Feynman integrals for non-smooth and rapidly growing potentials

arXiv:math-ph/0408032v1 20 Aug 2004

Rapidly Growing Potentials

Margarida de Faria

CCM, University of Madeira, P-9000-390 Fun hal mfariauma.pt

Maria João Oliveira Univ. Aberta, P-1269-001 Lisbon;

GFMUL, University of Lisbon, P-1649-003 Lisbon; BiBoS, University of Bielefeld, D-33501 Bielefeld

oliveira ii.f .ul.pt Ludwig Streit

BiBoS, University of Bielefeld, D-33501 Bielefeld; CCM, University of Madeira, P-9000-390 Fun hal

streitphysik.uni-bielefeld.de

Abstra t

TheFeynmanintegralfortheS hrödingerpropagatoris onstru ted asageneralizedfun tionofwhitenoise,foralinearspa eofpotentials spannedbymeasures andLapla e transformsof measures,i.e. lo ally singular aswell as rapidly growing at innity. Remarkably, all these propagators admit aperturbation expansion.

On a mathemati allevel of rigor, the onstru tion of Feynman integrals for quantum me hani al propagators will have to be done for spe i lasses of potentials. In parti ular, the Feynman integrand has been identied as a well-dened generalized fun tion in white noise spa e, e.g. for the following lasses of potentials:

-(signed) nite measures whi h are small at innity [10, 15℄ -Fourier transformsof measures [18℄

-Lapla e transformsof nite measures [13℄.

Potentials in the third spa e are lo ally smooth but may grow rapidly at innity, a prominent example is the Morse potential. On the other hand the rstof these lassesin ludeslo allysingularpotentialssu hastheDira deltafun tion. It isalsoimportantforthe onstru tionofFeynmanintegrals withboundary onditions[2℄. Hen eitwouldbedesirabletoadmitpotentials whi harelinear ombinationsofelementsfromtherstandthirdspa e. The present paper addresses this problem: we show the existen e of Feynman integralssolving the propagator equation for su h potentials.

II. White noise analysis

Inthisse tionwebrieyre allthe on eptsandresultsofwhitenoiseanalysis used throughout this work(see, e.g., [1℄,[4℄, [5℄,[9℄, [11℄,[12℄, [14℄, [16℄ fora detailed explanation).

The starting point of (one-dimensional) white noise analysis is the real Gelfand triple

S(R) ⊂ L

2

(R) ⊂ S

′

(R),

where

L

2

_{:= L}

2

_(R)

is the real Hilbert spa e of all square integrable fun -tions w.r.t. the Lebesgue measure,

S := S(R)

and

S

′

_{:= S}

′

_(R)

are the real S hwartz spa es of test fun tions and tempered distributions, respe tively. In the sequel we denote the norm on

L

2

by

| · |

, the orresponding inner produ t by

(·, ·)

, and the dual pairing between

S

′

and

S

by

h·, ·i

. The dual pairing

h·, ·i

and the inner produ t

(·, ·)

are onne ted by

hf, ξi = (f, ξ), f ∈ L

2

, ξ ∈ S.

Let

B

bethe

σ

-algebrageneratedby the ylindersetson

S

′

. Through the Minlos theorem one may dene the white noise measure spa e

(S

′

_{, B, µ)}

by giving the hara teristi fun tion

C(ξ) :=

Z

S

′

e

ihω,ξi

dµ(ω) = e

−

1

2

|ξ|

2

,

_{ξ ∈ S.}

Within this formalism a version of the (one-dimensional) Wiener Brownian motion isgiven by

B(t) :=

ω, 11

[0,t)

 ,

ω ∈ S

′

,

where

11

A

denotes the indi ator fun tionof a set

A

. Now letus onsider the omplex Hilbert spa e

L

2

_{(µ) := L}

2

_(S

′

_{, B, µ)}

. As this spa e quite often shows to be too small for appli ations, to pro eed further we shall onstru t a Gelfand triple around the spa e

L

2

_(µ)

. More pre isely,rstweshall hoose aspa eofwhite noisetestfun tions ontained in

L

2

_(µ)

and then we work on its larger dual spa e of distributions. In our asewewillusethespa e

(S)

−1

ofgeneralizedwhitenoisefun tionalsor Kon-dratiev distributions and itswell-known subspa e

(S)

′

of Hidadistributions (or generalizedBrownianfun tionals) with orrespondingGelfand triples

(S)

1

⊂ L

2

(µ) ⊂ (S)

−1

and

(S) ⊂ L

2

(µ) ⊂ (S)

′

.

Instead of reprodu ing the expli it onstru tion of

(S)

−1

and

(S)

′

(see, e.g., [1℄, [5℄), in Theorems 1 and 2 below we will dene both spa es by their

T

-transforms. Given a

Φ ∈ (S)

−1

, thereexist

p, q ∈ N

0

su h thatwe an dene for every

ξ ∈ U

p,q

:= {ξ ∈ S : 2

q

|ξ|

2

p

< 1}

the

T

-transform of

Φ

by

T Φ(ξ) := hhΦ, exp(i h·, ξi)ii .

(1)

Here

hh·, ·ii

denotes thedual pairingbetween

(S)

−1

and

(S)

1

whi hisdened asthebilinearextensionoftheinnerprodu ton

L

2

_(µ)

. Inparti ular,forHida distributions

Φ

, denition (1) extends to

ξ ∈ S

. By analyti ontinuation, thedenitionof

T

-transformmaybeextendedtotheunderlying omplexied spa e

S

C

of

S

.

In order to dene the spa es

(S)

−1

and

(S)

′

through their

T

-transforms we need the following two denitions.

Denition 1 A fun tion

F : U → C

isholomorphi onan open set

U ⊂ S

C

if

1. for all

θ

0

∈ U

and any

θ ∈ S

C

the mapping

C

∋ λ 7−→ F (λθ + θ

0

)

is holomorphi on some neighborhood of

0 ∈ C

,

2.

F

is lo ally bounded.

Denition 2 A fun tion

F : S → C

is alled a

U

-fun tional whenever 1. for every

ξ

1

, ξ

2

∈ S

the mapping

R

∋ λ 7−→ F (λξ

1

+ ξ

2

)

has an entire extension to

λ ∈ C

,

2. there exist onstants

K

1

, K

2

> 0

su h that

|F (zξ)| ≤ K

1

exp K

2

|z|

2

kξk

2

,

∀ z ∈ C, ξ ∈ S

for some ontinuous norm

k·k

on

S

.

We are now ready tostate the aforementioned hara terization results. Theorem 1 ([8℄) Let

0 ∈ U ⊂ S

C

be an open set and

F : U → C

be a holomorphi fun tion on

U

. Then there is a unique

Φ ∈ (S)

−1

su h that

T Φ = F

. Conversely, given a

Φ ∈ (S)

−1

the fun tion

T Φ

is holomorphi on someopen set in

S

C

ontaining 0. The orresponden e between

F

and

Φ

is a bije tion ifone identies holomorphi fun tions whi h oin ide on some open neighborhood of 0 in

S

C

.

Theorem 2 ([7℄,[17℄)The

T

-transformdenesabije tionbetweenthespa e

(S)

′

and the spa e of

U

-fun tionals.

As a onsequen e of Theorem 1 one may derive the next two statements. Therstone on ernsthe onvergen eofsequen esofgeneralizedwhitenoise fun tionalsandthese ondonetheBo hnerintegrationoffamiliesofthesame type of generalized fun tionals. Similar results exist for Hida distributions (see, e.g., [5℄).

Theorem 3 Let

(Φ

n

)

n∈N

beasequen ein

(S)

−1

su hthatthereare

p, q ∈ N

0

so that

1. all

T Φ

n

are holomorphi on

U

p,q

:= {θ ∈ S

C

: 2

q

_|θ|

2

p

< 1}

,

2. thereexistsa

C > 0

su hthat

|T Φ

n

(θ)| ≤ C

forall

θ ∈ U

p,q

andall

n ∈ N

, 3.

(T Φ

n

(θ))

n∈N

is a Cau hy sequen e in

C

for all

θ ∈ U

p,q

. Then

(Φ

n

)

n∈N

onverges strongly in

(S)

−1

Theorem 4 Let

(Λ, F, ν)

be a measure spa e and

λ 7−→ Φ

λ

be a mapping from

Λ

to

(S)

−1

. We assume that there exists a

U

p,q

⊂ S

C

,

p, q ∈ N

0

, su h that

1.

T Φ

λ

isholomorphi on

U

p,q

for every

λ ∈ Λ

,

2. the mapping

λ 7−→ T Φ

λ

(θ)

is measurable for every

θ ∈ U

p,q

, 3. there is a

C ∈ L

1

_{(Λ, F, ν)}

su hthat

|T Φ

λ

(θ)| ≤ C(λ),

∀ θ ∈ U

p,q

, ν − a.a. λ ∈ Λ.

Then there exist

p

′

_{, q}

′

_{∈ N}

0

, whi h only depend on

p, q

, su h that

Φ

λ

is Bo hner integrable. In parti ular,

Z

Λ

Φ

λ

dν(λ) ∈ (S)

−1

and

T

R

Λ

Φ

λ

dν(λ)

isholomorphi on

U

p

′

,q

′

. One has



Z

Λ

Φ

λ

dν(λ), ϕ





=

Z

Λ

hhΦ

λ

, ϕii dν(λ),

∀ ϕ ∈ (S)

1

.

III. The free Feynman integral

Wefollow[3℄and [6℄in viewing the Feynmanintegralas aweighted average over Brownian paths. We use a slight hange inthe denition of the paths, whi h are here modeled by

x(τ ) = x −

r

~

m

Z

t

τ

ω(s) ds := x −

r

~

m

ω, 11

(τ,t]

 ,

ω ∈ S

′

_.

That is,insteadof xing the startingpointof the paths,wex the endpoint

x

at time

t

. In the sequel we set

~

= m = 1

. Correspondingly, the Feynman integrandfor the free motionis dened by

I

0

:= I

0

(x, t|y, t

0

) := N exp

 i + 1

2

Z

R

ω

2

(τ ) dτ



δ(x(t

0

) − y),

where, informally,

N

is a normalizing fa tor, more pre isely,

N exp (·)

is a Gauss kernel (see, e.g., [5℄, [15℄). We re all that the Donsker delta fun tion

δ(x(t

0

) − y)

isused tox the startingpoint ofthe paths attime

t

0

< t

. The

T

-transform ofthe freeFeynman integrand

T I

0

(ξ) =

1

p2πi(t − t

0

)

exp



−

i

2

Z

R

ξ

2

(τ ) dτ



(2)

× exp

i

2(t − t

0

)

Z

t

t

0

ξ(τ ) dτ + x − y



2

!

is a

U

-fun tional and we use itto dene

I

0

as aHida distribution (see [3℄). Fromthephysi alpointofview, equality(2) learly shows thatthe Feyn-man integral

T I

0

(0)

is the free parti lepropagator

1

p2πi(t − t

0

)

exp



i

2(t − t

0

)

(x − y)

2



.

Besides this parti ular ase, even for nonzero

ξ

the

T

-transform of

I

0

has a physi al interpretation. Integrating formallyby parts we nd

T I

0

(ξ) =

Z

S

′

I

0

(ω) exp



−i

Z

t

t

0

x(τ ) ˙ξ(τ ) dτ



dµ(ω)

× exp



−

₂

i

Z

[t

0

,t]

c

ξ

2

_{(τ ) dτ + ixξ(t) − iyξ(t}

0

)



.

Theterm

exp



−i

R

t

t

0

x(τ ) ˙ξ(τ ) dτ



wouldthus orrespondtoatime-dependent potential

W (x, t) = ˙ξ(t)x

. In fa t,it is straighforward toverify that

Θ(t − t

0

) · T I

0

(ξ) = K

0

(ξ)

exp



−

₂

i

Z

[t

0

,t]

c

ξ

2

_{(τ ) dτ + ixξ(t) − iyξ(t}

0

)



,

where

Θ

isthe Heavisidefun tion and

K

₀

(ξ)

:= K

₀

(ξ)

_{(x, t|y, t}

0

) :=

Θ(t − t

0

)

p2πi|t − t

0

|

exp



−

i

2

Z

t

t

0

ξ

2

_{(τ ) dτ}



× exp

i

2|t − t

0

|

Z

t

t

0

ξ(τ ) dτ + x − y



2

!

× exp (iyξ(t

0

) − ixξ(t))

is the Green fun tion orresponding tothe potential

W

, i.e.,

K

(ξ)

0

obeys the S hrödinger equation



i∂

t

+

1

2

∂

2

x

− ˙ξ(t)x



Inthe sequel

K

1

denotes thelinear spa eofallpotentials

V

on

R

ofthe form

V (x) =

Z

R

e

αx

dm(α),

_{x ∈ R,}

where

m

isa omplex measureonthe Borelsets on

R

fulllingthe ondition

Z

R

e

C|α|

_{d |m| (α) < ∞,}

_{∀ C > 0}

(4)

( f. [13℄), and

K

2

denotes the spa e of all potentials

V

on

R

whi h are gen-eralized fun tionsof the type

V (x) =

Z

R

δ(x − y) dm(y),

x ∈ R,

where

dm(y) := V (y)dy

isa nitesigned Borelmeasure ofbounded support ( f. [10℄).

Remark 5 A Lebesgue dominated onvergen e argument shows that poten-tials in

K

1

are restri tions tothe realline of entire fun tions [13℄. In parti -ular, they are lo ally bounded and smooth.

Our aim isto denethe Feynmanintegrand

I := I

0

· exp



−i

Z

t

t

0

V (x(τ )) dτ



(5) for a potential

V

of the form

V = V

1

+ V

2

,

V

i

∈ K

i

,

V

1

(x) =

Z

R

e

αx

dm

1

(α),

V

2

(x) =

Z

R

δ(x − y) dm

2

(y),

(6) where

x(τ ) = x −

Z

t

τ

ω(s) ds,

_{ω ∈ S}

′

,

as before. In order to dothis, rst we must give a meaningto the heuristi expression (5). In Theorem7 itwillbe shown that

I

is indeedawell-dened generalized white noise fun tional. Se ondly, it has to be proven that the expe tation of

I

solves the S hrödinger equationfor the potential

V

.

This leads to

I =

∞

X

n=0

(−i)

n

n!

n

X

k=0

n

k



k!

Z

∆

k

d

k

τ

Z

t

t

0

d

n−k

s

Z

R

k

Z

R

n−k

I

0

exp

n−k

X

l=1

α

l

x(s

l

)

!

_k

Y

j=1

δ(x(τ

j

) − x

j

)

n−k

Y

l=1

dm

1

(α

l

)

k

Y

j=1

dm

2

(x

j

),

(7) where

∆

k

:= {(τ

1

, ..., τ

k

) : t

0

< τ

1

< ... < τ

k

< t}

. Intheaboveexpressionthe integrals over

∆

k

, R

k

and

[t

0

, t]

n−k

, R

n−k

disappear, respe tively, for

k = 0

and

k = n

. Our aim is to apply Theorems 3 and 4 to show the existen e of the above series and integrals. However, rst we have to establish the pointwise multipli ation of generalizedfun tionals

I

0

exp

n−k

X

l=1

α

l

x(s

l

)

!

_k

Y

j=1

δ(x(τ

j

) − x

j

)

as a well-dened generalized fun tional. Due to the hara terization result Theorem 2 itis enough to dene this produ t through its

T

-transform. Ar-guing informally,for

ξ ∈ S

weare ledto

T

I

0

exp

n−k

X

l=1

α

l

x(s

l

)

!

_k

Y

j=1

δ(x(τ

j

) − x

j

)

!

(ξ)

=

Z

S

′

I

0

exp

n−k

X

l=1

α

l

x(s

l

)

!

_k

Y

j=1

δ(x(τ

j

) − x

j

) exp (i hω, ξi) dµ(ω)

= exp

x

n−k

X

l=1

α

l

!

· T

I

0

k

Y

j=1

δ(x(τ

j

) − x

j

)

!

(ξ + i

n−k

X

l=1

α

l

11

(s

l

,t]

).

The produ t

I

0

Q

k

j=1

δ(x(τ

j

) − x

j

)

isa slightgeneralizationof the free Feyn-manintegrand

I

0

,withmorethanjustonedeltafun tion,andmaybedened

by its

T

-transform,

T

I

0

k

Y

j=1

δ(x(τ

j

) − x

j

)

!

(ξ)

= exp



−

i

2

Z

[t

0

,t]

c

ξ

2

_{(s)ds + ixξ(t) − iyξ(t}

0

)



k+1

Y

j=1

K

₀

(ξ)

(x

j

, τ

j

|x

j−1

, τ

j−1

)

= exp



−

₂

i

Z

R

ξ

2

(s)ds



k+1

Y

j=1

(

1

p2πi(τ

j

− τ

j−1

)

× exp





i

2 (τ

j

− τ

j−1

)

Z

τ

j

τ

j−1

ξ(s)ds + x

j

− x

j−1

!

2











.

(8) Here

τ

0

:= t

0

, x

0

:= y, τ

k+1

:= t

, and

x

k+1

:= x

. Clearly the expli it formula (8) is ontinuously extendable to all

ξ ∈ L

2

whi h allows an extension of

T



I

0

Q

k

j=1

δ(x(τ

j

) − x

j

)



to the argument

ξ + i

P

n−k

l=1

α

l

11

(s

l

,t]

. Proposition 6 The produ t

Φ

n,k

:= I

0

exp

n−k

X

l=1

α

l

x(s

l

)

!

_k

Y

j=1

δ(x(τ

j

) − x

j

)

dened by

T Φ

n,k

(ξ)

= T

I

0

k

Y

j=1

δ(x(τ

j

) − x

j

)

!

ξ + i

n−k

X

l=1

α

l

11

(s

l

,t]

!

exp

x

n−k

X

l=1

α

l

!

= exp





−

i

2

Z

R

ξ(s) + i

n−k

X

l=1

α

l

11

(s

l

,t]

(s)

!

2

ds





k+1

Y

j=1

1

p2πi(τ

j

− τ

j−1

)

× exp





k+1

X

j=1

i

2 (τ

j

− τ

j−1

)

Z

τ

j

τ

j−1

ξ(s) + i

n−k

X

l=1

α

l

11

(s

l

,t]

(s)

!

ds + x

j

− x

j−1

!

2





× exp

x

n−k

X

l=1

α

l

!

Proof. It is obvious that the latter expli it formulafullls the rst part of Denition 2, analyti ity. In order to prove that

Φ

n,k

is a Hida distribution by appli ation of Theorem 2, we only have toshow that

T Φ

n,k

alsoobeys a boundas inthe se ondpart of Denition 2. Forevery

θ ∈ S

C

we have

|T Φ

n,k

(θ)|

≤ exp

|x|

n−k

X

l=1

|α

l

|

!

×

k+1

Y

j=1

1

p2π(τ

j

− τ

j−1

)











exp

₋

i

2

Z

R

θ

2

(s)ds +

n−k

X

l=1

α

l

Z

R

θ(s)11

(s

l

,t]

(s)ds

!









×













exp





k+1

X

j=1

i

2 (τ

j

− τ

j−1

)

Z

τ

j

τ

j−1

θ(s)ds

!

2

















×











exp

k+1

X

j=1

1

τ

j−1

− τ

j

Z

τ

j

τ

j−1

θ(s)ds

!

_n−k

X

l=1

α

l

Z

τ

j

τ

j−1

11

(s

l

,t]

(s)ds

!!









×











exp

k+1

X

j=1

i (x

j

− x

j−1

)

τ

j

− τ

j−1

Z

τ

j

τ

j−1

θ(s) + i

n−k

X

l=1

α

l

11

(s

l

,t]

(s)

!

ds

!









whi h ismajorized by

|T Φ

n,k

(θ)| ≤

k+1

Y

j=1

1

p2π(τ

j

− τ

j−1

)

exp 2 kθk

2

× exp

|x| + t − t

0

+ kθk

2

n−k

X

l=1

|α

l

|

!

(9)

× exp

4 max

0≤j≤k+1

|x

j

|

n−k

X

l=1

|α

l

|

!

exp



max

0≤j≤k+1

|x

j

|

2



=: C(τ

1

, ..., τ

k

; α

1

, ..., α

n−k

; x

1

, ..., x

k

; θ) =: C

independent of

s

1

, ..., s

n−k

, where

kθk := sup

s∈[t

0

,t]

|θ(s)| +

Z

t

t

0







˙θ(s)





 ds + |θ|

is a ontinuous norm on

S

C

, f. Appendix below. This estimatefor

T Φ

n,k

is of the formrequired inDenition 2,whi h ompletes the proof.



A ording toProposition6, all

Φ

n,k

are Hidadistributions and thus also generalized white noisefun tionals with

T Φ

n,k

entire on

S

C

. Moreover, ea h

T Φ

n,k

(θ)

isameasurablefun tionof

τ

1

, ..., τ

k

; s

1

, ..., s

n−k

; α

1

, ..., α

n−k

; x

1

, ..., x

k

for every

θ ∈ S

C

. Hen e, inorder toapply Theorem 4toprovethe existen e of the integrals in

I

, we only have to nd a suitable integrable bound for

|T Φ

n,k

(θ)|

. Sin e the measure

m

1

fulllsthe integrability ondition (4) and the signedmeasure

m

2

isnite and has support ontained insome bounded interval

[−a, a]

,

a > 0

,one may inferthe integrabilityof

C

forevery

θ ∈ S

C

:











Z

∆

k

d

k

τ

Z

t

t

0

d

n−k

s

Z

R

k

k

Y

j=1

dm

2

(x

j

)

Z

R

n−k

n−k

Y

l=1

d |m

1

| (α

l

)C











≤ exp 2 kθk

2

+ b

2

(t − t

0

)

n−k

×

Z

∆

k

k+1

Y

j=1

1

p2π(τ

j

− τ

j−1

)

d

k

τ









Z

R

dm

2

(x)









k

×

Z

R

exp

_{|x| + 4b + t − t}

0

+ kθk

2

|α|
d |m

1

| (α)



n−k

,

where

b := max{a, |y|, |x|}

. Thus, a ording to Theorem 4, there exists an open set

U ⊂ S

C

independent of

n

su hthat

I

n,k

:=

Z

∆

k

d

k

τ

Z

t

t

0

d

n−k

s

Z

R

k

Z

R

n−k

Φ

n,k

n−k

Y

l=1

dm

1

(α

l

)

k

Y

j=1

dm

2

(x

j

) ∈ (S)

−1

for ea h

k ≤ n

and every

n ∈ N

, and all

T I

n,k

are holomorphi on

U

. To on ludetheexisten eof

I

weonlyhavetoprovethattheseriesin

n

onverges in

(S)

−1

in the strong sense. This follows from Theorem 3. In fa t, due to (7),for every

θ ∈ U

one has

|T I(θ)| ≤

∞

X

n=0

1

n!

n

X

k=0

n

k



k! |T I

n,k

(θ)|

where the right-hand side is upper bounded by the fa tor

exp 2 kθk

2

+ b

2

times the Cau hy produ t of the onvergent series

∞

X

n=0

1

n!



(t − t

0

)

Z

R

e(

|x|+4b+t−t

0

+kθk

2

)

|α|

d |m

1

| (α)



n

!

×

∞

X

n=0









Z

R

dm

2

(x)









n

Z

∆

n

n+1

Y

j=1

1

p2π(τ

j

− τ

j−1

)

d

n

τ

!

= exp



(t − t

0

)

Z

R

e(

|x|+4b+t−t

0

+kθk

2

)

|α|

d |m

1

| (α)



×

∞

X

n=0









Z

R

dm

2

(x)









n

Z

∆

n

n+1

Y

j=1

1

p2π(τ

j

− τ

j−1

)

d

n

τ.

We notethat the latter series onverges be ause

Z

∆

n

n+1

Y

j=1

1

p2π(τ

j

− τ

j−1

)

d

n

_{τ =}

 Γ (1/2)

_√

2π



n+1

(t − t

0

)

(n−1)/2

Γ

n+1

₂

is rapidly de reasing in

n

.

In this way we have proved the followingresult.

Theorem 7 For every

V

1

∈ K

1

and

V

2

∈ K

2

of the form (6),the

I :=

∞

X

n=0

(−i)

n

n!

n

X

k=0

n

k



k!

Z

∆

k

d

k

τ

Z

t

t

0

d

n−k

s

Z

R

k

Z

R

n−k

I

0

exp

n−k

X

l=1

α

l

x(s

l

)

!

_k

Y

j=1

δ(x(τ

j

) − x

j

)

n−k

Y

l=1

dm

1

(α

l

)

k

Y

j=1

dm

2

(x

j

),

exists as a generalized white noise fun tional. The series onverges strongly in

(S)

−1

and the integrals exist in the sense of Bo hner integrals. Therefore we may express the

T

-transform of

I

by

T I(θ) =

∞

X

n=0

(−i)

n

n!

n

X

k=0

n

k



k!

Z

∆

k

d

k

τ

Z

t

t

0

d

n−k

s

Z

R

k

Z

R

n−k

T

I

0

exp

n−k

X

l=1

α

l

x(s

l

)

!

_k

Y

j=1

δ(x(τ

j

) − x

j

)

!

(θ)

n−k

Y

l=1

dm

1

(α

l

)

k

Y

j=1

dm

2

(x

j

)

for every

θ

in a neighborhood

n

θ ∈ S

C

: 2

q

|θ|

2

_p

< 1

o

of zero, for some

p, q ∈

N

₀

A ording to Theorem 7,

I

is a well-dened generalized white noise fun -tional. In order to on lude that

I

denes a Feynman integrandit remains to showthat the expe tation

T I(0)

of

I

solves the S hrödingerequation for a potential

V = V

1

+ V

2

, V

i

∈ K

i

. As in the free motion ase we onsider, more generally,

K

(θ)

_{(x, t|y, t}

0

) := Θ(t − t

0

)T I(θ) exp

 i

2

Z

[t

0

,t]

c

θ

2

_{(τ ) dτ + iyθ(t}

0

) − ixθ(t)



.

Insertion of

T I(θ)

as given in Theorem 7, with

T

I

0

exp

n−k

X

l=1

α

l

x(s

l

)

!

_k

Y

j=1

δ(x(τ

j

) − x

j

)

!

as inProposition 4,yields

K

(θ)

_{(x, t|y, t}

0

) =

∞

X

n=0

K

_n

(θ)

_{(x, t|y, t}

0

),

with

K

_n

(θ)

_{(x, t|y, t}

0

) :=

(−i)

n

n!

Z

t

t

0

d

n

s

Z

R

n

n

Y

l=1

dm

1

(α

l

)K

0

(θ

n

)

(x, t|y, t

0

)

+

n−1

X

k=1

(−i)

n−k

(n − k)!

Z

t

t

0

d

n−k

s

Z

R

n−k

n−k

Y

l=1

dm

1

(α

l

)G

(θ

_k

n−k

)

(x, t|y, t

0

)

+G

(θ)

n

(x, t|y, t

0

),

(10) wherewehaveset

θ

n−k

:= θ

n−k

(s

1

, ..., s

n−k

, α

1

, ..., α

n−k

) := θ+i

P

n−k

l=1

α

l

11

(s

l

,t]

for

k = 0, ..., n − 1

,

θ

0

:= θ

, and

G

(θ

n−k

)

k

(x, t|y, t

0

) := (−i)

k

Z

∆

k

d

k

τ

Z

R

k

k

Y

j=1

dm

2

(x

j

)

k+1

Y

j=1

K

(θ

n−k

)

0

(x

j

, τ

j

|x

j−1

, τ

j−1

)

for

k = 1, ..., n, n > 0.

We expe t

K

(θ)

to be the propagator orresponding to the potential

Theorem 8

K

(θ)

_{(x, t|y, t}

0

)

isaGreenfun tionfortheS hroedingerequation



i∂

t

+

1

2

∂

2

x

− ˙θ(t)x − V (x)



K

(θ)

_{(x, t|y, t}

0

) = iδ(t − t

0

)δ(x − y).

(11)

In parti ular,

K(x, t|y, t

0

) := T I(0)

isa Feynmanintegral solving

i∂

t

K(x, t|y, t

0

) =



−

1

₂

∂

_x

2

+ V (x)



K(x, t|y, t

0

),

for

t > t

0

.

(12)

Remark 3

K

orresponds to a unitary evolution whenever

H = −

1

2

∂

x

2

+ V

has a unique self-adjoint extension.

Proof. Let us onsider an interval

[T

0

, T ]

su h that

[t

0

, t] ⊂ [T

0

, T ]

. Esti-matessimilartothosedoneintheproofofProposition6showthat

K

(θ)

n

(·, ·|y, t

0

)

islo allyintegrableon

R

× [T

0

, T ]

with respe t to

dm

2

× dt

and theLebesgue measure. Therefore, we may regard

K

(θ)

n

as a distribution on

D(Ω) :=

D(R × [T

0

, T ])

:

K

(θ)

n

(·, ·|y, t

0

), ϕ =

Z

R

dx

Z

T

T

0

dtK

(θ)

n

(x, t|y, t

0

)ϕ(x, t),

ϕ ∈ D(Ω).

And we may alsodene adistribution

V

2

K

(θ)

n

by setting

V

2

K

n

(θ)

(·, ·|y, t

0

), ϕ =

Z

R

dm

2

(x)

Z

T

T

0

dtK

_n

(θ)

_{(x, t|y, t}

0

)ϕ(x, t),

ϕ ∈ D(Ω).

To abbreviate we introdu e the notation

L := i∂

ˆ

t

+

1

2

∂

x

2

− ˙θ(t)x

and

L

ˆ

∗

for the dual operator. A ording to (10), observe that for any test fun tion

ϕ ∈ D(Ω)

one nds

D ˆ

_LK

(θ)

n

, ϕ

E

=

(−i)

n

n!

*

Z

·

t

0

d

n

s

Z

R

n

n

Y

l=1

dm

1

(α

l

)K

0

(θ

n

)

(·, ·|y, t

0

), ˆ

L

∗

ϕ

+

+

n−1

X

k=1

(−i)

n−k

(n − k)!

*

Z

·

t

0

d

n−k

s

Z

R

n−k

n−k

Y

l=1

dm

1

(α

l

)G

(θ

k

n−k

)

(·, ·|y, t

0

), ˆ

L

∗

ϕ

+

(13)

+

D

G

(θ)

_n

_{(·, ·|y, t}

0

), ˆ

L

∗

ϕ

E

,

(−i)

n

n!

*

Z

·

t

0

d

n

s

Z

R

n

n

Y

l=1

dm

1

(α

l

)K

0

(θ

n

)

(·, ·|y, t

0

), ˆ

L

∗

ϕ

+

(14)

=

(−i)

n−1

(n − 1)!

*

V

1

Z

·

t

0

d

n−1

s

Z

R

n−1

n−1

Y

l=1

dm

1

(α

l

)K

0

(θ

n−1

)

(·, ·|y, t

0

), ϕ

+

f. [13℄,and

D

G

(θ)

_n

_{(·, ·|y, t}

0

), ˆ

L

∗

ϕ

E

=

D

V

2

G

(θ)

n−1

(·, ·|y, t

0

), ϕ

E

(15) f.[15℄,[10℄. Thegeneri ase(13)isintermediatebetween (14)and(15)and is dealt with by a ombinationof the orresponding te hniques. This yields

*

Z

·

t

0

d

n−k

s

Z

R

n−k

n−k

Y

l=1

dm

1

(α

l

)G

(θ

k

n−k

)

(·, ·|y, t

0

), ˆ

L

∗

ϕ

+

= i(n − k)

*

V

1

Z

·

t

0

d

n−k−1

s

Z

R

n−k−1

n−k−1

Y

l=1

dm

1

(α

l

)G

(θ

_k

n−k−1

)

(·, ·|y, t

0

), ϕ

+

+

*

V

2

Z

·

t

0

d

n−k

s

Z

R

n−k

n−k

Y

l=1

dm

1

(α

l

)G

(θ

k−1

n−k

)

(·, ·|y, t

0

), ϕ

+

,

for any

k = 2, ..., n − 2

,

*

Z

·

t

0

d

n−1

s

Z

R

n−1

n−1

Y

l=1

dm

1

(α

l

)G

(θ

1

n−1

)

(·, ·|y, t

0

), ˆ

L

∗

ϕ

+

= i(n − 1)

*

V

1

Z

·

t

0

d

n−2

s

Z

R

n−2

n−2

Y

l=1

dm

1

(α

l

)G

(θ

1

n−2

)

(·, ·|y, t

0

), ϕ

+

+

*

V

2

Z

·

t

0

d

n−1

s

Z

R

n−1

n−1

Y

l=1

dm

1

(α

l

)K

0

(θ

n−1

)

(·, ·|y, t

0

), ϕ

+

,

and

Z

·

t

0

ds

Z

R

dm

1

(α

1

)G

(θ

n−1

1

)

(·, ·|y, t

0

), ˆ

L

∗

ϕ



= i

D

V

1

G

(θ)

n−1

(·, ·|y, t

0

), ϕ

E

+



V

2

Z

·

t

0

ds

Z

R

dm

1

(α

1

)G

(θ

n−2

1

)

(·, ·|y, t

0

), ϕ



.

D ˆ

_LK

(θ)

n

, ϕ

E

=

(−i)

n−1

(n − 1)!

*

(V

1

+ V

2

)

Z

·

t

0

d

n−1

s

Z

R

n−1

n−1

Y

l=1

dm

1

(α

l

)K

0

(θ

n−1

)

(·, ·|y, t

0

), ϕ

+

+

n−2

X

k=1

(−i)

n−k−1

(n − k − 1)!

*

V

1

Z

·

t

0

d

n−k−1

s

Z

R

n−k−1

n−k−1

Y

l=1

dm

1

(α

l

)G

(θ

k

n−k−1

)

(·, ·|y, t

0

), ϕ

+

+

n−1

X

k=2

(−i)

n−k

(n − k)!

*

V

2

Z

·

t

0

d

n−k

s

Z

R

n−k

n−k

Y

l=1

dm

1

(α

l

)G

(θ

_k−1

n−k

)

(·, ·|y, t

0

), ϕ

+

+

D

(V

1

+ V

2

) G

(θ)

n−1

(·, ·|y, t

0

), ϕ

E

,

whi h isequivalent to

D ˆ

_LK

(θ)

n

, ϕ

E

=

D

(V

1

+ V

2

) K

n−1

(θ)

, ϕ

E

,

_{ϕ ∈ D(Ω),}

for any

n ≥ 1

. Using(3) and summingover

n

, we obtain(11).



We on ludebyanobservationwhi hisobviousfromtheabove onstru -tion but somewhat unexpe ted given that the Hamiltonians with potentials in the lass

K

2

will in general not admit a perturbative expansion (see e.g. [13℄for more onthis).

Proposition 9 For any potential

V = g (V

1

+ V

2

)

with

V

i

∈ K

i

, the solution

K

of the propagator equation(12) is analyti in the oupling onstant

g

.

A knowledgments

M.J.O. would liketo express her gratitude to José Luís da Silva for helpful dis ussionsandalsothegeneroushospitalityofCustódiaDrumondandCCM duringaverypleasantstay atFun halduringthe MadeiraMathEn ounters XXIII. This work was supported by FCT POCTI, FEDER.

For the proof of Proposition6, we needto estimate

|T Φ

n,k

(θ)|

≤ exp

|x|

n−k

X

l=1

|α

l

|

!

×

k+1

Y

j=1

1

p2π(τ

j

− τ

j−1

)











exp

₋

i

2

Z

R

θ

2

(s)ds +

n−k

X

l=1

α

l

Z

R

θ(s)11

(s

l

,t]

(s)ds

!









×













exp





k+1

X

j=1

i

2 (τ

j

− τ

j−1

)

Z

τ

j

τ

j−1

θ(s)ds

!

2

















×











exp

k+1

X

j=1

1

τ

j−1

− τ

j

Z

τ

j

τ

j−1

θ(s)ds

!

_n−k

X

l=1

α

l

Z

τ

j

τ

j−1

11

(s

l

,t]

(s)ds

!!









×











exp

k+1

X

j=1

i (x

j

− x

j−1

)

τ

j

− τ

j−1

Z

τ

j

τ

j−1

θ(s) + i

n−k

X

l=1

α

l

11

(s

l

,t]

(s)

!

ds

!









We shall now estimate, onse utively, the exponents o uring in the above expression.

Usingthe Cau hy-S hwarz inequality we may approximate











exp

n−k

X

l=1

α

l

Z

R

θ(s)11

(s

l

,t]

(s)ds

!









≤ exp

n−k

X

l=1

|α

l

|

Z

R

|θ(s)|

2

ds



1/2

_√

t − s

l

!

≤ exp

√

t − t

0

|θ|

n−k

X

l=1

|α

l

|

!

and, similarly,













k+1

X

j=1

i

2 (τ

j

− τ

j−1

)

Z

τ

j

τ

j−1

θ(s)ds

!

2













≤

1

₂

|θ|

2

,











k+1

X

j=1

1

τ

j−1

− τ

j

Z

τ

j

τ

j−1

θ(s)ds

!

_n−k

X

l=1

α

l

Z

τ

j

τ

j−1

11

(s

l

,t]

(s)ds

!









≤

k+1

X

j=1

1

τ

j

− τ

j−1

Z

τ

j

τ

j−1

|θ(s)|ds

!

(τ

j

− τ

j−1

)

n−k

X

l=1

|α

l

|

=

n−k

X

l=1

|α

l

|

Z

t

t

0

|θ(s)|ds ≤

√

t − t

0

|θ|

n−k

X

l=1

|α

l

|,

wherewehaveagainusedtheCau hy-S hwarz inequalitytoobtainthelatter inequality.

Finally,inorder to estimatethe exponentialof the fun tion

k+1

X

j=1

i (x

j

− x

j−1

)

τ

j

− τ

j−1

Z

τ

j

τ

j−1

θ(s) + i

n−k

X

l=1

α

l

11

(s

l

,t]

(s)

!

ds

=

k+1

X

j=1

i (x

j

− x

j−1

)

τ

j

− τ

j−1

Z

τ

j

τ

j−1

θ(s)ds

+

n−k

X

l=1

α

l

k+1

X

j=1

x

j−1

− x

j

τ

j

− τ

j−1

Z

τ

j

τ

j−1

11

(s

l

,t]

(s)ds,

rst wepro eed as in[18℄, i.e.,

k+1

X

j=1

x

j

− x

j−1

τ

j

− τ

j−1

Z

τ

j

τ

j−1

θ(s)ds =

x

t − τ

k

Z

t

τ

k

θ(s)ds −

y

τ

1

− t

0

Z

τ

1

t

0

θ(s)ds

+

k

X

j=1

x

j

R

τ

j

τ

j−1

θ(s)ds

τ

j

− τ

j−1

−

R

τ

j+1

τ

j

θ(s)ds

τ

j+1

− τ

j

!

.

By the mean value theorem

k

X

j=1

x

j

R

τ

j

τ

j−1

θ(s)ds

τ

j

− τ

j−1

−

R

τ

j+1

τ

j

θ(s)ds

τ

j+1

− τ

j

!

=

k

X

j=1

x

j

(θ(r

j

) − θ(r

j+1

)) ,

where

r

j

∈ (τ

j−1

, τ

j

)

. Therefore











k+1

X

j=1

i (x

j

− x

j−1

)

τ

j

− τ

j−1

Z

τ

j

τ

j−1

θ(s)ds











≤ (|x| + |y|) sup

[t

0

,t]

|θ| + max

_1≤j≤k

|x

j

|

k

X

j=1











Z

r

j+1

r

j

˙θ(s)ds











≤ 2 max

0≤j≤k+1

|x

j

|

sup

_[t

₀

_,t]

|θ| +

Z

t

t

0







˙θ(s)





 ds

!

.

Now letus onsider the sum

n−k

X

l=1

α

l

k+1

X

j=1

x

j−1

− x

j

τ

j

− τ

j−1

Z

τ

j

τ

j−1

11

(s

l

,t]

(s)ds.

Sin e

s

l

∈ [t

0

, t]

, there is a

j

0

∈ {0, 1, ..., k}

su h that

s

l

∈ [τ

j

0

, τ

j

0

+1

]

. This

fa t allows to rewritethe se ond sum in the latter expression as

x

j

0

+1

− x + (x

j

0

+1

− x

j

0

)

s

l

− τ

j

0

+1

τ

j

0

+1

− τ

j

0

leading to











n−k

X

l=1

α

l

k+1

X

j=1

x

j−1

− x

j

τ

j

− τ

j−1

Z

τ

j

τ

j−1

11

(s

l

,t]

(s)ds











≤ 4 max

0≤j≤k+1

|x

j

|

n−k

X

l=1

|α

l

|.

Insertingthese estimates weobtain

|T Φ

n,k

(θ)|

≤ exp

|x|

n−k

X

l=1

|α

l

|

!

_k+1

Y

j=1

1

p2π(τ

j

− τ

j−1

)

exp |θ|

2

exp

2

√

_{t − t}

0

|θ|

n−k

X

l=1

|α

l

|

!

×exp

2 max

0≤j≤k+1

|x

j

|

sup

_[t

₀

_,t]

|θ| +

Z

t

t

0







˙θ(s)





 ds

!!

exp

4 max

0≤j≤k+1

|x

j

|

n−k

X

l=1

|α

l

|

!

.

Now weintrodu ethe norm

kθk := sup

s∈[t

0

,t]

|θ(s)| +

Z

t

t

0







˙θ(s)





 ds + |θ|

exp

_|x|

n−k

X

l=1

|α

l

|

!

_k+1

Y

j=1

1

p2π(τ

j

− τ

j−1

)

exp kθk

2

exp

2

√

_{t − t}

0

kθk

n−k

X

l=1

|α

l

|

!

× exp



2 max

0≤j≤k+1

|x

j

| kθk



exp

4 max

0≤j≤k+1

|x

j

|

n−k

X

l=1

|α

l

|

!

.

Then we use

√

t − t

0

kθk ≤

1

2

(t − t

0

+ kθk

2

)

and

2 max

0≤j≤k+1

|x

j

| kθk ≤ max

0≤j≤k+1

|x

j

|

2

+ kθk

2

to obtainthe desiredestimate(9).

Referen es

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[2℄ Bernido, Chr., Bernido,V. andStreit, L.,Feynman Paths withGeneral Boundary Conditions: TheirConstru tion in Terms of White Noise. In preparation.

[3℄ de Faria, M., Pottho, J. and Streit, L., The Feynman integrand as a Hida distribution. J. Math. Phys., 32,2123-2127(1991).

[4℄ Hida, T., Analysis of Brownian Fun tionals, volume 13 of Carleton Mathemati alLe ture Notes. Carleton, 1975.

[5℄ Hida, T., Kuo, H. H., Pottho, J. and Streit, L., White Noise. An InniteDimensionalCal ulus.KluwerA ademi Publishers,Dordre ht, 1993.

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W., Generalized fun tionals in Gaussian spa es: The hara terization theorem revisited. J. Fun t. Anal.,141, 301-318(1996).

[8℄ Kondratiev,Yu.G.,Leukert,P.andStreit,L.,Wi k al ulusinGaussian analysis. A ta Appl. Math.,44,269-294 (1996).

[9℄ Kondratiev, Yu. G.,Spa es of Test and Generalized Fun tions of an In-nite Numberof Variables.Master's thesis, University of Kiev, 1975. [10℄ Khandekar, D. C. and Streit, L., Constru tingthe Feynman integrand.

Ann. Physik, 1,49-55, (1992).

[11℄ Kubo, I. and Takenaka, S., Cal ulus on Gaussian white noise I. Pro . Japan A ad. Ser. A Math. S i.,56, 376-380(1980).

[12℄ Kubo, I. and Takenaka, S., Cal ulus on Gaussianwhite noise II. Pro . Japan A ad. Ser. A Math. S i.,56, 411-416(1980).

[13℄ Kuna, T., Streit, L. and Westerkamp, W., Feynman integrals for a lass ofexponentiallygrowingpotentials. J. Math.Phys.,39,4476-4491 (1998).

[14℄ Kuo,H.H., WhiteNoiseDistributionTheory.CRCPress,Bo aRaton, New York,London, and Tokyo,1996.

[15℄ Las he k,A.,Leukert,P.,Streit,L.andWesterkamp,W.,Quantum me- hani alpropagators in terms of Hida distributions. Rep. Math. Phys., 33, 221-232(1993).

[16℄ Obata,N., WhiteNoiseCal ulusandFo kSpa e,volume1577of LNM. Springer Verlag, Berlin,Heidelberg, and New York, 1994.

[17℄ Pottho, J. and Streit, L., A hara terization of Hida distributions. J. Fun t.Anal.,101, 212-229 (1991).

[18℄ Westerkamp, W., Re ent Results in Innite Dimensional Analysis and Appli ations to Feynman Integrals. PhD thesis, University of Bielefeld, 1995.