAetB
2
etA/2etBetA/2
3
I)
4
+&
d"
t
e-
&A
-413
eJt~ eJt~
1
etAI2etBetAl2
+
it3[A/2
t
B,
[A/2, B]]
-t3[A/2
+
B,
(42, B]]
etA/2etBetA12
1
tA/2 ,t
B
-1
A/2 -2t
B
e-
t
A/2
,t
B
,t A/2
-
3/2
e,Il~eJt~e-Jt~
e-d7~
I
\/~A~J~B~-J~A~-J~B
2
4
,tA/2,tBe-tA/2,-2tBe-tA/2etBetA/2
4
4
extA
eutB,ytAevtBeytAeutBextA
216
S.
T.
Kuroda
In
No.
8
u,
v,
x,
y
are given as
=
1.351***,
=
1
-
221
=
-1.702**.,
No.
1
is
the classical Lie formula and
No.
2
is its symmetrized
form used in
[4]
and
[2].
No.
4
is also well-known
([8],
p.
99;
see
also
[l]
[3]
for more recent developments).
No.
3
is
a
fourth order formula. Its feature is that the unit incre-
ment
F
contains only operators which remain bounded even when
A and
B
are unbounded. (There are fourth order formulas with
F
containing
A,
B
outside exponential factors. One example is
(2.10)
of
[4]
which has terms with double commutators sandwiched by expo-
nential factors.
Or,
even the Taylor expansion up to the third order
term may be regarded as such
a
formula.)
A
few words on the proof
of
No.
3
and
No.
5.
Our only tool is
brutal computations
of
Taylor coefficients. First, we try
to
improve
the order of error of
No.
2
by
1
and obtain
No.
6.
On the other
hand, as suggested by
(2.2.10)
of
[S],
which is
a
third order formula,
we obtain
No.
7.
No.
G
and
No.
7
contain the same double commu-
tator. They cancel each other to give No.
3.
The remainder
of
the
approximation
No.
4
is computed as -2-'t3/*[A
+
B,
[A,
B]]
+
O(t2).
We can replace
A,
B
by
-A,
-B
without changing
[A,
B].
Adding
these two formulas
we
obtain
No.
5.
When A and
B
are both skew-adjoint,
F
in
No.
1
and
No.
2
are unitary.
No.
3
does not have this advantage. Recently we have
found an order
4
formula
in
which
F
is
a
product of seven exponential
factors,
so
that
it
is unitary in skew-adjoint case. That is
No.
8.
The
proof of
No.
8
requires systematic computation of Taylor coefficients.
Remark
1.
The
Baker-Campbell-Hausdroff
formula expresses
exp(tA1) ".exp(tA,)
as
exp(C(t)) where the coefficients of the ex-
pansion of
C(t)
in
t
involves multi-commutators
of
Ak. The Zassen-
haus formula and its generalization (cf.
[7])
is
a
kind of product for-
mula, but again mu1 ti-commu tators appear
in
exponential functions.
For
our application it is important that only scalar multiples of A
Topics in the Spectral Methods in Numerical Computation
217
or
B
appear in each exponential factor. It is true that, say,
No.
2
is easily derived from the
B-C-H
formula, and possibly others, too.
We found, however, that
a
simple minded manipulation
of
Taylor
coefficients will be less
(or
at most equally) complicated
for
a
quick
derivation
of
higher order formulas.
A
more systematic analysis
of
these formulas with estimates will
be published elsewhere.
2.3
An Application
Formula
No.
5
may be used to solve numerically
a
Schrodinger oper-
ator with variable higher order coefficients by the method mentioned
in Section
1.
In this subsection we pretend that formulas like
No.
3
and
No.
5
remain valid also for unbounded operators. In fact, un-
der suitable assumptions on the smoothness of the coefficients, these
formulas are valid
if
O(tp)
is
interpreted with respect
to
a
suitable
norm.
Assuming for simplicity that the second order terms have con-
stant coefficients, we consider the operator
n
H
=
C(-i&
+
bk(z))2
+
q(z)
k=l
n
=
-A
-
C(2ibk(X)ak
+
iakbk
-
bi)+q(z)
(6)
k=
1
acting in
L2(Rn).
Here,
ak
=
&
and
bk,
q
are real functions. We
DUt
We now apply
No.
5.
Using the notation
~(t;
A,
B)
=
efAetEe-fAe-tB
218
S.
T.
Kuroda
and noting that repetitions of
No.
1
also give rise
to
an error
of
order
O(tz),
we obtain
n
k=I
+F(-G;
a;,
Bk)}
+
O(t2).
The result of
a
numerical test of this formula for
n
=
2
is promising.
The details will be left to future research.
Even when second order coefficients are variable,
a
similar
for-
mula can be derived, but it becomes rather complicated. We have
not yet tested the feasibility of such
a
formula in the numerical com-
putation.
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R. Chernoff,
Universally commutable operators are scalars,
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A. Fleck, Jr., and A Steiger,
Solutions of the
SchrGdinger equation
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Comput. Phys.
47
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Goldstein,
A
Lie product formula for one parameter
groups
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186
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R.
G.
Storer,
A
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T.
Kuroda and Toshio Suzuki,
A time-dependent method for
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Appl. Math.
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231-253.
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Topics in the Spectral Methods in Numerical Computation
219
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