CONDITIONAL INTEGRALS ON ABSTRACT WIENER AND HILBERT

J. Korean Math. Soc. 41 (2004), No. 2, pp. 319–344

CONDITIONAL INTEGRALS ON ABSTRACT WIENER AND HILBERT SPACES WITH APPLICATION TO FEYNMAN INTEGRALS Dong Myung Chung1 , Soon Ja Kang2 , and Kyung Pil Lim3 Abstract. In this paper, we define conditional integrals on abstract Wiener and Hilbert spaces and then obtain a formula for evaluating the integrals. We use this formula to establish the existence of conditional Feynman integrals for the classes Λq (B) and Λq (H) of functions on abstract Wiener and Hilbert spaces and then specialize this result to provide the fundamental solution to the Schr¨ odinger equation with the forced harmonic oscillator.

1. Introduction We are concerned with the fundamental solution to the Schrödinger equation for a quantum mechanical particle of mass m in Rn (1)

i~

~2 ∂ Γ(t, ~η ) = − △ Γ(t, ~η ) + V (~η )Γ(t, ~η ), ∂t 2m Γ(0, ~η ) = ψ(~η ), ~η ∈ Rn

where △ is the Laplacian on Rn , ~ is Planck’s constant and V is a suitable potential. According to Feynman [12], the fundamental solution ~ to the Schrödinger equation (1) such that K(t, ~η , 0, ξ) Z ~ ψ(ξ)d ~ ξ~ K(t, ~η , 0, ξ) Γ(t, ~η ) = Rn

Received January 23, 2003. 2000 Mathematics Subject Classification: 28C20. Key words and phrases: Feynman integrals, conditional analytic Feynman integral, abstract Wiener space, lifting, fundamental solution. 1 Research supported by the Sogang University Research Grant in 2001. 2 Research supported by Chonnam National University in the program, 2001. 3 Research supported by Korea Research Foundation Grant (KRF-2000-015DP0016).

320

Dong Myung Chung, Soon Ja Kang, and Kyung Pil Lim

can be expressed as the formal path integral: Z i ~ (2) K(t, ~η , 0, ξ) = exp S(x) D(x), ~ C~ [0,t] ξ,~ η

~ where Cξ,~ η , D(x) ~ η is the space of paths x such that x(0) = ξ and x(t) = ~ is a uniform “measure” which does not exist and S(x) is the action integral associated with the path x; i.e., # Z t " 2 m dx − V (x(s)) ds. S(x) = 2 ds 0 The basic problem of quantum mechanics is to find the solution Γ(t, ~η ) ~ to equation (1). or the fundamental solution K(t, ~η , 0, ξ) In [14], Gelfand and Yaglom made an attempt to give sense to the formal integral in equation (2) by introducing a Wiener measure with complex variance parameter. Unfortunately their attempt was failed as pointed out by Cameron in [2]. There has been several rigorous approaches to equation (2) to provide the fundamental solution to equation (1). See for examples [16, 21, 25]. In [7] and [10], the concept of the conditional Feynman integral has been introduced and is used to provide an method of getting the fundamental solution to the Schrödinger equation and to obtain the kernel of operator-valued Feynman integrals of various functions. The main purpose of this paper is to establish the existence of conditional analytic Feynman integral on abstract Wiener and Hilbert spaces for a wider classes of functions than the Fresnel class on abstract Wiener spaces considered in [6, 7]. The latter result establishes a formula for the conditional analytic Feynman integral of functions involving unbounded potentials and then use it to obtain the fundamental solution to the Schrödinger equation with the forced harmonic oscillator. This paper is organized as follows. In Section 1, we give an introduction for this paper. In Section 2, we recall some preliminary materials which will be needed in this paper. In Section 3, we define conditional integrals on abstract Wiener and Hilbert spaces and then obtain a formula for evaluating the integrals. In Section 4, we establish the existence of conditional analytic Feynman integrals for the classes Λq (B) and Λq (H) of functions on abstract Wiener and Hilbert spaces and use this result to provide the fundamental solution to the Schrödinger equation. In Section 5, we specialize the result of Section 4 to Feynman path integrals to obtain the fundamental solution to the Schrödinger equation with the

Conditional integrals on abstract Wiener and Hilbert spaces

potential V (t, ξ) = external force f (t).

mω 2 2 2 ξ

321

+ f (t)ξ, the forced harmonic oscillator with

2. Preliminaries Let H be a real separable infinite dimensional Hilbert space with inner product h·, ·i and norm | · |. Let P be the set of all orthogonal projections on H with finite dimensional range. For P1 , P2 ∈ P, we define P1 < P2 if P1 (H) ⊆ P2 (H). For P ∈ P, let and

CP = {P −1 B : B a Borel set in range of P } C=

[ P

CP .

A cylinder measure is a finitely additive nonnegative measure on (H, C) such that its restriction to CP is countably additive for all P ∈ P. The canonical Gauss measure m on H is the cylinder measure on (H, C) characterized by Z 2 /2

eihh,h1 i dm(h) = e−|h1 |

.

H

Let || · || be a measurable norm on H. It is well known that H is not complete with respect to || · || (see [22]). Let B denote the completion of H under || · || and let i denote the natural injection. The adjoint operator i∗ maps the strong dual B ∗ continuously, one-to-one, onto a dense subspace of H ∗ ≈ H. Gross proved that the induced measure mi−1 on the cylinder sets in B is indeed countably additive and hence extends to a countably additive measure ν on B-the Borel σ-field on B. The triple (H, B, ν) is called an abstract Wiener space and ν is called the abstract Wiener measure. For more details, see [22]. Let {ej |j ≥ 1} be a complete orthonormal set in H such that the ej ’s are in B ∗ . For each h ∈ H and x ∈ B, let (h, x˜) = lim

n→∞

n X j=1

hh, ej i(ej , x)

if the limit exists and is equal to 0 otherwise. It is shown that for each h(6= 0) in H, (h, ·˜) is a Gaussian random variable on B with mean zero, variance |h|2 , and that (h, λx˜) = λ(h, x˜) for all λ > 0. It is easy to see that if {h1 , h2 , . . . , hn } is a orthogonal set in H, then the random variable (hi , x˜)s are independent.

322


A function f defined on H of the form f (h) = φ(hh1 , hi, . . . , hhk , hi) is called a cylinder function, where hi ∈ H and φ is a complex valued Borel function on Rn . We denote by R(f ) the random variable φ((h1 , x˜), (h2 , x˜), . . . , (hk , x˜)) on (B, B, ν). This mapping is extended as follows. For more details, see [15, 21]. Definition. Let L(H, C, m) be the set of all complex valued continuous functions f on H such that the net {R(f ◦ P ) : P ∈ P} is Cauchy net in ν-probability. Furthermore, for f ∈ L(H, C, m), let R(f ) = lim ν − probability R(f ◦ P ). P ∈P

The mapping R will be called m-lifting. Definition. Let L1 (H, C, m) = and for f ∈

L1 (H,

Z f ∈ L(H, C, m) : |R(f )| dν < ∞ B

C, m), define Z Z f dm = H

R(f ) dν. B

Let A be a self-adjoint, trace class operator with eigenvalues {αk } and corresponding eigenfunctions {ek }. Let (x, Ax˜) = lim

n→∞

n X

2

αj (ej , x˜)

j=1

if the limit exists and is equal to 0 otherwise. Let X be an Rn -valued measurable function and Y a C-valued integrable function of (B, B(B), ν). Let PX be the probability distribution of X, i.e., for all A ∈ B(Rn ), PX (A) = ν(X −1 (A)). Then by RadonNikodym Theorem, there exists a Borel measurable and PX -integrable function ψ on (Rn , B(Rn ), PX ) such that Z Z ~ dPX , Y dν = ψ(ξ) X −1 (A)

A

~ ξ ∈ Rn is unique up to Borel for all A ∈ B(Rn ). The function ψ(ξ), ~ written E[Y |X = ξ], ~ is called the null sets in Rn . The function ψ(ξ),


323

conditional integral of Y on abstract Wiener space given vector-valued conditioning function X. Thus we have Z Z ~ dPX , (3) Y dν = E[Y |X = ξ] X −1 (A)

for all A ∈

A

B(Rn ).

3. A formula for conditional integral on abstract Wiener and Hilbert spaces In this section, we give a formula for evaluating conditional integrals on abstract Wiener and Hilbert spaces that include the results of Park and Skoug given in [23, 24] as special cases. 3.1. Conditional integrals on abstract Wiener spaces Let {g1 , g2 , . . . , gn } be an orthonormal set in H. Let B1 be the ndimensional subspace of H generated by {g1 , g2 , . . . , gn }. The mapping (4)

Q(x) =

n X (gj , x˜)gj j=1

defines a continuous operator on B. Since (gj , x˜) = hgj , xi for x ∈ H, Q is the lifting to B of the orthogonal projection of H onto B1 . Hence Q is a projection with the range B1 and B = B0 ⊕ B1 where B0 is the null space of Q. Let X : B → Rn be defined by (5) X(x) = ((g1 , x˜), (g2 , x˜), . . . , (gn , x˜)). Define [·] :

Rn

~ = → H by [ξ]

write [x] = [X(x)] =

n X j=1

n X j=1

ξj gj for ξ~ = (ξ1 , ξ2 , . . . , ξn ) ∈ Rn and we

(gj , x˜)gj for x ∈ B.

Lemma 3.1.1. Let F ∈ L1 (B, B(B), ν) and X be as in (5). Then Z Z ~ dν ◦ X −1 (ξ). ~ E F] = F (x)dν(x) = E F (x − [x] + [ξ]) B

(6)

Rn

Proof. Let Y and Z be B(B)-measurable functions on B defined by Y (x) = (I − Q)(x) = x − [X(x)], Z(x) = Q(x) = [X(x)].

324


Let B1 = Q(B) = {[X(x)]|x ∈ B} and B0 = Y (B). Since Y and Z are independent, we have Z Z F (x)dν(x) = F (x − [X(x)] + [X(x)])dν(x) B B Z F (y + z)d(ν ◦ Y −1 × ν ◦ Z −1 )(y, z) = Z 1 ZB0 ×B F (y + z)d(ν ◦ Y −1 )(y)d(ν ◦ Z −1 )(z). = B0

B1

Since Z(x) = Q(x) = [X(x)], by the change of variables formula and (6), we have Z Z F (y + z)d(ν ◦ Y −1 )(y)d(ν ◦ Z −1 )(z) Z B0Z B1 ~ ~ F (y + [ξ])d(ν ◦ Y −1 )(y)d(ν ◦ X −1 )(ξ) = n B0 R Z Z ~ ~ F (x − [x] + [ξ])dν(x)d(ν ◦ X −1 )(ξ) = n ZR B ~ ~ E[F (x − [x] + [ξ])]d(ν ◦ X −1 )(ξ). = Rn

Hence we complete the proof.

Lemma 3.1.2. Let F ∈ L1 (B, B(B), ν). Then Z Z ~ ~ F (x)dν(x) = E[F (x − [x] + [ξ])]d(ν ◦ X −1 )(ξ) X −1 (A)

for every A ∈

B(Rn ).

A

Proof. Using Lemma 3.1.1, for every A ∈ B(Rn ) and its indicator function IA , we have Z F (x)dν(x) X −1 (A) Z = IX −1 (A) (x)F (x)dν(x) ZB = {(IA ◦ X) · F }(x)dν(x) ZB ~ dν ◦ X −1 (ξ) ~ = E {(IA ◦ X) · F }(x − [x] + [ξ]) n ZR ~ ~ dν ◦ X −1 (ξ) ~ = E IA (X(x − [x] + [ξ]))F (x − [x] + [ξ]) Rn


= = =

Z

ZR ZR

n

n

A

325

~ (x − [x] + [ξ]) ~ dν ◦ X −1 (ξ) ~ E IA (ξ)F

~ ~ dν ◦ X −1 (ξ) ~ IA (ξ)E F (x − [x] + [ξ])

~ dν ◦ X −1 (ξ). ~ E F (x − [x] + [ξ])

Hence the proof is completed.

Theorem 3.1.3. Let F ∈ L1 (B, B(B), ν). Then ~ E F (x)|X(x) = ξ~ = E F (x − [x] + [ξ])

for a.e. ξ~ ∈ Rn .

Proof. By equation (3) and Lemma 3.1.2 , we have Z ~ E F (x)|X(x) = ξ~ d(ν ◦ X −1 )(ξ) A Z = F (x)dν(x) X −1 (A) Z ~ d(ν ◦ X −1 )(ξ) ~ = E F (x − [x] + [ξ]) A

for any A ∈

B(Rn ).

Hence the desired result is obtained.

Corollary 3.1.4. Let F ∈ L1 (B, B(B), ν). Then Z ~ ~ F (y + [ξ])dν E F (x)|X(x) = ξ = 0 (y) B0

where ν0 = ν ◦ Y −1 is the abstract Wiener measure on B0 induced by Y. 3.2. Conditional integrals on Hilbert spaces Let {g1 , g2 , . . . , gn } be an orthonormal set in H. Define an Rn -valued function y on H by (7)

y(h) = ((g1 , h), . . . , (gn , h)),

h ∈ H.

For any f ∈ L1 (H, C, m) and a function y on H, we define h h − →i − →i EH f | y(·) = ξ = E R(f ) | R(y)(·) = ξ . h − →i Such EH f | y = ξ is called a conditional integral of f on H given − → y= ξ.

326


Remark. Let E be a topological vector space. For λ > 0, define 1 = f (λ− 2 x), x ∈ E. Then for any C(or Rn )-valued function f on H with f λ ∈ L1 (H, C, m), h h − →i − →i EH f λ | y λ = ξ = E R(f λ ) | R(y λ ) = ξ ,

f λ (x)

for all λ > 0.

Theorem 3.2.1. Let f ∈ L1 (H, C, m). Then there exists a B(Rn )− → measurable function EH [f | y = ξ ] such that h h − → i − →i EH f | y(·) = ξ = EH f (· − [y(·)] + [ ξ ])

− → for all ξ ∈ Rn .

Proof. Since R(f ) ∈ L1 (B, B(B), ν)h for any f ∈ L1 (H,iC, m), there − → exists a B(Rn )-measurable function E R(f ) | R(y)(·) = ξ . Note that

R(y)(x) = ((g1 , x˜), (g2 , x˜), . . . , (gn , x˜)) = X(x), x ∈ B. h − →i Hence by the definition of EH f | y(·) = ξ and Theorem 3.1.3 , we get h h − →i − →i EH f | y(·) = ξ = E R(f ) | R(y)(·) = ξ h − →i = E R(f ) | X(·) = ξ h − → i = E R(f )(· − [X(·)] + [ ξ ]) h − → i = EH f (· − [y(·)] + [ ξ ]) . Hence the proof follows.

4. Conditional analytic Wiener and Feynman integrals In this section, we give an evaluation of the analytic Wiener and Feynman integrals of functions in the classes of Λq (B) and Λq (H) and show that the conditional analytic Feynman integral is used to provide the fundamental solution to the Schrödinger equation.


327

4.1. Conditional analytic Wiener and Feynman integrals on abstract Wiener spaces We begin with the definition of the conditional analytic Wiener and Feynman integral of a function F on B given a function X. Definition. Let X be an Rn -valued measurable function on B and let F be a C-valued measurable function on B such that the integral Z 1 λ E[F ] = F (λ− 2 x)dν(x) B

n exists λ asλ a finite number for all λ > 0. If for a.e. ~η ∈ R , Jλ (~η ) = E F |X = ~η exists for all λ > 0 and has an analytic continuation to C+ = {z ∈ C : Rez > 0}, denoted by Jλ∗ (~η ), then Jλ∗ is defined to be the conditional analytic Wiener integral of F on B given X with parameter λ ∈ C+ and we write

E anwλ [ F | X = ~η ] = Jλ∗ (~η ). If for fixed real q 6= 0, the limit lim E anwλ [ F | X = ~η ]

λ→−iq λ∈C+

exists for a.e. ~η ∈ Rn , then we denote the value of this limit by E anf q [F |X = ~η ] and call it the conditional analytic Feynman integral of F on B given X with parameter q. Let M(H) be the class of all C-valued Borel measures on H with bounded variation. Let F(H) be the class of all functions f on H of the form Z (8) f (h1 ) = eihh,h1 i dσ(h) H

for some σ ∈ M(H). F(H) is the Fresnel class of Albeverio and HoeghKrohn [1]. It is known [21] that each function of the form (8) can be extended to B uniquely by Z ˜ ei(h,x) dσ(h). F (x) = H

Given two C-valued measurable function F and G on B, F is said to be equal to G s-almost surely(s-a.s.) if for each α > 0, ν{x ∈ B|F (αx) 6= G(αx)} = 0 (for more detail, see [5, 19]). For a measurable function F

328


on B, let [F ] denote the equivalence class of functionals which are equal to F s-a.s. The class of equivalence classes defined by Z ˜ F(B) = {[F ]|F (x) = ei(h,x) dσ(h), σ ∈ M(H)} H

is called the Fresnel class of functions on B. It is known that F(B) forms a Banach algebra over the complex field and that F(H) and F(B) are isometrically isomorphic (see [1, 17, 21]). As customary, we will identify a function with its s-equivalence class and think of F(B) as a class of functions on B rather than as a class of equivalence classes. For any q ∈ R, q 6= 0, we denote by Λq (H) and Λq (B), respectively, the class of all functions g on H of the form given by Z i eihh,h1 i dσ(h1 ) (9) g(h) = exp{ (h, Ah)} 2 H and G on B of the form given by (10)

i G(x) = exp{ (x, Ax˜)} 2

Z

˜

ei(h,x) dσ(h)

H

for some σ ∈ M(H) and some self adjoint trace class operator A on H such that (I + 1q D)is invertible where D = (I −Q)A(I −Q) (see [11, 21]). It is known [21] that G is the m-lifting of g. For a self-adjoint trace class operator A with eigenvalues {αj }, the Fredholm determinant of (I + A), denoted by det(I + A), defined by det(I + A) =

∞ Y

(1 + αj )

j=1

and the Maslov index of (I + A), denote by ind(I+A), is the number of negative eigenvalues of (I + A). Let F be a C-valued measurable function on B such that Z 1 J(λ) = F (λ− 2 x)dν(x) B

exists for all real λ > 0. If J(λ) has an analytic continuation to C+ , denoted by J ∗ (λ), then J ∗ (λ) is defined to be the analytic Wiener integral of F on B with parameter λ, and for λ ∈ C+ we write E anwλ [F ] = J ∗ (λ). If for q 6= 0, lim E anwλ [F ] exists, we call the limit analytic Feynman λ→−iq λ∈C+

integral of F with parameter q and we denote it by E anf q [F ]. It is known [21] that E anf q [G] exists for all G ∈ Λq (B).


329

Theorem 4.1.1. Let G ∈ Λq (B) be given by (10) and let X be as in (5). Then the conditional analytic Feynman integral E anf q [ G | X = ξ~ ] exists and we have E anf q [ G | X = ξ~ ] ( − 1 Z 2 − πi Ind(I+ 1 D) i h[ξ],A[ 1 ~ ~ ξ]i ~ − i q · exp ihh, [ξ]i = det(I + D) e 2 · e2 q 2q H * + ) −1 1 ~ + h), (I − Q)(A[ξ] ~ + h) I+ D · (I − Q)(A[ξ] dσ(h). q Proof. For any λ > 0 and ξ~ ∈ Rn , we have √ ~ (11) E[Gλ (x − [x] + [ λξ])] √ √ i ˜ ~ ~ = E exp (x − [x] + [ λξ], A(x − [x] + [ λξ]) 2λ Z √ ~ ˜) √i (h,x−[x]+[ λξ] dσ(h) · e λ H √ √ i ˜ ~ ~ = E exp ((I − Q)x + [ λξ], A(I − Q)x + A[ λξ]) 2λ Z √ ~ ˜) √i (h,(I−Q)x+[ λξ] · e λ dσ(h) H √ √ √ i ˜ ˜ ~ ~ ~ (x, Dx) + 2((I − Q)A[ λξ], x) + h[ λξ], A[ λξ]i = E exp 2λ o Z n i ~ √ ((I−Q)h,x˜ )+ihh,[ξ]i λ · e dσ(h) H " ( Z i ~ i ~ ~ = exp h[ξ], A[ξ]i (x, Dx˜) eihh,[ξ]i E exp 2 2λ H )# i + √ (k, x˜) dσ(h) λ Z Z X ∞ 2 i i ~ ~ ~ αj (ej , x˜) h[ξ], A[ξ]i · eihh,[ξ]i exp = exp 2 2λ H B j=1 ∞ X i + √ hk, ej i(ej , x˜) dν(x) dσ(h) λ j=1

330


" Z Z ∞ ∞ Y iαj 2 i ~ 1 ~ ihh,[ ξ]i ~ √ = exp h[ξ], A[ξ]i · )y e (1 − 2 λ 2π H −∞ j=1 # ihk, ej i + √ ydy dσ(h) λ  − 1 2 Z ∞ Y i ~ iαj  ~ ~ = (1 − · e 2 h[ξ],A[ξ]i ) eihh,[ξ]i ·  λ  H j=1   ∞  1 X hk, ej i2  dσ(h) exp −  2λ 1 − λi αj 

j=1

~ + h) and {αj } and {ej } are eigenvalues and where k = (I − Q)(A[ξ] ∞ X |αj | < ∞, the infinite product eigenvectors of D, respectively. Since j=1

and series appeared in (11) converges absolutely. Hence by Theorem ~ such that 3.1.3, there exists a version Jλ (ξ) √ λ ~ ~ Jλ (ξ) = E G x − [x] + [ λξ]

for all λ > 0 and ξ~ ∈ Rn . ~ exists for all ξ~ ∈ Rn . We may We now show that E anwλ [G|X = ξ] αj assume that 1 + q < 0 for j = 1, 2, . . . , m, m = Ind(I + 1q D) and α 1 + qj > 0 for j ≥ m + 1. Since D is a trace class operator and I + 1q D is invertible, |αj | → 0, as j → ∞ and αj 6= −q for all j. Hence we can choose δ > 0 such that αj is not in [−q − δ, −q + δ] for all j. Let Ω = C+ ∪ {z ∈ C | Rez = 0, |q + Imz| ≤ δ}. For z ∈ Ω, let A1 (z) = A2 (z) =

m Y

1

1

z 2 (z − iαj )− 2 ,

j=1 ∞ Y

j=m+1

(1 −

iαj − 1 ) 2 z

and A3 (z) =

Z

exp H

  

~ − ihh, [ξ]i

 ∞ 2 X hk, ej i  1

2

j=1

z − iαj 

dσ(h).


331

By the similar arguments as in the proof of Theorem 3.2 in [21], we can show that A1 , A2 and A3 are continuous function on Ω and analytic in C+ . Thus i ~ ∗ ~ ~ Jz (ξ) = exp − h[ξ], A[ξ]i A1 (z)A2 (z)A3 (z) 2 ~ = is continuous on Ω and analytic in C+ . It is easy to see that Jλ∗ (ξ) ~ for real λ > 0 and hence E anwλ [G|X = ξ] ~ exists for all ξ~ ∈ Rn . Jλ (ξ) anf q ~ We finally show that E [G|X = ξ] exists. We note that ~ E anf q [ G | X = ξ~ ] = lim Jz∗ (ξ) z→−iq i ~ ~ A[ξ]i A1 (−iq)A2 (−iq)A3 (−iq). = exp − h[ξ], 2 But we get   − 1 m Y 2 π 1 α 1 + j  e− 2 iInd(I+ q D) , A1 (−iq) =  q j=1

1 ∞ Y αj − 2 A2 (−iq) = 1+ q j=m+1

and

as 1 +

αj ≥0 q

  ∞  2 X hk, ej i  i ~ A3 (−iq) = exp ihh, [ξ]i − dσ(h)  2 q + αj  H j=1 Z 1 −1 i ~ k, (I + D) k dσ(h). = exp ihh, [ξ]i − 2q q H Hence the desired result is obtained. Z

Corollary 4.1.2. Let σ be the measure concentrated at 0 ∈ H in Theorem 4.1.1. Then we have i (x, Ax˜) |X(x) = ξ~ E anf q exp 2 − 1 πi 1 i ~ 1 ~ 2 = det(I + D) e− 2 Ind(I+ q D) · e 2 h[ξ],A[ξ]i q * +) ( −1 1 i ~ I+ D ~ (I − Q)A[ξ], (I − Q)A[ξ] . · exp − 2q q

332


Corollary 4.1.3. [7] Let A = 0 in Theorem 4.1.1. Then we have Z n o E anf q exp i(h, x˜) dσ(h)|X(x) = ξ~ H    Z n  i  X 2 2 ~  = exp{ihh, [ξ]i} · exp − hgj , hi |h| − dσ(h).  2q  H j=1

In our next theorem, we need the following summation procedure (see [20, p.340]): Z

(12)

~ dξ~ = lim f (ξ)

A→∞

Rn

Z

(

) ~2 | ξ| ~ exp − f (ξ) dξ~ 2A Rn

whenever the expression on the right exists. Of course if f ∈ L1 (Rn ), it is clear using the dominated convergence theorem that Z Z ~ dξ~ = ~ dξ. ~ f (ξ) f (ξ) Rn

Rn

Theorem 4.1.4. Let G and X be as in Theorem 4.1.1. Then for all λ ∈ C+ , Z n/2 λ λ ~2 ~ dξ~ = E anwλ [G] (13) exp − |ξ| E anwλ [G|X = ξ] 2π 2 n R

and for all real q 6= 0, Z q n/2 iq ~ 2 ~ dξ~ = E anf q [G]. (14) |ξ| E anf q [G|X = ξ] exp 2πi 2 n R Proof. From Lemma 3.1.1, we have, for any real λ > 0, Z λ λ ~ ~ E[G ] = E G (x − [x] + [ξ]) dν ◦ (X λ )−1 (ξ). Rn

~ = Since dν ◦ (X λ )−1 (ξ)

λ n/2 exp 2π n/2

n

o ~ 2 dξ, ~ we have − λ2 |ξ|

λ ~2 ~ dξ. ~ exp − |ξ| E[Gλ |X λ = ξ] E[G ] = 2 n R n o ~ λ) = λ n/2 exp − λ |ξ| ~ 2 E anwλ [G|X]. As shown in the proof Let U (ξ, 2π 2 λ

Z

λ 2π

of Theorem 4.1.1, E anwλ [G|X] is analytic in C+ and continuous on Ω.


333

~ λ) is analytic in C+ and continuous on Ω. A simple appliThus U (ξ, cation of Morera’s Theorem gives the proof of equation (13). To prove equation (14), it suffices in view of equation (13) to show that Z Z ~ ~ ~ −iq) dξ. ~ lim U (ξ, λ) dξ = U (ξ, λ→−iq

Rn

Rn

But this follows from the use of the dominate convergence theorem that ) ( Z Z ~2 | ξ| ~ −iq) exp − ~ −iq) dξ~ = lim dξ~ U (ξ, U (ξ, A→∞ 2A n n R R ( ) Z ~2 | ξ| ~ λ) exp − = lim lim U (ξ, dξ~ A→∞ Rn λ→−iq 2A ( ) Z ~2 | ξ| ~ λ) exp − U (ξ, = lim lim dξ~ A→∞ λ→−iq Rn 2A ) ( Z ~2 | ξ| ~ λ) exp − dξ~ U (ξ, = lim lim λ→−iq A→∞ Rn 2A Z ~ λ) dξ. ~ U (ξ, = lim λ→−iq

Rn

Corollary 4.1.5. Let G and X be in Theorem 4.1.1. Let ψ be given by (15)

ψ(~η ) =

Z

Rn

exp{ih~u, ~η i}dµ(~u)

where µ is a C-valued Borel measure on Rn with bounded variation. For ~η ∈ Rn , let Kη~ be the function on B given by Kη~ (·) = G(·) ψ X(·) + ~η . Then for all q 6= 0, we have that

Γ(~η , q) ≡ E anf q [Kη~ ] ( ) Z n/2 ~ − ~η |2 q iq| ξ anf q ~ ξ. ~ = E G X = ξ~ − ~η exp ψ(ξ)d 2πi 2 Rn

334


Proof. We note that for ~η ∈ Rn , Kη~ belongs to Λq (B) and λ λ λ ~ ~ ~ E Kη~ |X + ~η = ξ = E G | X = ξ − ~η ψ(ξ) for all ξ~ ∈ Rn and λ > 0. The proof follows from this note and Theorem 4.1.4 Remark. Let ~ q) = E H(~η , ξ,

anf q

( ) n/2 ~ − ~η |2 q iq| ξ G|X = ξ~ − ~η exp . 2πi 2

~ q) with q = m is the fundamental Then Corollary 4.1.5 gives that H(~η , ξ, ~ solution to the Schrödinger equation (1) with ψ ∈ F(Rn ). 4.2. Conditional analytic Feynman integrals on Hilbert spaces In this section, we define conditional analytic Feynman integrals on H and give the evaluation of the conditional analytic Feynman integrals involving quadratic functions on H. Definition. Let f be a C-valued function on H such that for all real λ > 0, f λ ∈ L1 (H, C, m) and y be as in (7). Suppose that a.e ~ = EH [ f λ | y λ = ξ] ~ exists for all λ > 0 and has an analytic ξ~ ∈ Rn , Kλ (ξ) + ~ Then K ∗ (ξ) ~ = E angz [ f | y = ξ] ~ is continuation to C , denoted by Kz∗ (ξ). z called the conditional analytic Gauss integral of f on H with parameter z. If for q 6= 0, the limit ~ ~ = E anf q [f |y = ξ] lim E angz [f |y = ξ]

z→−iq λ∈C+

~ to be the conditional exists for a.e. ξ~ ∈ Rn , we define E anf q [f |y = ξ] analytic Feynman integral of f on H with parameter q. Remark. Suppose f satisfies that there exists a C-valued function F on B with the condition R(f λ ) = F λ , for all real λ > 0. Then for all real λ > 0, ~ = E angλ [f |y = ξ] ~ E anwλ [F |X = ξ] and hence ~ ~ = E anf q [f |y = ξ]. E anf q [F |X = ξ]


335

We now define analytic Feynman integrals for functions on H. Let f be a C-valued function on H such that for all real λ > 0, f λ ∈ Z L1 (H, C, m). Suppose that K(λ) = f λ dm exists for all λ > 0. If K(λ) H

has an analytic continuation to C+ , denoted by K ∗ (z), then K ∗ (z) is defined to be the analytic Gauss integral of f on H with parameter z, and for z ∈ C+ we write E angz [F ] = K ∗ (z). If for q 6= 0, lim E angz [f ] z→−iq z∈C+

exists, we call the limit analytic Feynman integral of f with parameter q and we denote it by E anf q [f ]. It is known [21] that E anf q [g] exists for all g ∈ Λq (H).

Theorem 4.2.1. Let g ∈ Λq (H) be given by (9) and let y be as in (7). Then the conditional analytic Feynman integral E anf q [ g | y = ξ~ ] exists and we have E anf q [ g | y = ξ~ ] ( − 1 Z 2 − πi Ind(I+ 1 D) i h[ξ],A[ 1 ~ ~ ξ]i ~ q · e2 = det(I + D) e 2 · exp ihh, [ξ]i q H * +) −1 i 1 ~ + h), (I − Q)(A[ξ] ~ + h) − I+ D (I − Q)(A[ξ] dσ(h). 2q q Theorem 4.2.2. Let g ∈ Λq (H) be given by (9) and let y be as in (7). Then for all λ ∈ C+ , Z n/2 λ λ 2 ~ dξ~ = E angλ [g] (16) exp − |ξ| E angλ [g|y = ξ] 2π 2 n R and for all real q 6= 0, Z q n/2 iq 2 ~ dξ~ = E anf q [g]. |ξ| E anf q [g|y = ξ] exp (17) 2 Rn 2πi Corollary 4.2.3. Let g ∈ Λq (H) be given by (9) and let y be as in (7). Let ψ be given by Z (18) ψ(~η ) = exp{ih~u, ~η i}dµ(~u) Rn

where µ is a C-valued Borel measure on Rn with bounded variation. For ~η ∈ Rn , let Lη~ be the function on H given by Lη~ (·) = g(·)ψ(y(·) + ~η ).

336


Then for all q 6= 0, we have that Γ(~η , q) ≡ E anf q (Lη~ ) ) ( Z q n/2 ~ − ~η |2 iq| ξ anf q ~ ξ. ~ = E [g|y = ξ~ − ~η ] exp ψ(ξ)d 2πi 2 Rn 5. Applications to Feynman path integrals We consider the case where B = C1 [0, T ], H = C1′ [0, T ] and ν is Wiener measure mw . Let S be the operator on C1′ [0, T ] defined by Z τ (19) Sf (τ ) = f (u)du. 0

Then S is a bounded linear operator and the adjoint operator S ∗ of S is given by Z τ ∗ f (u)du. (20) S f (τ ) = f (T )τ − 0

The operator A = given by

S∗S

is a self-adjoint trace class operator on C1′ [0, T ]

Af (τ ) = S

∗

Z

τ

f (u)du

0

=τ (21)

=

Z

Z

0 T

T

f (u)du −

Z

τ

0

Z

s

f (u)duds

0

min(τ, s)f (s)ds.

0

RT Furthermore hf, Agi = hSf, Sgi = 0 f (s)g(s)ds for all f, g ∈ C1′ [0, T ] and so A is positive definite, i.e.hf, Af i ≥ 0 for all f ∈ C1 [0, T ]. The real-valued function Z on [0, T ] × C1 [0, T ] defined by Z(t, x) ≡ z(t) = x(t) −

t x(T ) T

is called a pinned Wiener process on (C1 [0, T ], B(C1 [0, T ]), mw ) and z(0) = 0 and z(T ) = 0. This process {z(t) : t ∈ [0, T ]} is uniquely determined by the mean function E[z(t)] = 0 for every t ∈ [0, T ] and the covariance function E[z(s), z(t)] = k(s, t) = min{s, t} − st T . Let A be the integral


337

operator defined as above. Then it can be shown that the operator D = (I − Q)A(I − Q) on C1′ [0, T ] is expressed by Z T t Df (s) = k(s, t) f (t) − f (T ) dt, s ∈ [0, T ], f ∈ C1′ [0, T ]. T 0

and that the eigenvectors {en } and the eigenvalues {αn } of the operator D are given by √ nπ T2 2T sin s. (22) αn = 2 2 and en (s) = n π nπ T Lemma 5.0.1. For any real α, t ∈ [0, T ], ∞ nπ cosh √α(T − t) X T 1 √ (23) cos t = √ . − 2 2 2 n π + αT T 2αT 2 α sinh αT n=1

Proof. To prove this lemma, we use a known result that ∞ X cos (nx) 1 π cos (ax) (−1)n 2 = 2− , −π ≤ x ≤ π, 2 n −a 2a 2a sin (aπ) n=1

√

where a is not an integer. If we let a = i παT and x = π(TT−t) , then √ ∞ X π2 π2 nπ π 2 cosh α(T − t) n √ √ − (−1) 2 2 cos nπ − t = . n π + αT 2 T 2αT 2 2 αT sinh αT n=1

Hence we obtain ∞ X T n=1

n2 π 2 + αT 2

nπ cosh (√α(T − t)) 1 √ − cos t = √ . T 2αT 2 α sinh αT

Lemma 5.0.2. For a real number α, let ∞ X 1 en (s)en (t), R(s, t, α) = 1 + ααn n=1

s, t ∈ [0, T ]

where αn and en are as above. Then for each t ∈ [0, T ] √ √  sinh α(T − t) sinh αs   √ √ ,  α sinh α T R(s, t, α) = √ √  sinh α(T − s) sinh αt   √ √ , α sinh α T

0 ≤ s ≤ t; t ≤ s ≤ T.

338


Proof. Using (22) and Lemma 5.0.1 we have R(s, t, α) = =

∞ X

n=1 ∞ X

n=1

nπ nπ 2T n2 π 2 sin s sin t n2 π 2 + αT 2 n2 π 2 T T h i nπ nπ T cos (s − t) − cos (s + t) n2 π 2 + αT 2 T T

√ √ 1 √ = √ cosh α(T − |s − t|) − cosh α(T − |s + t|) 2 α sinh α T  √ √ sinh α (T − t) sinh αs   √ √ , s ≤ t;  √ α sinh α T √ = sinh α (T − s) sinh α t   √ √ , s ≥ t.  α sinh α T

For the partition 0 = t0 < t1 < · · · < tn = T , let gi ∈ C1′ [0, T ] be defined by Z τ 1 1 (u)du, i = 1, 2, . . . , n. (24) gi (τ ) = √ ti − ti−1 0 [ti−1 ,ti )

Then {g1 , g2 , . . . , gn } is an orthonormal set in C1′ [0, T ] and 1 (x(ti ) − x(ti−1 )). (gi , x˜) = √ ti − ti−1

We note that for x ∈ C1 [0, T ], (x(t1 ), x(t2 ), . . . , x(tn )) = (ξ1 , ξ2 , . . . , ξn ) 1 if and only if (gj , x˜) = (tj − tj−1 )− 2 (ξj − ξj−1 ) for all j = 1, 2, . . . , n, where ξ0 = 0. Theorem 5.0.3. Let F be measurable function on C1 [0, T ] defined by Z T Z Z T a 2 F (x) = exp − i h(s) x(s) ds dσ(h). exp ib x (s) ds · 2 t1 t1 C1′ [0,T ] Then for all (ξ1 , ξ2 ) ∈ R2 , E anf q [F | X = (ξ1 , ξ2 )] exists and is given by the formula E anf q [F |X = (ξ1 , ξ2 )] q 1  q  a sin a (T − t1 ) − 2  πi q q (T − t1 )  exp   = q a  2 π (T − t ) 1 q


339

 r  ia a × exp − q (ξ22 + ξ12 ) coth − (T − t1 )  2 −a q q  2 ξ ξ iq (ξ2 − ξ1 )  ia q 2 1 · − +q 2 T − t1  − a sinh − a (T − t1 ) q

q

q q − aq (t − t1 ) + ξ1 sinh − aq (T − t) q × h(t)dt C1′ [0,T ] t1 sinh − aq (T − t) q q q Z T Z t sinh − a (T − t) − a sinh − a (s − t1 ) 2 q q q ib q q − · q t1 t1 − aq sinh − aq (T − t1 ) ) Z

( Z exp ib

T

ξ2 sinh

h(s) h(t) ds dt dσ(h).

Proof. By using the integral operator A in (21), we can express F on classical Wiener space C1 [0, T ] as a function G on abstract Wiener space B which is given by oZ n a ˜ ˜ ei(bAh,x) dσ(h). G(x) = exp − i(x, Ax) 2 H

If g1 and g2 are taken as gi (τ ) = √

1 ti − ti−1

Z

τ 0

1[ti−1 ,ti ) (u)du,

i = 1, 2

where t0 = 0, t2 = T . Then we have h ξ2 − ξ1 ξ1 anf q anf q ˜ ˜ . G (g1 , x) = √ , (g2 , x) = √ E F X = (ξ1 , ξ2 ) = E t1 T − t1 By Theorem 4.1.1, we have ξ1 ξ2 − ξ1 E anf q G (g1 , x˜) = √ , (g2 , x˜) = √ t1 T − t1 − 1 n a o 2 πi a a ~ A[ξ]i ~ = det I − D exp − Ind I − D exp − ih[ξ], q 2 q 2 * ( −1 Z a ~ − bAh), ~ − i I− D (I − Q)(aA[ξ] × exp ihbAh, [ξ]i 2q q H

340


Eo ~ − bAh) dσ(h) (I − Q)(aA[ξ]

~ ) = ξ2 −ξ1 τ + ξ1 , 0 ≤ τ ≤ T − t1 and D = (I − Q)A(I − Q) where [ξ](τ T −t1 is the operator on C1′ [0, T − t1 ]. First we observe that − 1 1 Y ∞ a a (T − t1 )2 2 − 2 1− det I − D = q q (nπ)2 n=1 ∞ − 1 r 2 Y a (T − t1 )2 2 1− = · q (nπ)2 n=1 q 1 sin a (T − t1 ) − 2 q = q a (T − t ) 1 q and

 qa    (T − t ) 1 q πi πi a  . exp − Ind I − D = exp −   2  2 q π

By the simple computation, the followings are obtained Z T −t1 a ~ a ~ ~ 2 (s)ds − ih[ξ], A[ξ]i = − i [ξ] 2 2 0 a (25) = i(T − t1 )(ξ12 + ξ2 ξ1 + ξ22 ) 6 and Z T −t1 ~ ~ (26) ihbAh, [ξ]i = ib h(s + t1 )[ξ](s)ds. 0

Moreover we can get + * −1 i a ~ − bAh), (I − Q)(aA[ξ] ~ − bAh) − (I − Q)(aA[ξ] I− D 2q q ∞ i X hk, ej i2 2q 1 − aq αj j=1 Z T −t1 Z T −t1 a ~ i R s, t, − = − a[ξ](s) − bh(s + t1 ) 2q 0 q 0 ~ a[ξ](t) − bh(t + t1 ) dsdt

= −


ia2 = − 2q

T −t1

T −t1

341

a ~ ~ [ξ](s)[ξ](t)dsdt R s, t, − q 0 0 Z Z iab T −t1 T −t1 a ~ + R s, t, − [ξ](s)h(t + t1 )dsdt q 0 q 0 Z Z a ib2 T −t1 T −t1 h(s + t1 )h(t + t1 )dsdt R s, t, − − 2q 0 q 0 Z

Z

~ − bAh). But equation (25) and simple compuwhere k = (I − Q)(aA[ξ] tation, we have Z Z ia2 T −t1 T −t1 ia ~ a ~ ~ ~ − A[ξ]i [ξ](s)[ξ](t)dsdt − h[ξ], R s, t, − 2q 0 q 2 0 Z Z ia2 T −t1 T −t1 a ~ ~ = − [ξ](s)[ξ](t)dsdt R s, t, − 2q 0 q 0 Z ia T −t1 ~ 2 [ξ] (s)ds − 2 0 r ia a = − q − (T − t1 ) (ξ22 + ξ12 ) coth q 2 − aq ξ ξ iq (ξ2 − ξ1 )2 ia q 2 1 . · − +q 2 T − t1 − aq sinh − aq (T − t1 )

Also, by equation (26) and simple computation, we can get Z Z iab T −t1 T −t1 a ~ ~ h(s + t1 )[ξ](t)dsdt + ihbAh, [ξ]i R s, t, − q 0 q 0 Z Z a iab T −t1 T −t1 ~ R s, t, − h(s + t1 )[ξ](t)dsdt = q 0 q 0 Z T −t1 ~ + ib [ξ](t)h(t + t1 )dt 0 q q Z T ξ2 sinh − a (t − t1 ) + ξ1 sinh − a (T − t) q q q = ib h(t)dt t1 sinh − aq (T − t1 ) and

ib2 − 2q

Z

0

T −t1

Z

0

T −t1

a R s, t, − q

h(s + t1 )h(t + t1 )dtds

342

=−


ib2 q

Z

T

t1

Z

t

t1

sinh

q q − aq (T − t) sinh − aq (s − t1 ) q q h(s) h(t) ds dt. − aq sinh − aq (T − t1 )

By combining all the computation as above, we obtain the desired results. Corollary 5.0.4. [13, p.64] Let T − t1 < ωπ . The fundamental solution to the Schrödinger equation with the potential V (t, ξ) = mω 2 2 ′ 2 ξ + f (t)ξ where f ∈ C1 [0, T ]: ∂U i~ ∂ 2 U i mω 2 2 (27) = + ξ + f (t)ξ U − ∂t 2m ∂ξ 2 ~ 2

with the initial state U (0, ξ) = ψ(ξ), is given by

K(ξ2 , T ; ξ1 , t1 ) r imω mω exp = 2πi~ sin ω(T − t1 ) 2~ sin ω(T − t1 ) Z 2ξ2 T 2 2 × cos ω(T − t1 )(ξ2 + ξ1 ) − 2ξ2 ξ1 + f (t) sin ω(t − t1 )dt mω t1 Z 2ξ1 T + f (t) sin ω(T − t)dt mω t1 Z TZ t 2 (28) − 2 2 f (s)f (t) sin ω(T − t) sin ω(s − t1 )dsdt . m ω t1 t1 2

mω 1 Proof. We first note that if we let q = m ~ , a = ~ , b = ~ , the function F on C1 [0, T ] given in Theorem 5.0.3 is expressed as Z Z imω 2 T 2 i T F (x) = exp − x (s) ds + f (s) x(s) ds . 2~ t1 ~ t1 We also note that since T −t1 < ωπ , we see that ind det I − ω 2 D = 1. With these notes and Theorem 5.0.3, we can see that for (ξ2 , T, ξ1 , t1 , q) ∈ R × [0, T ] × R × [0, T ] × (R − {0}) with t1 < T ,

(29)

Γ(ξ2 , T ; ξ1 , t1 ; −iq) r iq(ξ2 − ξ1 )2 q E anf q [F |X = (ξ1 , ξ2 )] = exp 2πi(T − t1 ) 2(T − t1 )

is equal to (28). But in view of Corollary 4.1.5, (29) is the fundamental solution to the Schrödinger equation (27). Hence we complete the proof.


343

Remark. The Maslov index does not appear in the expression (27) which is given in Feynman and Hibbs’ book [13]. This is because they consider values of T − t1 < ωπ . References [1] S. Albeverio and R. Hoegh-Krohn, Mathematical theory of Feynman path integral, Lecture Notes Math. 523, Springer-Verlag, Berlin, 1976. [2] R. H. Cameron, A family of integrals serving to connect the Wiener and Feynamn integrals, J. Math. Phys. 39 (1960), 126–141. [3] R. H. Cameron and Storvick, Some Banach algebras of analytic Feynman integrable functionals, in analytic functions, Lecture Notes Math. 798, SpringerVerlag, berlin, New York, (1980), 18–67. [4] S. J. Chang and D. M. Chung, A class of conditional Wiener integrals, J. Korean Math. Soc. 30 (1993), no. 1, 161–172. [5] D. M. Chung, Scale-invariant measurability in abstract Wiener spaces, Pacific J. Math. 130 (1987), 27–40. , Conditional analytic Feynman integrals for the Fresnel class of functions [6] on abstract Wiener space, Ser. Probab. Statist. 1 (1990), 172–186. [7] , Conditional analytic Feynman integrals on abstract Wiener space, Proc. Amer. Math. Soc. 112 (1991), no. 2, 479–488. [8] D. M. Chung and S. J. Kang, Evaluation Formulas for Conditional Abstract Wiener Integrals, Stochastic Anal. Appl. 7 (1989), 125–144. [9] D. M. Chung and D. Skoug, Conditional Analytic Feynman integrals and a related Schr¨ odinger integral equation, SIAM J. Math. Anal. 20 (1989), no. 4, 950–965. [10] D. M. Chung, C. Park and D. Skoug, Operator-valued Feynman integrals via conditional Feynman integrals, Pacific J. Math. 146 (1990), no. 1, 21–42. [11] K. D. Elworthy and Truman, Feynman maps, Cameron-Martin formulas and anhamornic oscillators, Ann. Inst. Henri Poincarè, 1984. [12] R. P. Feynman, Space-time approach to non-relavistic quantum mechanics, Rev. Modern Phys. 20 (1948), 367–387. [13] R. P. Feynman and A. R. Hibbis, Quantum Mechanics and path integrals, Mc Grow-Hill, New York, 1965. [14] I. M. Gelfand and A. M. Yaglom, Integration in functional spaces, J. Math. Phys. 1 (1960), 48–69. [15] L. Gross, Measurable functions on Hilbert spaces, Tran. Amer. Math. Soc. 105 (1962), 372–390. [16] K. Ito, Generalized uniform complex measure in the Hilbertian metric space with their applications to Feynman path integrals, Proc. Fifth Berkely Symposium on Math. Stat. and Prob. (1976) II, part 4, 145–161. [17] G. W. Johnson, The equivalence of two approaches to the Feynman integral, J. Math. Phys. 23 (1982), 2090–2096. [18] G. W. Johnson and M. L. Lapidus, The Feynman integral and Feynman’s operational calculus, Oxford Mahtematical Monographs, Oxford Univ. press, 2000. [19] G. W. Johnson and D. L. Skoug, Scale-invariant measurability in Wiener space, Pacific J. Math. 83 (1979), 157–176.

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Dong Myung Chung Department of Mathematic Sogang University Seoul 121-742, Korea E-mail : [email protected] Soon Ja Kang Department of Mathematics Education Chonnam National University Kwangju, 500-757, Korea E-mail : [email protected] Kyung Pil Lim Department of Mathematics Sogang University Seoul 121-742, Korea E-mail : [email protected]