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James L. Bostick

Grants Officer

National Science Foundation

Washington, D. C. 0550

Technical M atters

I

G E O R G I A I N S T IT U T E O F T E C H N O L O G Y

OFFICE

OF CONTRACT AD MI NI STRATION

D AanD

SPONSORED PROJECT INITIATION

Date:

May 11,1976

Project Title:

"Applications of Nonstandard Analysis to Mathematical Physics"

Project No:

G-37-603

Sloan

Sponsor:

National Science Foundation

Agreement Period:

rom

/1/76

ntil

2/31/77

Project Director:

*12 month budget period plus 6 months for submission of required reports, etc.

Type Agreement:

Grant No. MCS76-07543

Amount:

$5,800

NS

1,803

GIT

( 3 - 3 7 - 3 1 3 )

$7,603

M

L

Reports

Required:

Annual

Letter Technical, Final Report

Sponsor C ontact Person (s):

Contractual M atters

(thru OCA )

Dr. William G. Rosen

Program Director

Mbdern Analysis and Probability

National Science Foundation

Washin gton , D. C. 20550

Defense Prio r ity Rat ing : NONE

Assigned to

Matheratics

COPIES TO:

Project Director

Division Chief EES)

School/Laboratory Director

Dean/DirectorEES

Accounting Office

Procurement Office

Security Coordinator OCA)

,11;orts Coordinator OCA)

Library, Technical Reports Section

Office of Computing Services

Director, Physical Plant

EES Information Office

Project File OCA)

Project Code GTRI)

Other

(School /Laboratory)

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Applications

of Nonstandard AnalysisMathematical Physics

GEORGIA INSTITUTE OF TECHNOLOGY

OFFICE OF CONTRACT ADMINISTRATION

S P O N S O R E D P R O J E C T T E R M I N A T IO N

Date:

V

Project Title:

Project No: G-37-603

Project Director: Dr. Alan D. Sloan

Sponsor:

National Science

Foundation

Effective Term ination Date:

2/31/79

12/31/79

Grant/Contract Closeout Actions Rem aining:

Final Invoice and Closing Docum ents

_x Final Fiscal Report via FCTR

_x Final Report of Inventions

Govt. Prop erty Inventory Related Certificate

Classified Material Certificate

Other

Assigned to: Mathematics

COPIES TO:

Project Director

Division Chief EES)

School/Laboratory Director

Dean/DirectorEES

Accounting Office

Procurement Office

Security Coordinator OCA)

/

eports Coordinator OCA)

CA-

4

1/79)

(

Scho

ol/

UN:34MAX

Library, Technical Reports Section

EES Information Office

Project File OCA)

Project Code GTRI)

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GEORGIA INSTITUTE OF TECHNOLOGY

ATLANTA GEORGIA 30332

MA1HEMATCS

J u n e 2 8 , 1 9 7 7

J a m e s L . B o s t i c k

G r a n t s O f f i c e s

N a t i o n a l S c i e n c e F o u n d a t i o n

Washington, D.C. 20550

D e a r M r . B o s t i c k :

T h i s i s t h e a nnu a l _tec l a ni c , a l__let t e r fo r N S F g ra n t

n u m b e r M C S 7 6 - 0 7 5 4 3 a w a r d e d t o G e o r g i a In s t i t u t e o f T e c h n o l o g y

wi t h D r . A l a n D . S l o a n a s p r o je c t d i r e c t o r .

T h e m a i n r e s u l t s f o t h e r e s e a r c h c o n d u c t e d f r o m 7 / 1 /7 6

t o 6 / 3 0 /7 7 u n d e r t h e a b o v e g r a n t c a n b e f o u n d i n t w o p a p e r s :

1.

" A n A p p l i c a t i o n o f t h e N o n s t a n d a r d T r o t t e r P r o d u c t

F o r m u l a " s u b m i t t e d t o t h e J o u r n a l o f M a t h e m a t i c a l P h y s i c s ; a n d

2.

" A N o t e o n Exp o n e n t i a l s o f D i s t r i b u t i o n s , " i n

p r e p a r a t i o n .

P r e p r i n t s a r e a v a i l a b l e o n r e q u e s t a n d r e p r i n t s w i l l

b e s e n t w h e n a v a i l a b l e .

T h e p r o j e c t d i r e c t o r w a s i n v i t e d t o t h e I ow a S y m p o s i u m

o n I n f i n i t e s i m a l s f r o m 6 /1 / 7 7 t o 6 /5/ 7 7 a n d d e l i v e r e d a n

h o u r a d d r e s s e n t i t l e d " N o n s t a n d a r d P e r t u r b a t i o n s . "

S i n c e r e l y .

A l a n D . S l o a n

A D S : m c

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GEORGIA INSTITUTE OF TECHNOLOGY

ATLANTA. GEORGIA 30332

J u n e 1 6 , 1 9 7 8

Mr. James L. Bostick

Grants Offices

National Science Foundation

Washington, D.C.

0 5 5 0

Dear Mr. Bostick:

This is the annual technical letter for NSF grant

number MCS 76-07543 awarded to Georgia Institute of

Technology with Dr. Alan D. Sloan as project director.

1.

Dr. Sloan's paper An Application of the Non-

standard Trotter Product Formula appeared in the Journal

of Mathematical Physics. Two reprints are enclosed.

2.

Dr. Sloan completed work on a paper entitled

A Note on Exponentials of Distributions which was

accepted for publication in the Pacific Journal of Mathe-

matics. Preprints are available on request.

3.

Dr. Sloan received an invitation to an Oberwolfach

conference July 2-July 8 in Germany. He prepared a lecture

on nonstandard aspects of evolution equations in Hilbert

s p a c e .

4. Dr. Sloan investigated Sobolev inequalities

connected with nonstandard distribution theory.

S i n c e r e l y ,

A l a n D . S l o a n

A D S : m c

Enclosure

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An application of the nonstandard Trotter product formula

Alan Sloan

School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332

(Received 21 March 1977)

The nonstandard Trotter product formula is used to extend the Feynman integral interpretation of

solutions to the Schrbdinger equation in the presence of a highly singular potential.

We seek an answer to the question of when a quantum

mechanical Hamiltonian

H

determines a system which

evolves according to a Feynman path integral.

I Speci-

fically, we want to know when there is a potential V so

that, upon setting

k =1,

and fixing

t >

0

exp(-

itH)u=limT,,, v

(u),

u in L2 (IR

3 ),

1)

where

T"

v (u) is defined as the integral

(T, v u)(x

0

)=-_- c, f

ro, exp[iS(x o

, x,

x

0

;n,t, V)]

xu(x

n

)dx dx,

and

c

o

= (27rit/rnn)-3012,

M>

X

X 7 7 .1)

,

and

X X

S(X0,

n

;

n,

t, V) =

-

1)

2

J .1

c,

112(

2

(t/n)

(x j)- .

An explic icomputation' shows that

T

0

,

v [exp(-

itHo

/n)

exp(-

itV/n)r,

where

H

= - A/

/ 1 2

on

L

2

(

1

R

3

)

5)

for

V

any real measurable function. Consequently,

Nelson 2

was able to employ the Trotter product form-

ula' to verify (1) in case

H

is the closure of the operator

sum,

H

0

+ V, and

V

is chosen so that

H

0

+ V

is essential-

ly self-adjoint. Nelson

2

and Faris .

' extended this type

of formula to more singular potentials by including an

additional limiting operation after analytic continuation

in the mass parameter

m. Although there has been

progress in extending Trotter's formula for exp[-

t(H o

+ V)],

5-1 when V is singular relative to

I/

, analogous

results are not known for the unitary groups exp[it(H

o

+ V)]. Through the use of elementary nonstandard tech-

niques we shall find a formula of the type (1) valid for

singular potentials.

For details on nonstandard analysis and notation we

refer the reader to Refs. 4 and 8-10. Here wd adopt

the convention that Xc:

*X for any set

X. If X is a topo-

logical space, a in

X, b in

*X, we write

a= b

if and only

if

b is in the monad of

a.

If 1 we say a

is the stand-

ard part of b

and that a and

b

are infinitely close. We

use the Euclidean topology on IR" and the norm topology

on L

2

(1R

3 )

a)

Supported in part by NSF MCS 76-97543.

Suppose

I/

is a self-adjoint operator on L 2 (IR

3

) which

can be approximated in the generalized strong sense by

bounded perturbations of H

o

; i.e. , there is a sequence

V

K

of bounded self-adjoint operators on L

2 (IR

3 ) such that

the resolvents of H

o +

VK converge strongly to those of

H. Then, (Ref. 11, 502), providing {H

o +

VK} is uniform-

ly bounded below, it follows that exp(- itH)

is the strong

limit of

exp[it(H + V

K

)]. Thus, for all K

in *IN- IN,

where N=-{1, 2, - }, and for all u in L

2

we have

exp(-

itH)uz exp[-

it (H + V u.

6)

By the Trotter product formula and the transfer prin-

ciple, for each

K

in *IN

exp[-

t(H

o +

VK)]=S-lina(exp(

itH

o

/n) exp(-

itV

K /n)]".

n in *111

7)

Combining (6) and (7) we find

exp(- itH)u=lim [exp(-

itH

o

/n) exp(- itV K /n)]".

8)

n in *IN

For all finite n and

K, (4) shows that

T

, v lc =[exp(-

itHo /n) exp(-

it

V

K /n)] ",

9)

so by the transfer principle (9) holds for all

n, K

in

*IN. Thus, for each fixed infinite

K,

(8) yields

exp(-

itH)u- lim T 0 v K (u). 10)

n n;

For each

n, K

in IN, (2) shows that

T

n

, vi c (u)

is given

by an action integral. The transfer principle then im-

plies that for all n,

K

in *N formulas (2) and (3) remain

valid with

V K replacing V. Consequently, we may con-

clude that

exp(-

itH)u(

) 0 lim c o f

xp[iS(.,x,x,pi,tV,)]

n in *IC

xu(x

)dx dx

n .

Lemma:

Let

X be a separable normed linear space.

Let f: X -*X

be bounded linear operators which are

uniformly bounded in the sense that there is an

M

in IN

such that

I

f II

for all

in *IN U {0}. If fo -f

point-

wise on

X a;1;2 _

N

n

*N, there is an /V in *IN such that

n

>N

impliesf

o (x)'-

f

(x) for all x in

X .

Proof:

Let C be a countable dense subset of X.

Let

X be a positive infinitesimal. Choose

N0

in *IN so that n

II fn(c) - fo(c)11 0 in R. There is a

c

in C

so that

(2 )

(3 )

(4 )

2495

ournal of Mathematical Physics, Vol. 18, No. 12, D ecember 1977

opyright 1977 Am erican Institute of Physics

495

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IIcxII < 5. Then for

n>N

II fn(x) fo(x)II

II fn(x) fn(c)II n(c) fo(c)II +.11 fo(c) fo(x)II

MIIx

cll +X+Mile xil

N. . E. D.

Combining this lemma with the previous conclusion,

noting that exp( itH) and the

T,,,v

ic

are uniformly

bounded by 1, yields the following:

Theorem:

If

VK

are bounded self-adjoint operators

such that Ho +

VK

converge to H in the generalized strong

sense as

n

00 in IN, then for each fixed positive infinite

integer

K there is an

N

in *IN such that for all n> N and

for all u in L

2

exp(itH)u()

- -'' cn "R

oo exp[iS(

,x, x;n, t, V K

)]

xu(x

n

)dxdx,

11)

in *L2

.

To compare the types of potentials covered by form-

ulas (1) and (11) we note that

2

(1) holds if

V

is in

LP

p =2,

whereas (11) holds whenever H is the gen-

eralized strong limit of bounded self-adjoint perturba-

tions of Ho

. Examples of such

H'

s can be found by de-

fining

H to be the form sum of H

o and V when either

(a ) V

is in

L

5

+ L

, p >1;

(b )

V> 0 is locally in L i outside a closed set of mea-

sure zero;

(c ) V

is a delta function distribution concentrated on

the surface of a compact C

1

hypersurface in Ie.

For the necessary approximation theorem in case (b)

see

Ref. 12 where the VK

's are defined by truncation.

In cases (a) and (c) see Ref. 13 where the

V K

's are giv-

en by regularization o f

V .

See Ref. 13 for additional

examples

Similar techniques apply in dimensions o ther than 3

and for other H o 's.

1

B. Feynman and A. Hibbs,

Quantum Mechanics and Path

Integrals

(McGraw-Hill, New York, 1965),

2 E. Nelson, "Feynman Integrals and the SchrOdinger Equa-

tion," J. Math. Phys.

5,

332-43 (1964).

3

H. Trotter, "Approximation of Semi-groups of Operators,"

Pacific

J.

Math. 8,

887-919 (1958).

4

M. Davis,

Applied Nonstandard Analysis

(Wiley-Interscienc

New York, 1977).

5

P, Chernoff, "Product Formulas, Nonlinear Semigroups and

Addition of U nbounded Operators," Mem. A m, Math. Soc.

140, 1-121 (1974).

6

W. Faris, "The Product Formula for Semi-groups defined by

Friedrics Extensions," Pacific J. Math, 21, 47-70 (1967).

7

T. Kato, "Trotter's Product Formula for an Arbitrary Pair

of Selfadjoint Contraction Semigroups," Berkeley preprint.

6

A. Robinson,

Nonstandard Analysis

(North-Holland, Amster

dam, 1974).

3

A. Robinson and A. Lightstone,

Non-Archimedian Fields

and Asymptotic Expansions

(North-Holland, Amsterdam,

1975).

'

0

K. Stroyan and W. Luxemburg, Introduction to the Theory

of Infinitesimals (Academic, N ew York, 1976),

11 T, Kato,

Perturbation Theory for Linear

Operators

(Springer, Berlin, 1966).

12W. Faris,

Self

d joint Operators

(Springer, Berlin, 1975).

13

1. Herbst and A. Sloan, "Perturbation of Translation In-

variant Positivity Preserving Semigroups on L

2

(EN)," to

appear in Trans. Am. Math, Soc.

2496

. Math. Phys., Vol. 18, No. 12, December 1977

lan Sloan 49

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EY-37-6

3

GEORGIA INSTITUTE OF TECHNOLOGY

ATLANTA GEORGIA 30332

MATHEMATICS

J u n e 1 8 , 1 9 7 9

Mr. James L. Bostick

Grants Offices

National Science Foundation

Washington, D. C. 20550

Dear Mr. Bostick:

This is the annual technical letter for NSF grant

MCS 76-07543 awarded to Georgia Institute of Technology

with Dr. Alan D. Sloan as project director. For the

period 7/1/78 to 6/30/7?

1 . Dr. Sloan delivered an invited talk at Oberwolfach,

Germany, July 2-July 8 on nonstandard aspects of evolution

e q u a t i o n s ;

2 . Dr. Sloan attended the International Congress of

Mathematicians in Helsinki in August travelling on a Mathe-

matics Research Council Travel Grant and discussed his work

on nonstandard analysis;

3. Dr. Sloan's paper A Note on Exponentials of Dis-

tributions" appeared in the Pacific Journal of Mathematics,

Vol. 79, No. 1;

4. Dr. Sloan prepared and submitted the manuscript

Strong Convergence of Schrodinger Propagators for publi-

c a t i o n ;

5.

Dr. Sloan continued work on nonstandard evolution

e q u a t i o n s .

6. Dr. Sloan initiated investigations into explicit

solutions of partial differential equations using non-

s t a n d a r d t e c h n i q u e s .

S i n c e r e l y ,

Alan Sloan

c m

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6-3 7-603

PACIFIC JOURNAL OF MATHEMATICS

Vol. 79, No. 1, 1978

A NOTE ON EXPONENTIALS OF DISTRIBUTIONS

ALAN SLOAN

Nonstandard analysis is used to discuss nonlinear func-

tions of distributions. An application is given to obtain

a

generalized Trotter product formula. The strong resolvent

topology is discussed from a nonstandard point of view.

Enlarging the real number system to include infinite and infini-

tesimal quantities enabled Laugwitz [5] to view the delta function

distribution as a point function. Independently Robinson [7] demon-

strated that distributions could be viewed as generalized polynomials.

Luxemburg [6] presented an alternate picture of distributions as

generalized functions within the context of Robinson's theory of

nonstandard analysis and it is a special case of this point of view

that we take here. Once one accepts distributions as generalized

functions, the composition map provides a natural method of defin-

ing nonlinear functions of distributions. Unfortunately, the diffi-

culties in the standard attempt to define a function,

f,

of a

distribution, r, (by first writing r as a limit, in some appropriate

sense, of smooth standard functions r

, next composing these

approximations with

f, and finally taking the limit of

f o

r ) still

remain in the nonstandard theory. In particular, this procedure

may not lead to a distribution and even when it does the distribu-

tion obtained may depend on the representation of the original

distribution as a generalized function. Thus, at present, there is

no comprehensive theory of nonlinear functions of distributions.

The development of such a theory may well proceed along

alternate tracks depending on the applications intended. One use

for distributions occurs in the study of perturbations of selfadjoint

operators in Hilbert space. In some perturbation problems the

pathology of obtaining compositions which do not represent distri-

butions may be avoided by considering only bounded functions of

distributions. For example, in the Trotter product formula one

works with bounded exponentials. Even though, in this case, the

exponentials of distributions may be identified with standard distri-

butions it is the conclusion of this paper that such an identification

should not be made. It

is the author's view that functions of

distributions should be regarded as generalized functions but not

distributions. The utility of this view is illustrated by a non-

standard version of the Trotter product formula.

For an introduction to nonstandard analysis and its relation to

distributions, see [10]. In [11] a square root for the delta function

2 0 7

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208

ALAN SLOAN

was defined nonstandardly.

Let

*X

be a 3-incomplete ultrapower of a structure

X

contain-

ing the real numbers,

R.

We shall use the convention that Ac*

for all sets

A

considered. If

T

is a topological space with u in

T

and v in

* T then we shall write

u

v if and only if v is in

fl

Y - 1 :

21

is an open set containing

u}.

In this case we say u is

the standard part of v, relative to the given topology and that v

is near standard. We write u = st(v). We always assume that

RP has the Euclidean topology and all Hilbert spaces have the norm

topology.

If f and g

are Lebesgue measurable on

*S

for some standard

set

S

which is measurable, we shall write

for

gdx, pro-

s

viding the integral exists. If, in fact, f

and

g

are standard func-

tions then

gdx,

by the transfer principle.

Let W be an open subset of

R.

Let

B.

be an increasing

sequence of relatively compact subsets of W whose union is W and

whose closures are in W. If _gr

(B.) is given the induc-

tive limit topology then the elements of

he dual of _0", are

the distributions on W. In order to define a function of a distri-

bution we first regularize the distribution.

Let

p

be a Cr (RP) function satisfying 0 5

p < 1, p(x) ----

1 if

II S 1 and

p(x) = 0 if II

x I I

. Set

7,,(x)

=

P(npx)(S p(n

o

t)dt)

where n o

is chosen so that

distance

(B,13:+1) >

2/n0

Choose

R .

to be a

Cr (R')

function satisfying

R . =

1 on B_ 1

, 0 S

13. < 1

and support

(SO

c

If

x

is in B. and

7(x

y) # 0 then Itx y II < 2/n

o

so that y

is in

f

f

7 (x ) denotes the function y

y) then

.) is in

C:'(B,,,,)

if x e B..

For any distribution T

on W define

a function

T. by

T.(x) = f3.(x)T(7,.(x

))

for n in

N.

Then

T. is in C,

-

(RP) and support

(T) c B.

By the

transfer principle T is in

*C:* (R')

and support

(T.) c *B for n in

*N.

Further, since

T T

in _0'

as n

c o

in N

it follows that

(1)

(f) T(f)

for all n in *N N

and f

in .0 where

T.(f) = .

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A NOTE ON EXPONENTIALS OF DISTRIBU TIONS

0 9

The

T , , ,

are called the regularizations of T.

DEFINITION 1. Fix n infinite in *N. Let

g

be a complex valued

function of a complex variable defined on the range of

T.

Then

we define

g(T)

to be

g o T.

Thus,

g(T)

is a generalized function in the sense that it is an

internal *C-valued function defined on * W. Since

T A

(x) = 0 if x is

not in we find that g(T)(x) = g(0) if x

is not in *B..

Certain generalized functions h: * j 1.'

C define distributions r

according to the rule

( 2 ) (f) =

s t ()

for

f

in 1.. Here, st (x,) is the standard part of X,, if X, is a finite

hypercomplex number, and undefined otherwise. Thus not every

function of a distribution will determine a distribution according

to (2). If

g is a bounded measurable function then g(T)

always

defines a distribution by (2).

If T is given by a locally unction F on W, (i.e., if T(f)

f) for all f in _0) then g of T is

naturally defined within

standard distribution theory as the distribution

f pro-

viding the composition is defined. A simple example suffices to

show that the distribution determined by

g(T)

need not be g o F.

EXAMPLE 2. Let

E

be the complement of the Cantor set, in

[0, 1], of measure 1/2 obtained by repeated deletions of 2n

- 1

open

intervals of length 2 - 2 ", n = 1, 2, from [0, 1]. Let

F

be the

characteristic function of

E

and

define

T(f) = .

Let W =

(-2, 2) in R

.

Choosing B, so that [0, 1] c B, gives

TT(x) = T(7.(x ))

F, 7.(x )>

V

(y)7.(x y)dy >

0 for all n in N

and all

x in [0, 1]. By the transfer principle

7 ,(x) > 0

for all n in

*N

and x

in

*[0, 1]. Let

g(x) = 1 if

x

0 and g(0) = 0. Then

g(T)(x) =

1 if

x is

in 10, 1] while

g o F(x) =

1 if

x is in

E and

g o F(x) =

0 if x is not in

E. Choosing

f

in Cr( 2, 2) with

f = 1

on [0,

1]

shows

g(T)(f) =

= 1 while

= Lg(F(x))f(x)dx

=

E

g(F(x))f(x)dx

= Lf(x)dx =

measure (E) = 1/2 .

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El0

ALAN SLOAN

Thaw

g F(f) =

g(T), f> = g(T)(f) .

The necessity of considering only bounded "smooth" g's is now

demonstrated, if one is to obtain a generalization of composition of

functions. In Lemma 5 we obtain such a generalization.

LEMMA 3.

Let f be in * Li(K, dx) where K is a subset of R.

Let iIf II 1

0.

Then M = {x in * f(x) r 0)

is contained in an

internal measurable subset P of *K and ?measure (P)

0.

Proof.

Let A m

= {x in

*K; I f (x)I >

1/m). Then

. , A..

Since

>

x >

meas (A

m

)/m ,

m

we find that meas (A

m

) P,$ 0 for each m

in N. By definition

A

i

c A,c

A,

.

Extend {A m

} to an

internal sequence. Let

I = {m

in *N: 2

m implies meas

(A

k

)

< 1/k and

A

k _, c A

k ).

I

is internal and

contains

N. Since N

is external there is an w in

(*N N)

n

I.

Choose P =

A,,. Then meas

(P)

0 and U. A,,, c

P.

LEMMA 4.

Let F be a locally

L'

function on W. Let T be

the distribution T(f) = . For every compact

K

in W

and

every infinite n in *N

IIT

Fll,

where

I I I L

is the L' norm on *K.

Proof.

Choose an open set

Q

with compact closure such that

KcQcQc W.

For n in N sufficiently large,

R. = 1

on Q, and

x in K

implies

support

(7(x

-))c Q. - Consequently, for n in

N

sufficiently large

Tm

(x)

=

( f m

*F)(x). But

7.*FFn

Li(K)

so for n infinite IIF

7.*F

I I l

0 while by definition T. =

7.*F

so }IF T.II, F .& 0.

LEMMA 5. Let

g be a bounded uniformly continuous function

defined on a closed interval containing 0 and the range of a locally

L' function F.

Let T

be the distribution defined

by F: T(f) =

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A NOTE ON EXI'ONENTIALS OF DISTRIBUTIONS

11

Proof.

We first must show

g(T)

is defined. There is a closed

interval on which

g

is defined and which contains the range of

F.

Since 0 < f3 < 1 and

7dx=1 it follows that the range of

R,(77,F)-= -

7

1

is in the closed interval and so the range of T is in the domain

of g for all n

in

N.

By transfer

g(T)

is defined for all n in

*N.

Fix n infinite and define

g(T) = g(T).

We are to prove that for every f

in 2'

F, f> .

Let K

be a compact set containing the support of

f.

Let

A be

an internal subset of

*K containing { x

in

*K: T(x) F(x)} and

having measure X

= 0. Since

g

is uniformly continuous,

x

in *K

A

implies

g(T(x)) g(F(x)). Let M in N be a bound for g. Let

E

be a positive real number. Let

in

be the Lebesgue measure of

K.

Then

I 1

A

g(T) g(F)I f

K A

dx + I g(T) g(F)I

< 2.111 ;

f

lcoX

lif

2c Ilf

1 1 - n t

Since

E

is arbitrary, this completes the proof.

f dx

EXAMPLE 6. Let

o

be the delta function

o(f)=f(0),

f

n

_"(11

).

Choose n

o

= n and

B = (n,

n). Then,

3(x)

= 7(x). For all

n

in

N,

support (3)c(-2/n, 2/n) and 0 3, < 0.9. Thus for

t > 0 in R,

0 < C

u * < 1 and if Ix I > 2/n then

6

-

""(x) = 1. By

the transfer

principle choosing n in

*N N

and setting

g(T) = g(T)

we find

0

1, f>, f

in .0 so

1 is the distribution determined by

c .

Next we wish to discuss exponentials of distributions in con-

nection with the Trotter product formula. This requires

a

discussion

of the topology of strong resolvent convergence, which we now

give. More details on this topology are to be found in the appendix.

DEFINITION. Let

SA(K)

be the set of all selfadjoint operators

on a Hilbert space

K.

A neighborhood basis of a selfadjoint operator

S

in the topology of strong resolvent convergence is given by the

collection

{G(S, E, 6): E is a

finite subset of

K, 0 < e < 00)

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211

LAN SLOAN

where

G(S, E, e)

T in

SA(K):

1 1 ( T

+ i)

-'h (S +

i)

- 1 101 n

via the spectral theorem. Let H. + V denote the standard self-

adjoint operator defined as the form sum of

H

and V in [1]. Then,

since (H, + + i)'

converges strongly to

(H, + V + i)', [1], we

conclude

+ V R d H

o

+

for all positive infinite integers, n.

See [1] and [4] for concrete

examples.

We now proceed to discuss

the Trotter product formula for

given form sums. A discussion of form sums may be found in

[1].

Here H. is a selfadjoint operator on a

Hilbert space

K and

is the selfadjoint operator on K

given

as the form sum of H,

and

V

in the following three cases:

4

Case

1. V is

a selfadjoint operator and the operator sum

H,+

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A NOTE ON EXPONENTIALS OF DISTRIBUTIONS 15

V is essentially selfadjoint.

Case

2.

H

and V

are bounded below selfadjoint operators

with dense form domain intersection; and

Case

3.

V

is a small Hermitian form perturbation of

H

.

In Case 1, H

is just the closure of H

+ V

and the Trotter

product formula holds [12]:

( 6 )

= strong limit (e"o'"e''''''')

uniformly for t in compact subsets of [0, c-.), where x =

f

H V

are bounded below then (6) also holds with X = 1.

In Case 2 Kato has shown [4] that (6) holds for x = 1 but

in general (6) is not known for X ---

i

and in fact there has been

essentially no progress in extending (6) for X = i beyond Case 1.

In Case 3, not only is Trotter's product formula unknown for

= i and X = 1 but in general it makes no sense. For

example, if

H = d 2 /dx 2

and

V = 8,

the Dirac delta function on

Lz(RI),

then

e'

7

"

is undefined in any standard sense. In some

examples of Case 3 such as

H

= d

2 /de

and V(x)

=

ez cos (ex)

the

formula (6) makes sense but its validity is unknown for x = 1.

Elementary nonstandard analysis provides a convenient frame-

work for obtaining a type of product formula in the cases discussed

above.

THEOREM 10.

A Product Formula:

Let H

, H be in SA(K) and V in *SA(K) where K is a separa-

ble Hilbert space. Suppose the closure [H

+ V] of the operator

sum of H

and V is in *SA(K). Also assume

Hd

[H + V].

Then there is an

N

in *N such that for all n >

N, h

in

K and

nonnegative finite t in *R:

(7 )

1tNh 2 2 1 - 1 0/se

it isyth

where X =

If also, H, H

, V > c

for some

c

in R

then (7) holds with X= 1.

Proof.

By Lemma 8

(

8

)

ltH h F = 4 , enuro+vjh

for all h in K and all finite t in *R.

Fix a to be a positive infinitesimal. Fix

h in K and M

in

N.

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216

LAN SLOAN

It follows from the transferred Trotter product formula of Case 1

that there is a

Q(h, M)

in

*N,

(we suppress the a dependence), such

that

q > Q(h, M)

implies

(

9

)

l e

:itplo+V]h e

2fIlc,

, q

e;0 11q)9h I

Q(h)

implies

(10)

e itIlh e

1t110

,

g

e)tVTh

for all finite

t in 10,

Let

C

be a countable basis for

K . Let

Q

be an upper bound

for

{Q(h): h

is in C},

which again exists by [8]. Let

q > Q , t in

be finite and

h

in

K

be arbitrary. Given

1 3 in (0, hoose

k in C

so that I

h k II

0 in R.

Choose b >

0 in

R so that

e -"dt < el211hIl

Let

I

be the collection of z's in *[0,

hich satisfy

t,

8

in [0, b], It sl < z

imply

(e

ur

ec/(2(1 C

a.))

I

is internal. The semigroup property and (13) show that

I

con-

tains all positive infinitesimals. Consequently, there is a

a in

I n

(0, 00).

Let a

k

= k for k = 0, 1, , n, where n in N

is chosen so that

n8

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T E

IXPONENTIALS OF DISTRIBUTIONS

2 3

+

dt

L c .

0 (e

-

sc /2(1

))dt +

2

E

By Lemma 21,

e -

"e"Thdt

0

is near standard. From the Laplace transform formulas

(T --T-

t

0

we conclude that

(T ic)-'h

is near standard. We have thus veri-

fied (a) of Theorem 23. We next verify (b).

Let s > 0, s a 0. Choose ,63 >

0 to be infinite in such a way

that e

. Then

+ ie Trih hll

e+"'Thdt h

e " `T h hH d t

0

+ 2HI

t

a0

since 0 s in J, where

n=1

t t,

T

n

( A , t , $ ) =

A(t

1

)A(t

n

) d t

n

dt

1

S S

and by U (s,t) for s > t in J, gives a unitary propagator. Here

the integrals are strong while the sum converges in operator

norm. x(t) = U(t,$)x s

gives a norm differentiable function which

is a solution to equation (1) for all x s in

H

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I

7

T h e m a i n i n g r e d i e n t i n t h e p r o o f o f t h e o r e m 3 i s a n a p p l i -

c a t i o n o f t h e u n i f o r m b o u n d e d n e s s p r i n c i p l e t o p r o v e

L e m m a 4 [10]:

Let A be a D-generator and K a compact interval

in J. Then sup IA(t) I I < m

-

LEK

L e m m a 5 :

Let A be a bounded linear operator and B an internal

f i n i t e l y b o u n d e d l i n e a r o p e r a t o r . I f g , h E H, g t i h a n d A g , % 'B g

t h e n A h

t i

B h .

P r o o f :

lAh-Bh1 1 s in T, and x in D(H

1/2

)

1A

-1/2

( t

c (t-s) -1) (

1 + -

1 )/c.

)

( u

A

(st)-U

A

( s

'

t))xlf

K

T

( e

m n

n

n

Here A

n

i s d e f i n e d a s i n ( 8 ) .

Proof: Fix s in T and for t > s in T define a

n

(t) = U

A

(s,t)x.

n

Fix positive integers m and n. Define

p(t) = a

n

( t )

-

am

( t )

w(t)=(p(t),A

- 1 (t ) p(t ) )

-

dA

1

(t) = G(t ) .

In general we denote dt y f so that = -iA

n

a

n

+ iAm

a

m

and

PIA

-1

p) + (P.GP) P , A

-1.

-1

Since A = A

n

-

we may write

a n d

d t

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l(P,GP)

-

1 < cw

1 5 )

4

w = (-iA

n

a

n

+ i A

m

a

,

A

1

P ) + ( P , G P ) + ( P , A

-1

(-iA

n

a

n

+ i A

m

a

m

) )

= i ( A n a n ' ( A n

1

- -

) p ) - i ( A

m

a (A

-1

-

m 1

) p) +(p, Gp)

m

+ i (p, (A

-1

-

1

) A

m a m

-

i

(p, (A

n

1

-

) A

n

p)

m

= i (a

n

, P) -

A

n

an

,p)

a

m

, P)

+ I l

i

(

A

m

a

m

, p )

+

(p,Gp)

- i(P,a

n

) +

1

4

11

( p , A

n a n ) + i( p , a m )

-

Il-l

( p , A m a m )

= i(a

n

-a

m

, p ) - i ( p , a

n

-a

m

) + ( p , G p ) +

1

( p , - i A

m

a

m

)

1

- (-iA

n

a

n'

p ) + - (- i A

m

a

m

, p ) -

p , - i A

n

a

n

)

( A

n , p ) 11-

( L m

, P )

-

1

1

-

( P , 1 1 )

1

+

1 7 1

(pa

m

)

T h u s

w = (p , G p ) +

1

2

Re(A

m

, p )

-

12

Re(k i , p )

N e x t o b s e r v e t h a t s i n c e

I

/2

/2 1/2_-1/2 .1

l(P,GP )1 = I (A

, A A

i

I I

A1 /2 G A

1/2I1

IIA

-1/2 1 3 1 1 2

= 11 A

1/2

G A

1/2

11(p,A

- 1

p )

we o b t a i n

= i(pP) - i ( P , P ) + ( P

G P ) -

( 1 4 )

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1 5

Also, for r in

11 ,

1(L

r'

P)-

=

1 /2 ar

(H+1)

1 /2

p ) 1

IlL

r

1 1 _

1

1101

+ 1

1 . ;

r

1 1 _ 1

(

a

n

1 1 + 1

la.11+1

)

so that from (11), (12) and (13) there follows

l (

r

(t),P(t))1 < 2K

2

T

Substituting (15) and (16) into (14) we find

t

I < cw(t) +

K (

n

+

d t

for all t > s in T. Here K is the finite constant 8K

T

.

To finish the proof we observe that w(s) = (p(s),A

- 1

(s)p(s)) =

( a n (s) -am

(s),A 1

(s) p(s ) ) = (x-x , A

- 1

(s ) p ( s ) = 0 .

i n c e w > 0 , ( 1 7 )

i m p l i e s .

K 1

e

c( t - s )

1

0 < w(t) < ( +

- c n in

n m

Q.E.D.

Proof of Theorem 9: Fix a finite s in

*

J and x in D(H

1 /2

)

D e f i n e a

n

as in the proof of the technical lemma and a as in theo-

1

rem 8. Similarly, define B

n

(t) = B(t )(1 + n 1 B(t))

-

so that B is

an internal D-generator with

1 1 B n . (t ) 1 1 < n .

18) .

(1 6 )

(17)

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16

Let b

n

be the solution to the Schrodinger equation (1) with gen-

erator B

n

and initial state x. Since

- 1

1

1

1

n

(t)= A(t)

'\ B(t)

B

n

( t )

n

it follows from bounded stability, theorem 7, that

a

n

(t)

ti

b

n

(t ) ,

d

n E IN, V finite t in J.

19)

Let I

o

= {mE

< M

*

J i lt 0 arbitrary in 7R and x

i n D ( H

1/2

) such that Ilx-yll < E . Then for g in H, using (26)

I ( U

A

y -U

B

y,g)I

< I ( U

A

( y -x) , g ) I +

I ( U

A

-U

B

)x,g)I

1U (x

-

Y)/g)I

2Il

y -

x 1

1 g

1 0, t >

s and x in

D ( H

1 / 2

). Let T

be a compact interval containing t and s. Choose K

T

a s i n ( 1 1 ) ,

(12) and (13). Choose N to be an integer larger than

8K(e

c(t- s)

-1)/2Ec. Let m > N. For any n, let B

m

( n ) b e t h e m

t h

Yoshida approximation to A

n

and C the m

t h

Yoshida approximation

t o

A .

Then by the Technical Lemma and the triangle inequality

I I A

1 / 2 ( t ) ( U

A

(s,t)-U

A (s,t) )x

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2 1

Definition: A Y-generator on J is a function A() from J into

the selfadjoint operators on

H

such that there are positive

numbers E and M satisfying

(i ) A(t) > E + 1 ;

(ii )

t

A(t)+E)

- 1

is strongly differentiable with deri-

vative G(t) ;

(iii)

G is strongly continuous; and

(iv)

II(A(t ) +E ) B(t )I 1 < M .

Then according to a theorem of Yoshida [15, 429] there is a uni-

tary propagator UA

such that for all x in D(H(s)), defining

a(t) = U(t,$)x gives a solution to the Schrodinger equation in

the sense that

d a

(t) = -iA(t)a(t).

d t

The techniques given above lead to the following theorem and

c o r o l l a r y .

Theorem: Let A be a Y-generator with constants E and M. Let B

be an internal Y-generator with constants E and M. If

(A(t)+E)

- 1 :1;(B(t)+E)

- 1 for all finite t in

*

J then

U

A ( t ,

1 ,

B

(t , s ) f o r a l l f i n i t e t , s i n J .

Corollary: Let A,A n

be Y-generators with constants E,M for

n =

Let An

(t) converge to A(t) in the strong resol-

vent topology for each t in J.Then U

A

(t,$) converges strongly

n

UA '

t s ) f o r e a c h t , s i n J .

2. In corollary 10 and the last corollary, the convergence

of progators is uniform in any finite interval of J.

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itrtivigsrws

IV. Perturbations of a Time Independent Generator.

In this section H

0

will be a non-negative selfadjoint oper-

ator on H. If F is a Hermitian form Q(F) will denote the form

domain of F. In particular Q(H ) = D(H

1 /2

) .

0

Definition: A K-perturbation of H

0

is a Hermitian form valued

function, V, defined on J with the properties

(i)

Q (I I

0

)C

Q

(V(t)

) .

(ii ) There are constants 0 < a < 1, 0 < b < co such that

1 one can

treat the problem of switching from one potential to another.

By choosing f(t) = sin wt one obtains periodically varying

p o t e n t i a l s .

One obtains considerable technical advantage in continuing

the study of time dependent singular perturbations of -A by

introducing infinite regularizations and by accepting infini-

tesimal errors. This naturally occurs because U

H +V +E

h a s t h e

0 k

*

cQ

generator H

0

+ V

k

+ E defined as an operator sum with V

k

in C

c

while U

H +V+E

has the generator H

0

+ V + E defined as a form sum.

0

For example, since

vk

e

*

L

Pn

) for all p > 1, the product

formula of Faris [2] is valid and one can express the solution

u of the Schrodinger equation (1) as

r i ,

r a

m-1( A

m

()

u ( t )

i t

t

t

us

j=0

(38)

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2 7

with u(0) = u

s

f o r u

s

i n Q(H

0 ),for m sufficiently large in IN.

S i n c e L

2

( I R

n

i

is separable one m can be chosen so that (38)

holds for all u

s

i n Q(H

0 ) and all finite t (see [13] for details

of this argument). Following Faris, (38) leads to a Feynman

p a t h i n t e g r a l :

*

For each pair of points (x

0 ,xm ) in IR

2 n

we may view

*

n (m-2)

as a space of polygonal paths connecting x

0

and x

m

.

These paths are not standard, but if x

0

and xm

are finite and if

w is any standard path connecting x

0

and x

m

; i.e.

w

: [0,t] + AP and w(0) = st(x

0

),w(t) = st(x

m

) , t h e n t h e r e

are polygonal paths

w

1

k ,

so that w(s) rk, w

I

( s ) f o r a l l s i n [0 , t].

There are many more polygonal paths than standard paths. Since

*

f o r e a c h t , V

k

(t) is a C

c function we let (V

k

(t) )(y ) = V

k

(y,t).

For each polygonal path x = (x 1

,,x

m

_

1

) in

*

I R

n ( m - 2 )

l e t

2

(x.-x.

jt) t

S(x

o

,x,x

m

;m,t,V) =X

j=1

2

k

m m

x. -- .

2(t/m)

Let u(,t)E L

2

(IRP) be the state at time t of the quantum

mechanical system with time dependent Hamiltonian H

0

+ V( t )

with state at time 0 given by u(-,0)E Q(H

o )

h u s

( U

t 0))(u(-,0)) = u(-,t). Define the amplitude of the

H

0

corresponding quantum mechanical particle to go from x

m

at time

0 t o x

o

at time t by

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4 0 1 , 1

1

1 ,M ,

K((x

0

,t),(x

m'

0)) =

4 2 7 4 t90..47t

28

(39)

iS(x,x,x

m '

-m,t,V

k

)

e

(x

m

,O )dx

*

1 R

(m-2 ) -n

2

s A .

-3n/2 ly /4s .

S i n c e e

s c o n v o l u t i o n b y (4 f f i s )

n I R

n

, (38)

i m p l i e s

u(.,t) ti

*

I R

n

K(( . , t ) , (x

m

,0))u(x

n

, O ) d x

m

40)

i n * 2

n L (IR

n

) .

(39) and (40) are a formulation of the Feynman path integral

for particles moving in singular and time dependent potentials.

d

2

Since adding a delta function to -A =-

2 in L

2

(2

3

) is

d y

equivalent to imposing a boundary condition on the maximally

d

2

2

d e f i n e d

it follows that the problem of smoothly

dy

changing boundary conditions in time can be given a Feynman path

i n t e g r a l i n t e r p r e t a ti o n .

We conclude the discussion of this example by noting that,

in general, the solution to equation (1) is shown to exist by

proving certain approximate solutions converge weakly. However,

by accepting infinitesimal errors one can explicitly construct

the propagator. For this purpose we enter the interaction

i t H

0

itH

0

representation defining V(t) = e

k

(t)e

here k remains

an infinite positive integer. Then t V(t) is an internal

D-generator so that

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t

p - 1

t

1

U(t,$) = 1 +

-i) P f

f

. . f

t d t . . . d t

1

P

p=1

s

converges in norm to a unitary propagator and

- i t H

s H

U (t,$)=

e0

5(t,$)e

0

a

29

is almost the propagator for equation

(1 ) i n t h e s e n s e t h a t

U( t , $ ) x s

r

t ; U a (t , $ )x

s

f o r a l l x

s

i n Q(H

0

) and t,s finite. For the standard details,

see [10]. Thus we have defined time ordered exponentials of

certain quadratic form valued functions.

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R e f e r e n c e s

3 0

1 . P. Chernoff, Product Formulas, Nonlinear Semigroups and

Addition of Unbounded Operators, Mem. Am. Math. Soc.

1 4 0 1 - 1 2 1 ( 1 97 4 ) .

2.

W.

Faris, Product Formulas for Perturbations of Linear

Propagators, J. Functional Analysis, 1, 93-108 (1907).

3. I .

Herbst and A. Sloan, Perturbation of Translation Invariant

Positivity Preserving Semigroups on L

2

( a t

') , T r a n s .

Am. Math. Soc., 236, 325-360 (1978).

4. T.

Kato, Integration of the Equation of Evolution in a

Banach Space, J. Math. Soc. Japan, 5, 208-234 (1953).

5.

T.

Kato, Perturbation Theory for Linear Operators, Springer,

B e r l i n , 1 9 6 6 .

6.

T.

Kato, On Linear Differential Equations in Banach Spaces,

Comm. Pure and Applied Math., IX., 479-486 (1956).

7.

T.

Kato, Linear Evolution Equations of Hyperbolic Type, J.

Fac. Sci. Univ. Tokyo, I, 17, 241-258 (1970).

8.

J.

Kisynski, Sur les Operateurs de Green des Problems de

Cauchy Abstraits, Studia Math. XXIII, 285-328 (1964).

9.

E.

Nelson, Internal Set Theory: A

New Approach to Non

-

s t a n -

dard Analysis, Bull. Am. Math. Soc. 83, 1165-1198

1977).

10.

M.

R e e d a n d B. Simon, Methods of Modern Mathematical Physics

II Fourier Analysis, Self-Adjointness, Academic Press,

New York (1975).

1 1.

B . S

imon, Quantum Mechanics for Hamiltonians Defined as

Quadratic Forms, Princeton Un. Press, Princeton, (1971).

1 2.

A.

Sloan, An Application of the Nonstandard Trotter Product

F o r m u l a ,

J. Math. Phys. 18, 2495-2496 (1977).

1 3.

A.

Sloan, A Note on Exponentials of Distributions, to appear,

Pacific

J.

o f M a t h .

14.

K.

Stroyan and W. Luxemburg, Introduction to the Theory of